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										<sp count="4"/>EXAMPLES</link>
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				<p style="Sous-titre" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Conical Surface Regression and Analysis</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">by Xavier Colonna de Lega</p>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Zygo Corporation</p>
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				<p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Introduction</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">This worksheet illustrates how to calculate the parameters defining a conical surface that best matches a set of (x,y,z) points in a least-squares sense. The deviation of the original data points with respect to the best-fit cone is then calculated to create radial and tangential profiles of the surface deviation.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Valves used for hydraulic and fuel injection systems in cars and trucks are typically made of a moving ball or needle that mates to a conical surface. The region on the cone where the two surfaces contact in the closed position is called the valve seat. In order for the seat to be an effective sealing surface its deviation from a perfect cone is tightly toleranced. Dedicated profilers are used on the production floor and in the QC lab to verify that these tolerances are met. The data used in this worksheet are the result of a measurement of a machined valve seat with a surface profiling interferometer. </p>
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					<b>Coordinate system</b>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The (X,Y,Z) coordinate system is defined by the interferometer geometry (see the figure below). The center is the point P. The axes are defined by the unitless <b>X</b>, <b>Y</b> and <b>Z</b> normed vectors.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">In the figure the conical surface is perfectly aligned to the instrument: its axis defined by the cone center point C and the unit vector <b>D</b>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Finally, the location of the cone center on the Z-axis is such that the normal to the surface at a point M belonging to the inspection diameter passes through the center of the coordinate system P. In other words, the conical surface is tangent to a sphere of radius MP centered at P. The nominal distance MP is defined by the instrument. It is called "radius" in the worksheet.</p>
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				<region region-id="161" left="30" top="111" width="162.75" height="12.75" align-x="89.25" align-y="120" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
					<math optimize="false" disable-calc="false">
						<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
							<ml:function>
								<ml:id xml:space="preserve">Range</ml:id>
								<ml:boundVars>
									<ml:id xml:space="preserve">array</ml:id>
								</ml:boundVars>
							</ml:function>
							<ml:apply>
								<ml:minus/>
								<ml:apply>
									<ml:id xml:space="preserve">max</ml:id>
									<ml:id xml:space="preserve">array</ml:id>
								</ml:apply>
								<ml:apply>
									<ml:id xml:space="preserve">min</ml:id>
									<ml:id xml:space="preserve">array</ml:id>
								</ml:apply>
							</ml:apply>
						</ml:define>
					</math>
				</region>
			</area>
			<rendering item-idref="7"/>
		</region>
		<region region-id="177" left="36" top="710.25" width="96.75" height="12" align-x="57.75" align-y="720" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Coordinate system:</p>
			</text>
		</region>
		<region region-id="5299" left="36" top="740.25" width="65.25" height="12" align-x="44.25" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Unit vectors:</p>
			</text>
		</region>
		<region region-id="5682" left="144" top="724.5" width="39" height="45.75" align-x="156.75" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve" font="4">X</ml:id>
					<ml:matrix rows="3" cols="1">
						<ml:real>1</ml:real>
						<ml:real>0</ml:real>
						<ml:real>0</ml:real>
					</ml:matrix>
				</ml:define>
			</math>
			<rendering item-idref="8"/>
		</region>
		<region region-id="5301" left="216" top="724.5" width="39.75" height="45.75" align-x="229.5" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve" font="4">Y</ml:id>
					<ml:matrix rows="3" cols="1">
						<ml:real>0</ml:real>
						<ml:real>1</ml:real>
						<ml:real>0</ml:real>
					</ml:matrix>
				</ml:define>
			</math>
			<rendering item-idref="9"/>
		</region>
		<region region-id="5302" left="276" top="740.25" width="44.25" height="13.5" align-x="287.25" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve" font="4">Z</ml:id>
					<ml:apply>
						<ml:crossProduct/>
						<ml:id xml:space="preserve" font="4">X</ml:id>
						<ml:id xml:space="preserve" font="4">Y</ml:id>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="10"/>
		</region>
		<region region-id="5303" left="36" top="794.25" width="79.5" height="12" align-x="48.75" align-y="804" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Center position:</p>
			</text>
		</region>
		<region region-id="5683" left="144" top="778.5" width="47.25" height="45.75" align-x="155.25" align-y="804" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">P</ml:id>
					<ml:apply>
						<ml:mult style="default"/>
						<ml:matrix rows="3" cols="1">
							<ml:real>0</ml:real>
							<ml:real>0</ml:real>
							<ml:real>0</ml:real>
						</ml:matrix>
						<ml:id xml:space="preserve">m</ml:id>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="11"/>
		</region>
		<region region-id="201" left="36" top="836.25" width="169.5" height="12" align-x="52.5" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Nominal cone semi included angle:</p>
			</text>
		</region>
		<region region-id="5684" left="216" top="837" width="48" height="16.5" align-x="230.25" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#00ff80" tag="">
			<math optimize="false" disable-calc="false">
				<ml:globalDefine xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve" subscript="0">γ</ml:id>
					<ml:apply>
						<ml:mult style="default"/>
						<ml:real>45</ml:real>
						<ml:id xml:space="preserve">deg</ml:id>
					</ml:apply>
				</ml:globalDefine>
			</math>
			<rendering item-idref="12"/>
		</region>
		<region region-id="202" left="36" top="866.25" width="351.75" height="36" align-x="45.75" align-y="876" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Nominal cone direction: we define the normed vector <f family="Arial" charset="0">
						<b>
							<inlineAttr line-through="false">
								<c val="#000">D</c>
							</inlineAttr>
						</b>
					</f> by two angles<f family="Symbol" charset="2" size="10">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000080">
								<sp/>
							</c>
						</inlineAttr>
					</f>
					<f family="Symbol" charset="2" size="10">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">a</c>
						</inlineAttr>
					</f>
					<c val="#000"> and </c>
					<f family="Symbol" charset="2" size="10">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">b</c>
						</inlineAttr>
					</f> that <f family="Arial" charset="0">correspond to two successive rotations of the cone axis about the X and Y axes.</f>
				</p>
			</text>
		</region>
		<region region-id="5305" left="36" top="914.25" width="90" height="12" align-x="52.5" align-y="924" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Nominal rotations:</p>
			</text>
		</region>
		<region region-id="5685" left="144" top="915" width="50.25" height="16.5" align-x="159.75" align-y="924" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve" subscript="0">α</ml:id>
					<ml:apply>
						<ml:mult style="default"/>
						<ml:real>0</ml:real>
						<ml:id xml:space="preserve">mrad</ml:id>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="13"/>
		</region>
		<region region-id="6174" left="216" top="915" width="50.25" height="16.5" align-x="231.75" align-y="924" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve" subscript="0">β</ml:id>
					<ml:apply>
						<ml:mult style="default"/>
						<ml:real>0</ml:real>
						<ml:id xml:space="preserve">mrad</ml:id>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="14"/>
		</region>
		<region region-id="5308" left="36" top="956.25" width="79.5" height="12" align-x="52.5" align-y="966" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Rotation matrix:</p>
			</text>
		</region>
		<region region-id="5686" left="144" top="940.5" width="204.75" height="45.75" align-x="183.75" align-y="966" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:function>
						<ml:id xml:space="preserve">rot</ml:id>
						<ml:boundVars>
							<ml:id xml:space="preserve">α</ml:id>
							<ml:id xml:space="preserve">β</ml:id>
						</ml:boundVars>
					</ml:function>
					<ml:apply>
						<ml:mult style="default"/>
						<ml:matrix rows="3" cols="3">
							<ml:real>1</ml:real>
							<ml:real>0</ml:real>
							<ml:real>0</ml:real>
							<ml:real>0</ml:real>
							<ml:apply fixity="prefix">
								<ml:id xml:space="preserve">cos</ml:id>
								<ml:id xml:space="preserve">α</ml:id>
							</ml:apply>
							<ml:apply fixity="prefix">
								<ml:id xml:space="preserve">sin</ml:id>
								<ml:id xml:space="preserve">α</ml:id>
							</ml:apply>
							<ml:real>0</ml:real>
							<ml:apply>
								<ml:neg/>
								<ml:apply fixity="prefix">
									<ml:id xml:space="preserve">sin</ml:id>
									<ml:id xml:space="preserve">α</ml:id>
								</ml:apply>
							</ml:apply>
							<ml:apply fixity="prefix">
								<ml:id xml:space="preserve">cos</ml:id>
								<ml:id xml:space="preserve">α</ml:id>
							</ml:apply>
						</ml:matrix>
						<ml:matrix rows="3" cols="3">
							<ml:apply fixity="prefix">
								<ml:id xml:space="preserve">cos</ml:id>
								<ml:id xml:space="preserve">β</ml:id>
							</ml:apply>
							<ml:real>0</ml:real>
							<ml:apply fixity="prefix">
								<ml:id xml:space="preserve">sin</ml:id>
								<ml:id xml:space="preserve">β</ml:id>
							</ml:apply>
							<ml:real>0</ml:real>
							<ml:real>1</ml:real>
							<ml:real>0</ml:real>
							<ml:apply>
								<ml:neg/>
								<ml:apply fixity="prefix">
									<ml:id xml:space="preserve">sin</ml:id>
									<ml:id xml:space="preserve">β</ml:id>
								</ml:apply>
							</ml:apply>
							<ml:real>0</ml:real>
							<ml:apply fixity="prefix">
								<ml:id xml:space="preserve">cos</ml:id>
								<ml:id xml:space="preserve">β</ml:id>
							</ml:apply>
						</ml:matrix>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="15"/>
		</region>
		<region region-id="5310" left="36" top="1004.25" width="116.25" height="12" align-x="52.5" align-y="1014" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Nominal cone direction:</p>
			</text>
		</region>
		<region region-id="6175" left="216" top="1004.25" width="89.25" height="17.25" align-x="234" align-y="1014" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve" font="4" subscript="0">D</ml:id>
					<ml:apply>
						<ml:mult style="default"/>
						<ml:apply>
							<ml:id xml:space="preserve">rot</ml:id>
							<ml:sequence>
								<ml:id xml:space="preserve" subscript="0">α</ml:id>
								<ml:id xml:space="preserve" subscript="0">β</ml:id>
							</ml:sequence>
						</ml:apply>
						<ml:parens>
							<ml:apply>
								<ml:neg/>
								<ml:id xml:space="preserve" font="4">Z</ml:id>
							</ml:apply>
						</ml:parens>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="16"/>
		</region>
		<region region-id="203" left="36" top="1034.25" width="129" height="12" align-x="49.5" align-y="1044" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Radius of tangent sphere:</p>
			</text>
		</region>
		<region region-id="6176" left="216" top="1035" width="64.5" height="12.75" align-x="246" align-y="1044" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#00ff80" tag="">
			<math optimize="false" disable-calc="false">
				<ml:globalDefine xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">radius</ml:id>
					<ml:apply>
						<ml:mult style="default"/>
						<ml:real>2.0</ml:real>
						<ml:id xml:space="preserve">mm</ml:id>
					</ml:apply>
				</ml:globalDefine>
			</math>
			<rendering item-idref="17"/>
		</region>
		<region region-id="205" left="36" top="1064.25" width="146.25" height="12" align-x="45.75" align-y="1074" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Nominal cone center or apex:</p>
			</text>
		</region>
		<region region-id="6177" left="216" top="1058.25" width="77.25" height="31.5" align-x="232.5" align-y="1074" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve" subscript="0">C</ml:id>
					<ml:apply>
						<ml:plus/>
						<ml:id xml:space="preserve">P</ml:id>
						<ml:apply>
							<ml:mult style="default"/>
							<ml:apply>
								<ml:div/>
								<ml:id xml:space="preserve">radius</ml:id>
								<ml:apply fixity="prefix">
									<ml:id xml:space="preserve">sin</ml:id>
									<ml:id xml:space="preserve" subscript="0">γ</ml:id>
								</ml:apply>
							</ml:apply>
							<ml:id xml:space="preserve" font="4">Z</ml:id>
						</ml:apply>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="18"/>
		</region>
		<region region-id="5687" left="0" top="1092" width="6000" height="6" align-x="0" align-y="1092" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="207" left="36" top="1106.25" width="362.25" height="96" align-x="39.75" align-y="1116" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">We have defined the conical surface and its location in space by 6 scalar values: the semi included angle <f family="Symbol" charset="2" size="10">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">g</inlineAttr>
					</f>, the (x,y,z) coordinates of the cone center or apex and the two rotation angles describing the normed vector <f family="Verdana" charset="0">
						<b>
							<inlineAttr line-through="false">D</inlineAttr>
						</b>
					</f>.</p>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"/>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">In practice, the measured data corresponds to a part that is not perfectly aligned to the instrument. The least-squares fit cone provides estimates for these 6 parameters. </p>
			</text>
		</region>
		<region region-id="251" left="36" top="1213.5" width="153.75" height="12.75" align-x="48.75" align-y="1224" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Import measured data</p>
			</text>
		</region>
		<region region-id="5688" left="36" top="1232.25" width="177" height="12" align-x="45.75" align-y="1242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Load measured (x,y,z) coordinates:</p>
			</text>
		</region>
		<region region-id="5689" left="36" top="1257" width="263.25" height="12.75" align-x="101.25" align-y="1266" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">MeasuredXYZ</ml:id>
					<ml:apply>
						<ml:mult style="default"/>
						<ml:apply>
							<ml:id xml:space="preserve">READBIN</ml:id>
							<ml:sequence>
								<ml:str xml:space="preserve">ConeXYZ.bin</ml:str>
								<ml:str xml:space="preserve">float</ml:str>
								<ml:real>0</ml:real>
								<ml:real>3</ml:real>
								<ml:real>0</ml:real>
							</ml:sequence>
						</ml:apply>
						<ml:id xml:space="preserve">mm</ml:id>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="19"/>
		</region>
		<region region-id="6147" left="36" top="1280.25" width="195" height="12" align-x="51.75" align-y="1290" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Number of (x,y,z) measurement points:</p>
			</text>
		</region>
		<region region-id="6148" left="240" top="1281" width="126.75" height="12.75" align-x="331.5" align-y="1290" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:id xml:space="preserve">rows</ml:id>
						<ml:id xml:space="preserve">MeasuredXYZ</ml:id>
					</ml:apply>
					<result xmlns="http://schemas.mathsoft.com/math30">
						<ml:real>51518</ml:real>
					</result>
				</ml:eval>
				<resultFormat>
					<general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="6"/>
					<matrix display-style="auto" expand-nested-arrays="false"/>
					<unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/>
				</resultFormat>
			</math>
			<rendering item-idref="20"/>
		</region>
		<region region-id="6149" left="36" top="1304.25" width="102" height="12" align-x="48.75" align-y="1314" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Extract coordinates:</p>
			</text>
		</region>
		<region region-id="6150" left="36" top="1323.75" width="87.75" height="18" align-x="45.75" align-y="1338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">x</ml:id>
					<ml:apply>
						<ml:matcol/>
						<ml:id xml:space="preserve">MeasuredXYZ</ml:id>
						<ml:real>0</ml:real>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="21"/>
		</region>
		<region region-id="6151" left="144" top="1323.75" width="89.25" height="18" align-x="155.25" align-y="1338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">y</ml:id>
					<ml:apply>
						<ml:matcol/>
						<ml:id xml:space="preserve">MeasuredXYZ</ml:id>
						<ml:real>1</ml:real>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="22"/>
		</region>
		<region region-id="6152" left="252" top="1323.75" width="87.75" height="18" align-x="261.75" align-y="1338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">z</ml:id>
					<ml:apply>
						<ml:matcol/>
						<ml:id xml:space="preserve">MeasuredXYZ</ml:id>
						<ml:real>2</ml:real>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="23"/>
		</region>
		<region region-id="276" left="36" top="1357.5" width="170.25" height="12.75" align-x="36" align-y="1368" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Best-fit conical surface</p>
			</text>
		</region>
		<region region-id="278" left="36" top="1382.25" width="327" height="48" align-x="36" align-y="1392" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">We first define a merit function to minimize. Here we choose to calculate the sum of the squared distances from the measured data points to the best-fit cone. The distance is measured along the local cone normal.</p>
			</text>
		</region>
		<region region-id="6091" left="234" top="1452" width="132" height="125.25" align-x="234" align-y="1452" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<picture>
				<metafile mapping-mode="8" x-extent="130.7622" y-extent="123.874" item-idref="24"/>
			</picture>
			<rendering item-idref="25"/>
		</region>
		<region region-id="437" left="36" top="1448.25" width="194.25" height="36" align-x="36" align-y="1458" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="true" lock-width="true">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The figures at right illustrates the calculation of the distance HM for a given measured point M.</p>
			</text>
		</region>
		<region region-id="445" left="36" top="1496.25" width="48" height="12" align-x="42.75" align-y="1506" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">We have:</p>
			</text>
		</region>
		<region region-id="449" left="78" top="1496.25" width="78.75" height="13.5" align-x="109.5" align-y="1506" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:absval/>
						<ml:id xml:space="preserve" font="4">CM'</ml:id>
					</ml:apply>
					<ml:apply>
						<ml:greaterThan/>
						<ml:apply>
							<ml:mult style="default"/>
							<ml:id xml:space="preserve" font="4">CM</ml:id>
							<ml:id xml:space="preserve" font="4">D</ml:id>
						</ml:apply>
						<ml:real>0</ml:real>
					</ml:apply>
				</ml:apply>
			</math>
			<rendering item-idref="26"/>
		</region>
		<region region-id="4820" left="36" top="1520.25" width="199.5" height="24" align-x="43.5" align-y="1530" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The signed length M'M" is defined as positive.</p>
			</text>
		</region>
		<region region-id="6068" left="36" top="1556.25" width="88.5" height="13.5" align-x="63.75" align-y="1566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:id xml:space="preserve">M'M''</ml:id>
					<ml:apply>
						<ml:mult style="default"/>
						<ml:apply>
							<ml:absval/>
							<ml:id xml:space="preserve" font="4">CM'</ml:id>
						</ml:apply>
						<ml:apply>
							<ml:id xml:space="preserve">tan</ml:id>
							<ml:id xml:space="preserve">γ</ml:id>
						</ml:apply>
					</ml:apply>
				</ml:apply>
			</math>
			<rendering item-idref="27"/>
		</region>
		<region region-id="4829" left="36" top="1574.25" width="191.25" height="12" align-x="43.5" align-y="1584" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The signed length M'M is also positive.</p>
			</text>
		</region>
		<region region-id="6186" left="0" top="1596" width="6000" height="6" align-x="0" align-y="1596" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="4830" left="36" top="1608.75" width="162.75" height="21" align-x="99.75" align-y="1626" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:equal/>
						<ml:id xml:space="preserve">M'M</ml:id>
						<ml:apply>
							<ml:absval/>
							<ml:id xml:space="preserve" font="4">MM'</ml:id>
						</ml:apply>
					</ml:apply>
					<ml:apply>
						<ml:sqrt/>
						<ml:apply>
							<ml:minus/>
							<ml:apply>
								<ml:pow/>
								<ml:parens>
									<ml:apply>
										<ml:absval/>
										<ml:id xml:space="preserve" font="4">CM</ml:id>
									</ml:apply>
								</ml:parens>
								<ml:real>2</ml:real>
							</ml:apply>
							<ml:apply>
								<ml:pow/>
								<ml:parens>
									<ml:apply>
										<ml:absval/>
										<ml:id xml:space="preserve" font="4">CM'</ml:id>
									</ml:apply>
								</ml:parens>
								<ml:real>2</ml:real>
							</ml:apply>
						</ml:apply>
					</ml:apply>
				</ml:apply>
			</math>
			<rendering item-idref="28"/>
		</region>
		<region region-id="6092" left="246" top="1626" width="114.75" height="127.5" align-x="246" align-y="1626" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<picture>
				<metafile mapping-mode="8" x-extent="113.4992" y-extent="126.1134" item-idref="29"/>
			</picture>
			<rendering item-idref="30"/>
		</region>
		<region region-id="6093" left="36" top="1640.25" width="204.75" height="24" align-x="44.25" align-y="1650" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This yields an expression for the signed length MM".</p>
			</text>
		</region>
		<region region-id="4834" left="36" top="1671" width="86.25" height="12.75" align-x="62.25" align-y="1680" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:id xml:space="preserve">MM''</ml:id>
					<ml:apply>
						<ml:minus/>
						<ml:id xml:space="preserve">M'M''</ml:id>
						<ml:id xml:space="preserve">M'M</ml:id>
					</ml:apply>
				</ml:apply>
			</math>
			<rendering item-idref="31"/>
		</region>
		<region region-id="4840" left="36" top="1700.25" width="48" height="12" align-x="42.75" align-y="1710" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">We have:</p>
			</text>
		</region>
		<region region-id="4841" left="96" top="1694.25" width="62.25" height="27.75" align-x="122.25" align-y="1710" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:id xml:space="preserve">MM''</ml:id>
					<ml:apply>
						<ml:div/>
						<ml:id xml:space="preserve">MH</ml:id>
						<ml:apply>
							<ml:id xml:space="preserve">cos</ml:id>
							<ml:id xml:space="preserve">γ</ml:id>
						</ml:apply>
					</ml:apply>
				</ml:apply>
			</math>
			<rendering item-idref="32"/>
		</region>
		<region region-id="4845" left="36" top="1730.25" width="191.25" height="24" align-x="39" align-y="1740" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If we define the surface deviation as HM this signed quantity is given by:</p>
			</text>
		</region>
		<region region-id="4851" left="36" top="1764.75" width="213.75" height="21" align-x="57" align-y="1782" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:id xml:space="preserve">HM</ml:id>
					<ml:apply>
						<ml:minus/>
						<ml:apply>
							<ml:mult style="default"/>
							<ml:apply>
								<ml:id xml:space="preserve">cos</ml:id>
								<ml:id xml:space="preserve">γ</ml:id>
							</ml:apply>
							<ml:apply>
								<ml:sqrt/>
								<ml:apply>
									<ml:minus/>
									<ml:apply>
										<ml:pow/>
										<ml:parens>
											<ml:apply>
												<ml:absval/>
												<ml:id xml:space="preserve" font="4">CM</ml:id>
											</ml:apply>
										</ml:parens>
										<ml:real>2</ml:real>
									</ml:apply>
									<ml:apply>
										<ml:pow/>
										<ml:parens>
											<ml:apply>
												<ml:absval/>
												<ml:id xml:space="preserve" font="4">CM'</ml:id>
											</ml:apply>
										</ml:parens>
										<ml:real>2</ml:real>
									</ml:apply>
								</ml:apply>
							</ml:apply>
						</ml:apply>
						<ml:apply>
							<ml:mult style="default"/>
							<ml:apply>
								<ml:id xml:space="preserve">sin</ml:id>
								<ml:id xml:space="preserve">γ</ml:id>
							</ml:apply>
							<ml:apply>
								<ml:absval/>
								<ml:id xml:space="preserve" font="4">CM'</ml:id>
							</ml:apply>
						</ml:apply>
					</ml:apply>
				</ml:apply>
			</math>
			<rendering item-idref="33"/>
		</region>
		<region region-id="5820" left="36" top="1796.25" width="299.25" height="51" align-x="43.5" align-y="1806" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The sum calculation has been optimized in this program. It turns out that calculation with vectors is slower than scalar calculations. The least square sum is scaled to mm<f family="Verdana" charset="0">
						<inlineAttr line-through="false">
							<sup>2</sup>
						</inlineAttr>
					</f> in order to avoid large exponents.</p>
			</text>
		</region>
		<region region-id="5821" left="0" top="1860" width="6000" height="6" align-x="0" align-y="1860" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="495" left="36" top="1868.25" width="348" height="285.75" align-x="182.25" align-y="1884" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:function>
						<ml:id xml:space="preserve">Residuals</ml:id>
						<ml:boundVars>
							<ml:id xml:space="preserve">points</ml:id>
							<ml:id xml:space="preserve">cx</ml:id>
							<ml:id xml:space="preserve">cy</ml:id>
							<ml:id xml:space="preserve">cz</ml:id>
							<ml:id xml:space="preserve">α</ml:id>
							<ml:id xml:space="preserve">β</ml:id>
							<ml:id xml:space="preserve">γ</ml:id>
						</ml:boundVars>
					</ml:function>
					<ml:program>
						<ml:localDefine>
							<ml:id xml:space="preserve">sγ</ml:id>
							<ml:apply>
								<ml:div/>
								<ml:apply>
									<ml:id xml:space="preserve">sin</ml:id>
									<ml:id xml:space="preserve">γ</ml:id>
								</ml:apply>
								<ml:id xml:space="preserve">mm</ml:id>
							</ml:apply>
						</ml:localDefine>
						<ml:localDefine>
							<ml:id xml:space="preserve">cγ</ml:id>
							<ml:apply>
								<ml:div/>
								<ml:apply>
									<ml:id xml:space="preserve">cos</ml:id>
									<ml:id xml:space="preserve">γ</ml:id>
								</ml:apply>
								<ml:id xml:space="preserve">mm</ml:id>
							</ml:apply>
						</ml:localDefine>
						<ml:localDefine>
							<ml:matrix rows="3" cols="1">
								<ml:id xml:space="preserve">Dx</ml:id>
								<ml:id xml:space="preserve">Dy</ml:id>
								<ml:id xml:space="preserve">Dz</ml:id>
							</ml:matrix>
							<ml:matrix rows="3" cols="1">
								<ml:apply>
									<ml:id xml:space="preserve">sin</ml:id>
									<ml:id xml:space="preserve">β</ml:id>
								</ml:apply>
								<ml:apply>
									<ml:mult style="default"/>
									<ml:apply>
										<ml:id xml:space="preserve">sin</ml:id>
										<ml:id xml:space="preserve">α</ml:id>
									</ml:apply>
									<ml:apply>
										<ml:id xml:space="preserve">cos</ml:id>
										<ml:id xml:space="preserve">β</ml:id>
									</ml:apply>
								</ml:apply>
								<ml:apply>
									<ml:mult style="default"/>
									<ml:apply>
										<ml:neg/>
										<ml:apply>
											<ml:id xml:space="preserve">cos</ml:id>
											<ml:id xml:space="preserve">α</ml:id>
										</ml:apply>
									</ml:apply>
									<ml:apply>
										<ml:id xml:space="preserve">cos</ml:id>
										<ml:id xml:space="preserve">β</ml:id>
									</ml:apply>
								</ml:apply>
							</ml:matrix>
						</ml:localDefine>
						<ml:str xml:space="preserve">Loop on all data points</ml:str>
						<ml:for>
							<ml:id xml:space="preserve">k</ml:id>
							<ml:range>
								<ml:real>0</ml:real>
								<ml:apply>
									<ml:minus/>
									<ml:apply>
										<ml:id xml:space="preserve">rows</ml:id>
										<ml:id xml:space="preserve">points</ml:id>
									</ml:apply>
									<ml:real>1</ml:real>
								</ml:apply>
							</ml:range>
							<ml:program>
								<ml:localDefine>
									<ml:id xml:space="preserve">CMx</ml:id>
									<ml:apply>
										<ml:minus/>
										<ml:apply>
											<ml:indexer/>
											<ml:id xml:space="preserve">points</ml:id>
											<ml:sequence>
												<ml:id xml:space="preserve">k</ml:id>
												<ml:real>0</ml:real>
											</ml:sequence>
										</ml:apply>
										<ml:id xml:space="preserve">cx</ml:id>
									</ml:apply>
								</ml:localDefine>
								<ml:localDefine>
									<ml:id xml:space="preserve">CMy</ml:id>
									<ml:apply>
										<ml:minus/>
										<ml:apply>
											<ml:indexer/>
											<ml:id xml:space="preserve">points</ml:id>
											<ml:sequence>
												<ml:id xml:space="preserve">k</ml:id>
												<ml:real>1</ml:real>
											</ml:sequence>
										</ml:apply>
										<ml:id xml:space="preserve">cy</ml:id>
									</ml:apply>
								</ml:localDefine>
								<ml:localDefine>
									<ml:id xml:space="preserve">CMz</ml:id>
									<ml:apply>
										<ml:minus/>
										<ml:apply>
											<ml:indexer/>
											<ml:id xml:space="preserve">points</ml:id>
											<ml:sequence>
												<ml:id xml:space="preserve">k</ml:id>
												<ml:real>2</ml:real>
											</ml:sequence>
										</ml:apply>
										<ml:id xml:space="preserve">cz</ml:id>
									</ml:apply>
								</ml:localDefine>
								<ml:localDefine>
									<ml:id xml:space="preserve">CM2</ml:id>
									<ml:apply>
										<ml:plus/>
										<ml:apply>
											<ml:plus/>
											<ml:apply>
												<ml:pow/>
												<ml:id xml:space="preserve">CMx</ml:id>
												<ml:real>2</ml:real>
											</ml:apply>
											<ml:apply>
												<ml:pow/>
												<ml:id xml:space="preserve">CMy</ml:id>
												<ml:real>2</ml:real>
											</ml:apply>
										</ml:apply>
										<ml:apply>
											<ml:pow/>
											<ml:id xml:space="preserve">CMz</ml:id>
											<ml:real>2</ml:real>
										</ml:apply>
									</ml:apply>
								</ml:localDefine>
								<ml:localDefine>
									<ml:id xml:space="preserve">cmz</ml:id>
									<ml:apply>
										<ml:plus/>
										<ml:apply>
											<ml:plus/>
											<ml:apply>
												<ml:mult style="default"/>
												<ml:id xml:space="preserve">CMx</ml:id>
												<ml:id xml:space="preserve">Dx</ml:id>
											</ml:apply>
											<ml:apply>
												<ml:mult style="default"/>
												<ml:id xml:space="preserve">CMy</ml:id>
												<ml:id xml:space="preserve">Dy</ml:id>
											</ml:apply>
										</ml:apply>
										<ml:apply>
											<ml:mult style="default"/>
											<ml:id xml:space="preserve">CMz</ml:id>
											<ml:id xml:space="preserve">Dz</ml:id>
										</ml:apply>
									</ml:apply>
								</ml:localDefine>
								<ml:localDefine>
									<ml:apply>
										<ml:indexer/>
										<ml:id xml:space="preserve">HM</ml:id>
										<ml:id xml:space="preserve">k</ml:id>
									</ml:apply>
									<ml:apply>
										<ml:minus/>
										<ml:apply>
											<ml:mult style="default"/>
											<ml:id xml:space="preserve">cγ</ml:id>
											<ml:apply>
												<ml:sqrt/>
												<ml:apply>
													<ml:minus/>
													<ml:id xml:space="preserve">CM2</ml:id>
													<ml:apply>
														<ml:mult style="default"/>
														<ml:id xml:space="preserve">cmz</ml:id>
														<ml:id xml:space="preserve">cmz</ml:id>
													</ml:apply>
												</ml:apply>
											</ml:apply>
										</ml:apply>
										<ml:apply>
											<ml:mult style="default"/>
											<ml:id xml:space="preserve">sγ</ml:id>
											<ml:id xml:space="preserve">cmz</ml:id>
										</ml:apply>
									</ml:apply>
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						<ml:return>
							<ml:id xml:space="preserve">HM</ml:id>
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		<region region-id="296" left="36" top="2174.25" width="120.75" height="12" align-x="48" align-y="2184" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Guess values for the fit:</p>
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			<math optimize="false" disable-calc="false">
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						<ml:id xml:space="preserve">mrad</ml:id>
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					<result xmlns="http://schemas.mathsoft.com/math30">
						<ml:real>0</ml:real>
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			<math optimize="false" disable-calc="false">
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					<ml:id xml:space="preserve" subscript="0">β</ml:id>
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			<math optimize="false" disable-calc="false">
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					<ml:id xml:space="preserve" subscript="0">γ</ml:id>
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						<ml:id xml:space="preserve">deg</ml:id>
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					<result xmlns="http://schemas.mathsoft.com/math30">
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							<ml:real>45</ml:real>
							<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
								<unitReference unit="degree"/>
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						<ml:id xml:space="preserve">mm</ml:id>
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								<ml:real>0</ml:real>
								<ml:real>0</ml:real>
								<ml:real>2.8284271247461903</ml:real>
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								<unitReference unit="millimeter"/>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:matrix rows="3" cols="1">
						<ml:id xml:space="preserve" subscript="0">cx</ml:id>
						<ml:id xml:space="preserve" subscript="0">cy</ml:id>
						<ml:id xml:space="preserve" subscript="0">cz</ml:id>
					</ml:matrix>
					<ml:id xml:space="preserve" subscript="0">C</ml:id>
				</ml:define>
			</math>
			<rendering item-idref="39"/>
		</region>
		<region region-id="302" left="36" top="2288.25" width="241.5" height="12" align-x="53.25" align-y="2298" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Minimize the sum of squares using a solve block:</p>
			</text>
		</region>
		<region region-id="512" left="36" top="2312.25" width="139.5" height="12" align-x="42" align-y="2322" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Set convergence tolerance:</p>
			</text>
		</region>
		<region region-id="509" left="204" top="2307" width="52.5" height="18.75" align-x="228" align-y="2322" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">TOL</ml:id>
					<ml:apply>
						<ml:pow/>
						<ml:real>10</ml:real>
						<ml:real>-4</ml:real>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="40"/>
		</region>
		<region region-id="5828" left="0" top="2334" width="6000" height="6" align-x="0" align-y="2334" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
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		<region region-id="5707" left="36" top="2348.25" width="331.5" height="12" align-x="41.25" align-y="2358" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">On slower computers we can use a data subset to perform the fit.</p>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">sub</ml:id>
					<ml:real>20</ml:real>
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		</region>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
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						<ml:apply>
							<ml:minus/>
							<ml:apply>
								<ml:id xml:space="preserve">rows</ml:id>
								<ml:id xml:space="preserve">MeasuredXYZ</ml:id>
							</ml:apply>
							<ml:real>1</ml:real>
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					</ml:range>
				</ml:define>
			</math>
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		<region region-id="5733" left="234" top="2385.75" width="88.5" height="18" align-x="250.5" align-y="2400" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">T0</ml:id>
					<ml:apply>
						<ml:transpose/>
						<ml:id xml:space="preserve">MeasuredXYZ</ml:id>
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				</ml:define>
			</math>
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		<region region-id="5734" left="72" top="2412.75" width="129.75" height="33" align-x="174.75" align-y="2442" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
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					<ml:apply>
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						<ml:apply>
							<ml:div/>
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							<ml:id xml:space="preserve">sub</ml:id>
						</ml:apply>
					</ml:apply>
					<ml:apply>
						<ml:matcol/>
						<ml:id xml:space="preserve">T0</ml:id>
						<ml:id xml:space="preserve">i</ml:id>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="44"/>
		</region>
		<region region-id="5735" left="234" top="2427.75" width="167.25" height="18" align-x="314.25" align-y="2442" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">subMeasuredXYZ</ml:id>
					<ml:apply>
						<ml:transpose/>
						<ml:id xml:space="preserve">subMeasuredXYZ</ml:id>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="45"/>
		</region>
		<region region-id="5813" left="36" top="2461.5" width="198.75" height="12.75" align-x="36" align-y="2472" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Sum of Squares Solve Block</p>
			</text>
		</region>
		<region region-id="5709" left="36" top="2486.25" width="351.75" height="48" align-x="46.5" align-y="2496" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Enable one or the other equality statement, depending on if you wish to use the truncated data or the full set. The Quasi-Newton algorithm gives the best results here. The right-click option on <b>Minerr</b> was used to force this choice.</p>
			</text>
		</region>
		<region region-id="303" left="36" top="2547" width="24" height="12.75" align-x="47.25" align-y="2556" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id>
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			<math optimize="false" disable-calc="false">
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					<ml:apply>
						<ml:id xml:space="preserve">Residuals</ml:id>
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							<ml:id xml:space="preserve" subscript="0">cy</ml:id>
							<ml:id xml:space="preserve" subscript="0">cz</ml:id>
							<ml:id xml:space="preserve" subscript="0">α</ml:id>
							<ml:id xml:space="preserve" subscript="0">β</ml:id>
							<ml:id xml:space="preserve" subscript="0">γ</ml:id>
						</ml:sequence>
					</ml:apply>
					<ml:real>0</ml:real>
				</ml:apply>
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					<ml:real>0</ml:real>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
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						<ml:id xml:space="preserve">cx</ml:id>
						<ml:id xml:space="preserve">cy</ml:id>
						<ml:id xml:space="preserve">cz</ml:id>
						<ml:id xml:space="preserve">α</ml:id>
						<ml:id xml:space="preserve">β</ml:id>
						<ml:id xml:space="preserve">γ</ml:id>
					</ml:matrix>
					<ml:apply>
						<ml:transpose/>
						<ml:apply>
							<ml:Minerr auto-method="false" method="levenberg" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/>
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								<ml:id xml:space="preserve" subscript="0">cx</ml:id>
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					</ml:apply>
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		</region>
		<region region-id="314" left="36" top="2654.25" width="192" height="12" align-x="47.25" align-y="2664" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Verify improvement of sum of squares:</p>
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		<region region-id="6214" left="72" top="2679" width="267" height="16.5" align-x="115.5" align-y="2688" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve" subscript="Initial">Res</ml:id>
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						<ml:id xml:space="preserve">Residuals</ml:id>
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							<ml:id xml:space="preserve">subMeasuredXYZ</ml:id>
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							<ml:id xml:space="preserve" subscript="0">cy</ml:id>
							<ml:id xml:space="preserve" subscript="0">cz</ml:id>
							<ml:id xml:space="preserve" subscript="0">α</ml:id>
							<ml:id xml:space="preserve" subscript="0">β</ml:id>
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			<math optimize="false" disable-calc="false">
				<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:vectorSum/>
						<ml:apply>
							<ml:pow/>
							<ml:id xml:space="preserve" subscript="Initial">Res</ml:id>
							<ml:real>2</ml:real>
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					</ml:apply>
					<result xmlns="http://schemas.mathsoft.com/math30">
						<ml:real>22.065087978862532</ml:real>
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				<resultFormat>
					<general precision="15" show-trailing-zeros="true" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/>
					<matrix display-style="auto" expand-nested-arrays="false"/>
					<unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
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					<ml:apply>
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						<ml:apply>
							<ml:pow/>
							<ml:id xml:space="preserve" subscript="Final">Res</ml:id>
							<ml:real>2</ml:real>
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						<ml:real>0.0005177020435843876</ml:real>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Best fit cone included angle:</p>
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					<general precision="3" show-trailing-zeros="true" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/>
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						</b>
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					<b>
						<inlineAttr font-style="normal" line-through="false">:</inlineAttr>
					</b> A typical cone angle tolerance is on the order of 1°.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Tilt of cone axis:</p>
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							<ml:real>-0.2960581656466596</ml:real>
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								<unitReference unit="degree"/>
							</unitMonomial>
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							<ml:real>0.24427254110587707</ml:real>
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				<resultFormat>
					<general precision="3" show-trailing-zeros="true" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/>
					<matrix display-style="auto" expand-nested-arrays="false"/>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Best fit cone parameters:</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Direction:</p>
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						<ml:apply>
							<ml:id xml:space="preserve">rot</ml:id>
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								<ml:id xml:space="preserve">α</ml:id>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Cone center:</p>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">C</ml:id>
					<ml:matrix rows="3" cols="1">
						<ml:id xml:space="preserve">cx</ml:id>
						<ml:id xml:space="preserve">cy</ml:id>
						<ml:id xml:space="preserve">cz</ml:id>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Displacement from nominal:</p>
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								<ml:real>0.012939550661145422</ml:real>
								<ml:real>-0.13307251102491871</ml:real>
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								<unitReference unit="millimeter"/>
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				<p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">3D plot of measured data and fitted cone</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Angular width of displayed best fit cone surface:</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The light-blue wireframe surface represents a section of the best fit cone.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This step consists in calculating the distance of each data point to the best-fit cone along the local normal. This distance h is associated to the (X,Y) projection of the data point onto a plane perpendicular to the cone axis. </p>
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										<ml:localDefine>
											<ml:id xml:space="preserve">ω</ml:id>
											<ml:apply>
												<ml:id xml:space="preserve">atan2</ml:id>
												<ml:sequence>
													<ml:apply>
														<ml:mult style="default"/>
														<ml:id xml:space="preserve" font="4">CM</ml:id>
														<ml:id xml:space="preserve" font="4">U</ml:id>
													</ml:apply>
													<ml:apply>
														<ml:mult style="default"/>
														<ml:id xml:space="preserve" font="4">CM</ml:id>
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												</ml:sequence>
											</ml:apply>
										</ml:localDefine>
										<ml:localDefine>
											<ml:apply>
												<ml:indexer/>
												<ml:id xml:space="preserve">Flat</ml:id>
												<ml:sequence>
													<ml:id xml:space="preserve">k</ml:id>
													<ml:real>1</ml:real>
												</ml:sequence>
											</ml:apply>
											<ml:apply>
												<ml:mult style="default"/>
												<ml:id xml:space="preserve">MM'</ml:id>
												<ml:apply>
													<ml:id xml:space="preserve">cos</ml:id>
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											</ml:apply>
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										<ml:localDefine>
											<ml:apply>
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												<ml:id xml:space="preserve">Flat</ml:id>
												<ml:sequence>
													<ml:id xml:space="preserve">k</ml:id>
													<ml:real>2</ml:real>
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											</ml:apply>
											<ml:apply>
												<ml:mult style="default"/>
												<ml:id xml:space="preserve">MM'</ml:id>
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										<ml:localDefine>
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												<ml:sequence>
													<ml:id xml:space="preserve">k</ml:id>
													<ml:real>0</ml:real>
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										<ml:localDefine>
											<ml:apply>
												<ml:indexer/>
												<ml:id xml:space="preserve">Flat</ml:id>
												<ml:sequence>
													<ml:id xml:space="preserve">k</ml:id>
													<ml:real>3</ml:real>
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											</ml:apply>
											<ml:apply>
												<ml:div/>
												<ml:id xml:space="preserve">MM'</ml:id>
												<ml:id xml:space="preserve">sγ</ml:id>
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										<ml:localDefine>
											<ml:apply>
												<ml:indexer/>
												<ml:id xml:space="preserve">Flat</ml:id>
												<ml:sequence>
													<ml:id xml:space="preserve">k</ml:id>
													<ml:real>4</ml:real>
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											</ml:apply>
											<ml:apply>
												<ml:mult style="default"/>
												<ml:id xml:space="preserve">ω</ml:id>
												<ml:id xml:space="preserve">m</ml:id>
											</ml:apply>
										</ml:localDefine>
									</ml:program>
								</ml:for>
								<ml:localDefine>
									<ml:id xml:space="preserve">Flat</ml:id>
									<ml:apply>
										<ml:id xml:space="preserve">csort</ml:id>
										<ml:sequence>
											<ml:id xml:space="preserve">Flat</ml:id>
											<ml:real>4</ml:real>
										</ml:sequence>
									</ml:apply>
								</ml:localDefine>
								<ml:return>
									<ml:id xml:space="preserve">Flat</ml:id>
								</ml:return>
							</ml:program>
						</ml:define>
					</math>
				</region>
			</area>
			<rendering item-idref="64"/>
		</region>
		<region region-id="5835" left="0" top="3486" width="6000" height="6" align-x="0" align-y="3486" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="593" left="36" top="3494.25" width="307.5" height="84" align-x="43.5" align-y="3504" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Each line in the residual matrix contains the height deviation d from the best-fit cone, X, Y, <f family="Symbol" charset="2" size="10">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">r</inlineAttr>
					</f> and <f family="Symbol" charset="2" size="10">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">w</inlineAttr>
					</f>. (X,Y) are the rectangular coordinates of the projection of the data point onto a plane perpendicular to the best-fit cone axis. (<f family="Symbol" charset="2" size="10">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">r</inlineAttr>
					</f>,<f family="Symbol" charset="2" size="10">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">w</inlineAttr>
					</f>) are the distance of the data point from the cone center and the angular position of the data point projection in the plane XY.</p>
			</text>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">Residuals</ml:id>
					<ml:apply>
						<ml:id xml:space="preserve">GetResiduals</ml:id>
						<ml:sequence>
							<ml:id xml:space="preserve">MeasuredXYZ</ml:id>
							<ml:id xml:space="preserve">C</ml:id>
							<ml:id xml:space="preserve" font="4">D</ml:id>
							<ml:id xml:space="preserve">γ</ml:id>
						</ml:sequence>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="65"/>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Surface deviation:</p>
			</text>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">δ</ml:id>
					<ml:apply>
						<ml:matcol/>
						<ml:id xml:space="preserve">Residuals</ml:id>
						<ml:real>0</ml:real>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="66"/>
		</region>
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			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Look at the distribution of values:</p>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">hh</ml:id>
					<ml:apply>
						<ml:id xml:space="preserve">histogram</ml:id>
						<ml:sequence>
							<ml:real>1000</ml:real>
							<ml:apply>
								<ml:mult style="default"/>
								<ml:id xml:space="preserve">δ</ml:id>
								<ml:apply>
									<ml:pow/>
									<ml:id xml:space="preserve">nm</ml:id>
									<ml:real>-1</ml:real>
								</ml:apply>
							</ml:apply>
						</ml:sequence>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="67"/>
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		<region region-id="4984" left="30" top="3678" width="342.75" height="170.25" align-x="30" align-y="3678" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<plot disable-calc="false" item-idref="68"/>
			<rendering item-idref="69"/>
		</region>
		<region region-id="620" left="36" top="3847.5" width="196.5" height="12.75" align-x="45.75" align-y="3858" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Cone flatness deviation</p>
			</text>
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		<region region-id="5841" left="36" top="3866.25" width="163.5" height="12" align-x="45" align-y="3876" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Peak to valley surface deviation:</p>
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			<math optimize="false" disable-calc="false">
				<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:id xml:space="preserve">Range</ml:id>
						<ml:id xml:space="preserve">δ</ml:id>
					</ml:apply>
					<ml:unitOverride>
						<ml:id xml:space="preserve">μm</ml:id>
					</ml:unitOverride>
					<result xmlns="http://schemas.mathsoft.com/math30">
						<unitedValue>
							<ml:real>3.6611831928622109</ml:real>
							<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
								<unitReference unit="micron"/>
							</unitMonomial>
						</unitedValue>
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			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Projected points coordinates:</p>
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			<math optimize="false" disable-calc="false">
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					<ml:id xml:space="preserve">Px</ml:id>
					<ml:apply>
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						<ml:real>1</ml:real>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">Py</ml:id>
					<ml:apply>
						<ml:matcol/>
						<ml:id xml:space="preserve">Residuals</ml:id>
						<ml:real>2</ml:real>
					</ml:apply>
				</ml:define>
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		</region>
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					<math optimize="false" disable-calc="false">
						<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
							<ml:function>
								<ml:id xml:space="preserve">CircleAtR</ml:id>
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							<ml:program>
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											<ml:real>360</ml:real>
											<ml:real>5</ml:real>
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												<ml:apply>
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													<ml:real>5</ml:real>
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											</ml:apply>
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										<ml:localDefine>
											<ml:apply>
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												<ml:sequence>
													<ml:id xml:space="preserve">k</ml:id>
													<ml:real>0</ml:real>
												</ml:sequence>
											</ml:apply>
											<ml:apply>
												<ml:mult style="default"/>
												<ml:apply>
													<ml:div/>
													<ml:id xml:space="preserve">φ</ml:id>
													<ml:real>2</ml:real>
												</ml:apply>
												<ml:apply>
													<ml:id xml:space="preserve">cos</ml:id>
													<ml:id xml:space="preserve">ω</ml:id>
												</ml:apply>
											</ml:apply>
										</ml:localDefine>
										<ml:localDefine>
											<ml:apply>
												<ml:indexer/>
												<ml:id xml:space="preserve">out</ml:id>
												<ml:sequence>
													<ml:id xml:space="preserve">k</ml:id>
													<ml:real>1</ml:real>
												</ml:sequence>
											</ml:apply>
											<ml:apply>
												<ml:mult style="default"/>
												<ml:apply>
													<ml:div/>
													<ml:id xml:space="preserve">φ</ml:id>
													<ml:real>2</ml:real>
												</ml:apply>
												<ml:apply>
													<ml:id xml:space="preserve">sin</ml:id>
													<ml:id xml:space="preserve">ω</ml:id>
												</ml:apply>
											</ml:apply>
										</ml:localDefine>
										<ml:localDefine>
											<ml:apply>
												<ml:indexer/>
												<ml:id xml:space="preserve">out</ml:id>
												<ml:sequence>
													<ml:id xml:space="preserve">k</ml:id>
													<ml:real>2</ml:real>
												</ml:sequence>
											</ml:apply>
											<ml:apply>
												<ml:mult style="default"/>
												<ml:real>0</ml:real>
												<ml:id xml:space="preserve">mm</ml:id>
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									</ml:program>
								</ml:for>
								<ml:return>
									<ml:id xml:space="preserve">out</ml:id>
								</ml:return>
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					</math>
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					<math optimize="false" disable-calc="false">
						<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
							<ml:function>
								<ml:id xml:space="preserve">LineAtψ</ml:id>
								<ml:boundVars>
									<ml:id xml:space="preserve">ψ</ml:id>
									<ml:id xml:space="preserve">Sx</ml:id>
									<ml:id xml:space="preserve">Sy</ml:id>
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							<ml:program>
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											<ml:apply>
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												<ml:apply>
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											</ml:apply>
											<ml:apply>
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												<ml:apply>
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											</ml:apply>
										</ml:sequence>
									</ml:apply>
								</ml:localDefine>
								<ml:localDefine>
									<ml:id xml:space="preserve">ψ</ml:id>
									<ml:apply>
										<ml:id xml:space="preserve">atan2</ml:id>
										<ml:sequence>
											<ml:apply>
												<ml:id xml:space="preserve">cos</ml:id>
												<ml:id xml:space="preserve">ψ</ml:id>
											</ml:apply>
											<ml:apply>
												<ml:id xml:space="preserve">sin</ml:id>
												<ml:id xml:space="preserve">ψ</ml:id>
											</ml:apply>
										</ml:sequence>
									</ml:apply>
								</ml:localDefine>
								<ml:matrix rows="2" cols="3">
									<ml:real>0</ml:real>
									<ml:apply>
										<ml:mult style="default"/>
										<ml:id xml:space="preserve">d</ml:id>
										<ml:apply>
											<ml:id xml:space="preserve">cos</ml:id>
											<ml:id xml:space="preserve">ψ</ml:id>
										</ml:apply>
									</ml:apply>
									<ml:real>0</ml:real>
									<ml:apply>
										<ml:mult style="default"/>
										<ml:id xml:space="preserve">d</ml:id>
										<ml:apply>
											<ml:id xml:space="preserve">sin</ml:id>
											<ml:id xml:space="preserve">ψ</ml:id>
										</ml:apply>
									</ml:apply>
									<ml:real>0</ml:real>
									<ml:real>0</ml:real>
								</ml:matrix>
							</ml:program>
						</ml:define>
					</math>
				</region>
				<region region-id="651" left="30" top="200.25" width="95.25" height="12" align-x="36" align-y="210" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
					<text use-page-width="false" push-down="false" lock-width="true">
						<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">For display of profiles:</p>
					</text>
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					<math optimize="false" disable-calc="false">
						<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
							<ml:id xml:space="preserve">str</ml:id>
							<ml:apply>
								<ml:id xml:space="preserve">LineAtψ</ml:id>
								<ml:sequence>
									<ml:id xml:space="preserve">ψ</ml:id>
									<ml:id xml:space="preserve">Px</ml:id>
									<ml:id xml:space="preserve">Py</ml:id>
								</ml:sequence>
							</ml:apply>
						</ml:define>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">This surface is plotted using a 3D scatter plot because the surface plot can not handle data with holes. The style for the 3 plots is scatter plot.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The blue line position is defined by the angle <f family="Symbol" charset="2" size="10">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">y</inlineAttr>
					</f>. The so-called straightness profile is generated as the intersection of the surface with a plane that contains the blue segment and is perpendicular to the plane of the figure.</p>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"/>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Similarly, the red trace is the location where we want to extract a roundness profile, which is the intersection of the measured surface with a cylinder of diameter <f family="Symbol" charset="2" size="10">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">f</inlineAttr>
					</f> that has its axis perpendicular to the plane of the figure.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">In the following we will use the projection shown above to extract a straightness and a roundness profile. Note that the projection introduces geometrical distortions that we will ignore in this worksheet.</p>
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				<p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Straightness profile</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Straightness profile at angular position:</p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">We select a subset of data points based on their proximity to the generatrix of interest. Here we retain points whose angular position is within +/-1° of the generatrix direction <f family="Symbol" charset="2" size="10">
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					</f>.</p>
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						<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Select the points that are within +/-<f family="Arial" charset="0" size="10">
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							</f>y from the direction y. We use the function <b>
								<u>
									<link href="./matching.xmcd" title="Links to section" popup="false">Match</link>
								</u>
							</b>.</p>
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								<ml:str xml:space="preserve">Get angular position of projected data points</ml:str>
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								<ml:str xml:space="preserve">Calculate angular distance to chosen direction</ml:str>
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								<ml:str xml:space="preserve">Select data points that are close enough to profile direction</ml:str>
								<ml:str xml:space="preserve">Match provides a list of positions in matrix Data</ml:str>
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									<ml:id xml:space="preserve">msg</ml:id>
									<ml:str xml:space="preserve">Could not find matching points</ml:str>
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								<ml:str xml:space="preserve">Allocate memory in one step</ml:str>
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									<ml:real>0</ml:real>
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								<ml:str xml:space="preserve">Create point selection</ml:str>
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								</ml:for>
								<ml:str xml:space="preserve">Finally, sort according to distance from cone center</ml:str>
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										<ml:id xml:space="preserve">csort</ml:id>
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						<p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Interpolated straightness profile </p>
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						<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">(profile along a generatrix) The input data are in the format:</p>
						<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Data = [ Residual along normal, X-projection on plane perpendicular to cone axis, Y-projection, distance to cone center along generatrix ]</p>
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						<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">These data points have been selected based on their proximity to the generatrix of interest. Output data is in the same format and contains the interpolated profile. We use <b>
								<u>
									<link href="./regress_multi.xmcd" title="Links to section" popup="false">loess</link>
								</u>
							</b> for the interpolation.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">We use <b>
						<u>
							<link href="./regress_multi.xmcd" title="Links to section" popup="false">loess</link>
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					</b> to create an interpolated profile. The <f family="Verdana" charset="0">
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					</f> proximity parameter is chosen to be small to limit lowpass filtering:</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Interpolation sampling step:</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">We also create an interpolated profile by fitting a plane to the four closest neighbors of each interpolation location. The corresponding function needs data in a new coordinate system. The first parameter X is the position of the data points projected on the plane of the profile. The second parameter Y is the distance of the data points to the plane of the profile. The third parameter Z is the surface deviation. The last two parameters are the maximum distance from the profile plane for a point to be valid and the sampling step for the interpolation.</p>
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																		<ml:indexer/>
																		<ml:id xml:space="preserve">neighbors</ml:id>
																		<ml:sequence>
																			<ml:id xml:space="preserve">ii</ml:id>
																			<ml:real>0</ml:real>
																		</ml:sequence>
																	</ml:apply>
																</ml:apply>
															</ml:lambda>
															<ml:bounds>
																<ml:real>0</ml:real>
																<ml:real>3</ml:real>
															</ml:bounds>
														</ml:apply>
													</ml:localDefine>
												</ml:matrix>
												<ml:matrix rows="1" cols="2">
													<ml:localDefine>
														<ml:id xml:space="preserve">Sxz</ml:id>
														<ml:apply>
															<ml:summation/>
															<ml:lambda>
																<ml:boundVars>
																	<ml:id xml:space="preserve">ii</ml:id>
																</ml:boundVars>
																<ml:parens>
																	<ml:apply>
																		<ml:mult style="default"/>
																		<ml:parens>
																			<ml:apply>
																				<ml:minus/>
																				<ml:apply>
																					<ml:indexer/>
																					<ml:id xml:space="preserve">X</ml:id>
																					<ml:apply>
																						<ml:indexer/>
																						<ml:id xml:space="preserve">neighbors</ml:id>
																						<ml:sequence>
																							<ml:id xml:space="preserve">ii</ml:id>
																							<ml:real>0</ml:real>
																						</ml:sequence>
																					</ml:apply>
																				</ml:apply>
																				<ml:id xml:space="preserve">x</ml:id>
																			</ml:apply>
																		</ml:parens>
																		<ml:apply>
																			<ml:indexer/>
																			<ml:id xml:space="preserve">Z</ml:id>
																			<ml:apply>
																				<ml:indexer/>
																				<ml:id xml:space="preserve">neighbors</ml:id>
																				<ml:sequence>
																					<ml:id xml:space="preserve">ii</ml:id>
																					<ml:real>0</ml:real>
																				</ml:sequence>
																			</ml:apply>
																		</ml:apply>
																	</ml:apply>
																</ml:parens>
															</ml:lambda>
															<ml:bounds>
																<ml:real>0</ml:real>
																<ml:real>3</ml:real>
															</ml:bounds>
														</ml:apply>
													</ml:localDefine>
													<ml:localDefine>
														<ml:id xml:space="preserve">Syz</ml:id>
														<ml:apply>
															<ml:summation/>
															<ml:lambda>
																<ml:boundVars>
																	<ml:id xml:space="preserve">ii</ml:id>
																</ml:boundVars>
																<ml:parens>
																	<ml:apply>
																		<ml:mult style="default"/>
																		<ml:apply>
																			<ml:indexer/>
																			<ml:id xml:space="preserve">Y</ml:id>
																			<ml:apply>
																				<ml:indexer/>
																				<ml:id xml:space="preserve">neighbors</ml:id>
																				<ml:sequence>
																					<ml:id xml:space="preserve">ii</ml:id>
																					<ml:real>0</ml:real>
																				</ml:sequence>
																			</ml:apply>
																		</ml:apply>
																		<ml:apply>
																			<ml:indexer/>
																			<ml:id xml:space="preserve">Z</ml:id>
																			<ml:apply>
																				<ml:indexer/>
																				<ml:id xml:space="preserve">neighbors</ml:id>
																				<ml:sequence>
																					<ml:id xml:space="preserve">ii</ml:id>
																					<ml:real>0</ml:real>
																				</ml:sequence>
																			</ml:apply>
																		</ml:apply>
																	</ml:apply>
																</ml:parens>
															</ml:lambda>
															<ml:bounds>
																<ml:real>0</ml:real>
																<ml:real>3</ml:real>
															</ml:bounds>
														</ml:apply>
													</ml:localDefine>
												</ml:matrix>
												<ml:str xml:space="preserve">Calculate interpolated value</ml:str>
												<ml:localDefine>
													<ml:apply>
														<ml:indexer/>
														<ml:id xml:space="preserve">pt</ml:id>
														<ml:sequence>
															<ml:id xml:space="preserve">k</ml:id>
															<ml:real>0</ml:real>
														</ml:sequence>
													</ml:apply>
													<ml:id xml:space="preserve">x</ml:id>
												</ml:localDefine>
												<ml:localDefine>
													<ml:apply>
														<ml:indexer/>
														<ml:id xml:space="preserve">pt</ml:id>
														<ml:sequence>
															<ml:id xml:space="preserve">k</ml:id>
															<ml:real>1</ml:real>
														</ml:sequence>
													</ml:apply>
													<ml:apply>
														<ml:div/>
														<ml:apply>
															<ml:minus/>
															<ml:apply>
																<ml:plus/>
																<ml:apply>
																	<ml:minus/>
																	<ml:apply>
																		<ml:plus/>
																		<ml:apply>
																			<ml:plus/>
																			<ml:apply>
																				<ml:mult style="default"/>
																				<ml:apply>
																					<ml:mult style="default"/>
																					<ml:apply>
																						<ml:neg/>
																						<ml:id xml:space="preserve">Sx</ml:id>
																					</ml:apply>
																					<ml:id xml:space="preserve">Sxz</ml:id>
																				</ml:apply>
																				<ml:id xml:space="preserve">Sy2</ml:id>
																			</ml:apply>
																			<ml:apply>
																				<ml:mult style="default"/>
																				<ml:apply>
																					<ml:mult style="default"/>
																					<ml:id xml:space="preserve">Sx</ml:id>
																					<ml:id xml:space="preserve">Sxy</ml:id>
																				</ml:apply>
																				<ml:id xml:space="preserve">Syz</ml:id>
																			</ml:apply>
																		</ml:apply>
																		<ml:apply>
																			<ml:mult style="default"/>
																			<ml:apply>
																				<ml:mult style="default"/>
																				<ml:id xml:space="preserve">Sy</ml:id>
																				<ml:id xml:space="preserve">Sxy</ml:id>
																			</ml:apply>
																			<ml:id xml:space="preserve">Sxz</ml:id>
																		</ml:apply>
																	</ml:apply>
																	<ml:apply>
																		<ml:mult style="default"/>
																		<ml:apply>
																			<ml:mult style="default"/>
																			<ml:id xml:space="preserve">Sy</ml:id>
																			<ml:id xml:space="preserve">Syz</ml:id>
																		</ml:apply>
																		<ml:id xml:space="preserve">Sx2</ml:id>
																	</ml:apply>
																</ml:apply>
																<ml:apply>
																	<ml:mult style="default"/>
																	<ml:apply>
																		<ml:mult style="default"/>
																		<ml:id xml:space="preserve">Sz</ml:id>
																		<ml:id xml:space="preserve">Sx2</ml:id>
																	</ml:apply>
																	<ml:id xml:space="preserve">Sy2</ml:id>
																</ml:apply>
															</ml:apply>
															<ml:apply>
																<ml:mult style="default"/>
																<ml:id xml:space="preserve">Sz</ml:id>
																<ml:apply>
																	<ml:pow/>
																	<ml:id xml:space="preserve">Sxy</ml:id>
																	<ml:real>2</ml:real>
																</ml:apply>
															</ml:apply>
														</ml:apply>
														<ml:apply>
															<ml:minus/>
															<ml:apply>
																<ml:plus/>
																<ml:apply>
																	<ml:minus/>
																	<ml:apply>
																		<ml:minus/>
																		<ml:apply>
																			<ml:mult style="default"/>
																			<ml:apply>
																				<ml:mult style="default"/>
																				<ml:id xml:space="preserve">Sx2</ml:id>
																				<ml:id xml:space="preserve">Sy2</ml:id>
																			</ml:apply>
																			<ml:real>4</ml:real>
																		</ml:apply>
																		<ml:apply>
																			<ml:mult style="default"/>
																			<ml:id xml:space="preserve">Sx2</ml:id>
																			<ml:apply>
																				<ml:pow/>
																				<ml:id xml:space="preserve">Sy</ml:id>
																				<ml:real>2</ml:real>
																			</ml:apply>
																		</ml:apply>
																	</ml:apply>
																	<ml:apply>
																		<ml:mult style="default"/>
																		<ml:apply>
																			<ml:pow/>
																			<ml:id xml:space="preserve">Sxy</ml:id>
																			<ml:real>2</ml:real>
																		</ml:apply>
																		<ml:real>4</ml:real>
																	</ml:apply>
																</ml:apply>
																<ml:apply>
																	<ml:mult style="default"/>
																	<ml:apply>
																		<ml:mult style="default"/>
																		<ml:apply>
																			<ml:mult style="default"/>
																			<ml:real>2</ml:real>
																			<ml:id xml:space="preserve">Sxy</ml:id>
																		</ml:apply>
																		<ml:id xml:space="preserve">Sx</ml:id>
																	</ml:apply>
																	<ml:id xml:space="preserve">Sy</ml:id>
																</ml:apply>
															</ml:apply>
															<ml:apply>
																<ml:mult style="default"/>
																<ml:apply>
																	<ml:pow/>
																	<ml:id xml:space="preserve">Sx</ml:id>
																	<ml:real>2</ml:real>
																</ml:apply>
																<ml:id xml:space="preserve">Sy2</ml:id>
															</ml:apply>
														</ml:apply>
													</ml:apply>
												</ml:localDefine>
												<ml:localDefine>
													<ml:id xml:space="preserve">k</ml:id>
													<ml:apply>
														<ml:plus/>
														<ml:id xml:space="preserve">k</ml:id>
														<ml:real>1</ml:real>
													</ml:apply>
												</ml:localDefine>
											</ml:program>
										</ml:ifThen>
										<ml:localDefine>
											<ml:id xml:space="preserve">x</ml:id>
											<ml:apply>
												<ml:plus/>
												<ml:id xml:space="preserve">x</ml:id>
												<ml:id xml:space="preserve">step</ml:id>
											</ml:apply>
										</ml:localDefine>
										<ml:localDefine>
											<ml:id xml:space="preserve">idx</ml:id>
											<ml:apply>
												<ml:id xml:space="preserve">round</ml:id>
												<ml:apply>
													<ml:div/>
													<ml:apply>
														<ml:plus/>
														<ml:id xml:space="preserve">i1</ml:id>
														<ml:id xml:space="preserve">i2</ml:id>
													</ml:apply>
													<ml:real>2</ml:real>
												</ml:apply>
											</ml:apply>
										</ml:localDefine>
										<ml:while>
											<ml:apply>
												<ml:mult style="default"/>
												<ml:parens>
													<ml:apply>
														<ml:lessThan/>
														<ml:apply>
															<ml:indexer/>
															<ml:id xml:space="preserve">X</ml:id>
															<ml:id xml:space="preserve">idx</ml:id>
														</ml:apply>
														<ml:id xml:space="preserve">x</ml:id>
													</ml:apply>
												</ml:parens>
												<ml:parens>
													<ml:apply>
														<ml:lessThan/>
														<ml:id xml:space="preserve">idx</ml:id>
														<ml:apply>
															<ml:id xml:space="preserve">last</ml:id>
															<ml:id xml:space="preserve">X</ml:id>
														</ml:apply>
													</ml:apply>
												</ml:parens>
											</ml:apply>
											<ml:localDefine>
												<ml:id xml:space="preserve">idx</ml:id>
												<ml:apply>
													<ml:plus/>
													<ml:id xml:space="preserve">idx</ml:id>
													<ml:real>1</ml:real>
												</ml:apply>
											</ml:localDefine>
										</ml:while>
										<ml:while>
											<ml:apply>
												<ml:mult style="default"/>
												<ml:parens>
													<ml:apply>
														<ml:greaterThan/>
														<ml:apply>
															<ml:indexer/>
															<ml:id xml:space="preserve">X</ml:id>
															<ml:id xml:space="preserve">idx</ml:id>
														</ml:apply>
														<ml:id xml:space="preserve">x</ml:id>
													</ml:apply>
												</ml:parens>
												<ml:parens>
													<ml:apply>
														<ml:greaterThan/>
														<ml:id xml:space="preserve">idx</ml:id>
														<ml:real>0</ml:real>
													</ml:apply>
												</ml:parens>
											</ml:apply>
											<ml:localDefine>
												<ml:id xml:space="preserve">idx</ml:id>
												<ml:apply>
													<ml:minus/>
													<ml:id xml:space="preserve">idx</ml:id>
													<ml:real>1</ml:real>
												</ml:apply>
											</ml:localDefine>
										</ml:while>
									</ml:program>
								</ml:while>
								<ml:return>
									<ml:id xml:space="preserve">pt</ml:id>
								</ml:return>
							</ml:program>
						</ml:define>
					</math>
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			</area>
			<rendering item-idref="91"/>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">X</ml:id>
					<ml:id xml:space="preserve">subρ</ml:id>
				</ml:define>
			</math>
			<rendering item-idref="92"/>
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		<region region-id="5900" left="144" top="4971" width="130.5" height="18.75" align-x="156.75" align-y="4986" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">Y</ml:id>
					<ml:apply>
						<ml:vectorize/>
						<ml:parens>
							<ml:apply>
								<ml:mult style="default"/>
								<ml:id xml:space="preserve">subρ</ml:id>
								<ml:apply>
									<ml:id xml:space="preserve">mod</ml:id>
									<ml:sequence>
										<ml:apply>
											<ml:minus/>
											<ml:id xml:space="preserve">subω</ml:id>
											<ml:id xml:space="preserve">ψ</ml:id>
										</ml:apply>
										<ml:apply>
											<ml:mult style="default"/>
											<ml:real>2</ml:real>
											<ml:id xml:space="preserve">π</ml:id>
										</ml:apply>
									</ml:sequence>
								</ml:apply>
							</ml:apply>
						</ml:parens>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="93"/>
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		<region region-id="5901" left="72" top="5001" width="40.5" height="12.75" align-x="84" align-y="5010" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">Z</ml:id>
					<ml:id xml:space="preserve">subZ</ml:id>
				</ml:define>
			</math>
			<rendering item-idref="94"/>
		</region>
		<region region-id="5902" left="72" top="5025" width="315" height="12.75" align-x="129.75" align-y="5034" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">Straightness'</ml:id>
					<ml:apply>
						<ml:id xml:space="preserve">Interpolate4PtPlane</ml:id>
						<ml:sequence>
							<ml:id xml:space="preserve">X</ml:id>
							<ml:id xml:space="preserve">Y</ml:id>
							<ml:id xml:space="preserve">Z</ml:id>
							<ml:apply>
								<ml:id xml:space="preserve">min</ml:id>
								<ml:id xml:space="preserve">subρ</ml:id>
							</ml:apply>
							<ml:apply>
								<ml:id xml:space="preserve">max</ml:id>
								<ml:id xml:space="preserve">subρ</ml:id>
							</ml:apply>
							<ml:apply>
								<ml:mult style="default"/>
								<ml:real>10</ml:real>
								<ml:id xml:space="preserve">μm</ml:id>
							</ml:apply>
							<ml:id xml:space="preserve">dρ</ml:id>
						</ml:sequence>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="95"/>
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		<region region-id="5903" left="72" top="5049.75" width="91.5" height="18" align-x="93" align-y="5064" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">stZ'</ml:id>
					<ml:apply>
						<ml:matcol/>
						<ml:id xml:space="preserve">Straightness'</ml:id>
						<ml:real>1</ml:real>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="96"/>
		</region>
		<region region-id="5904" left="198" top="5049.75" width="90" height="18" align-x="217.5" align-y="5064" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">stρ'</ml:id>
					<ml:apply>
						<ml:matcol/>
						<ml:id xml:space="preserve">Straightness'</ml:id>
						<ml:real>0</ml:real>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="97"/>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:id xml:space="preserve">Range</ml:id>
						<ml:id xml:space="preserve">stZ'</ml:id>
					</ml:apply>
					<ml:unitOverride>
						<ml:id xml:space="preserve">μm</ml:id>
					</ml:unitOverride>
					<result xmlns="http://schemas.mathsoft.com/math30">
						<unitedValue>
							<ml:real>1.8614291520747146</ml:real>
							<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
								<unitReference unit="micron"/>
							</unitMonomial>
						</unitedValue>
					</result>
				</ml:eval>
			</math>
			<rendering item-idref="98"/>
		</region>
		<region region-id="5628" left="6" top="5100" width="411.75" height="227.25" align-x="6" align-y="5100" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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			<rendering item-idref="100"/>
		</region>
		<region region-id="5670" left="36" top="5329.5" width="149.25" height="12.75" align-x="58.5" align-y="5340" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="RoundnessProfile">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Roundness profile</p>
			</text>
		</region>
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			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Cone diameter where roundness is measured:</p>
			</text>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:globalDefine xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">φ</ml:id>
					<ml:apply>
						<ml:mult style="default"/>
						<ml:real>2.68</ml:real>
						<ml:id xml:space="preserve">mm</ml:id>
					</ml:apply>
				</ml:globalDefine>
			</math>
			<rendering item-idref="101"/>
		</region>
		<region region-id="5979" left="0" top="5400" width="6008.25" align-x="0" align-y="5400" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="" height="824.25">
			<area is-collapsed="true" name="Roundness" show-name="true" show-border="false" show-icon="true" show-timestamp="true" allow-expand="false" is-locked="false" timestamp="" top-lock-id="4898" bottom-lock-id="4901">
				<region region-id="5033" left="24" top="33" width="415.5" height="282" align-x="134.25" align-y="42" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
					<math optimize="false" disable-calc="false">
						<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
							<ml:function>
								<ml:id xml:space="preserve">SelectDataRd</ml:id>
								<ml:boundVars>
									<ml:id xml:space="preserve">φ</ml:id>
									<ml:id xml:space="preserve">Data</ml:id>
									<ml:id xml:space="preserve">Δr</ml:id>
								</ml:boundVars>
							</ml:function>
							<ml:program>
								<ml:str xml:space="preserve">Get radial distance of each point to specified radius</ml:str>
								<ml:localDefine>
									<ml:id xml:space="preserve">dr</ml:id>
									<ml:apply>
										<ml:mult style="default"/>
										<ml:parens>
											<ml:apply>
												<ml:minus/>
												<ml:apply>
													<ml:mult style="default"/>
													<ml:apply>
														<ml:matcol/>
														<ml:id xml:space="preserve">Data</ml:id>
														<ml:real>3</ml:real>
													</ml:apply>
													<ml:apply>
														<ml:id xml:space="preserve">sin</ml:id>
														<ml:id xml:space="preserve">γ</ml:id>
													</ml:apply>
												</ml:apply>
												<ml:apply>
													<ml:div/>
													<ml:id xml:space="preserve">φ</ml:id>
													<ml:real>2</ml:real>
												</ml:apply>
											</ml:apply>
										</ml:parens>
										<ml:apply>
											<ml:pow/>
											<ml:id xml:space="preserve">μm</ml:id>
											<ml:real>-1</ml:real>
										</ml:apply>
									</ml:apply>
								</ml:localDefine>
								<ml:str xml:space="preserve">Select data points that are close enough to profile direction</ml:str>
								<ml:str xml:space="preserve">Scale the distance to micrometer so that there are no TOL issues</ml:str>
								<ml:localDefine>
									<ml:id xml:space="preserve">msg</ml:id>
									<ml:str xml:space="preserve">Could not find matching points</ml:str>
								</ml:localDefine>
								<ml:tryCatch>
									<ml:localDefine>
										<ml:id xml:space="preserve">Positions</ml:id>
										<ml:apply>
											<ml:id xml:space="preserve">Match</ml:id>
											<ml:sequence>
												<ml:apply>
													<ml:mult style="default"/>
													<ml:matrix rows="2" cols="1">
														<ml:apply>
															<ml:neg/>
															<ml:id xml:space="preserve">Δr</ml:id>
														</ml:apply>
														<ml:id xml:space="preserve">Δr</ml:id>
													</ml:matrix>
													<ml:apply>
														<ml:pow/>
														<ml:id xml:space="preserve">μm</ml:id>
														<ml:real>-1</ml:real>
													</ml:apply>
												</ml:apply>
												<ml:id xml:space="preserve">dr</ml:id>
												<ml:str xml:space="preserve">range</ml:str>
											</ml:sequence>
										</ml:apply>
									</ml:localDefine>
									<ml:return>
										<ml:id xml:space="preserve">msg</ml:id>
									</ml:return>
								</ml:tryCatch>
								<ml:str xml:space="preserve">Allocate memory in one step</ml:str>
								<ml:localDefine>
									<ml:apply>
										<ml:indexer/>
										<ml:id xml:space="preserve">SelectData</ml:id>
										<ml:sequence>
											<ml:real>3</ml:real>
											<ml:apply>
												<ml:id xml:space="preserve">last</ml:id>
												<ml:id xml:space="preserve">Positions</ml:id>
											</ml:apply>
										</ml:sequence>
									</ml:apply>
									<ml:real>0</ml:real>
								</ml:localDefine>
								<ml:localDefine>
									<ml:id xml:space="preserve">TData</ml:id>
									<ml:apply>
										<ml:transpose/>
										<ml:parens>
											<ml:id xml:space="preserve">Data</ml:id>
										</ml:parens>
									</ml:apply>
								</ml:localDefine>
								<ml:str xml:space="preserve">Create point selection</ml:str>
								<ml:for>
									<ml:id xml:space="preserve">i</ml:id>
									<ml:range>
										<ml:real>0</ml:real>
										<ml:apply>
											<ml:id xml:space="preserve">last</ml:id>
											<ml:id xml:space="preserve">Positions</ml:id>
										</ml:apply>
									</ml:range>
									<ml:localDefine>
										<ml:apply>
											<ml:matcol/>
											<ml:id xml:space="preserve">SelectData</ml:id>
											<ml:id xml:space="preserve">i</ml:id>
										</ml:apply>
										<ml:apply>
											<ml:matcol/>
											<ml:id xml:space="preserve">TData</ml:id>
											<ml:apply>
												<ml:indexer/>
												<ml:id xml:space="preserve">Positions</ml:id>
												<ml:id xml:space="preserve">i</ml:id>
											</ml:apply>
										</ml:apply>
									</ml:localDefine>
								</ml:for>
								<ml:str xml:space="preserve">Finally, sort according to angular position on the strip of data</ml:str>
								<ml:return>
									<ml:apply>
										<ml:id xml:space="preserve">csort</ml:id>
										<ml:sequence>
											<ml:apply>
												<ml:transpose/>
												<ml:parens>
													<ml:id xml:space="preserve">SelectData</ml:id>
												</ml:parens>
											</ml:apply>
											<ml:real>4</ml:real>
										</ml:sequence>
									</ml:apply>
								</ml:return>
							</ml:program>
						</ml:define>
					</math>
				</region>
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					<math optimize="false" disable-calc="false">
						<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
							<ml:function>
								<ml:id xml:space="preserve">RdProf</ml:id>
								<ml:boundVars>
									<ml:id xml:space="preserve">φ</ml:id>
									<ml:id xml:space="preserve">Data</ml:id>
									<ml:id xml:space="preserve">γ</ml:id>
									<ml:id xml:space="preserve">lp</ml:id>
									<ml:id xml:space="preserve">dp</ml:id>
								</ml:boundVars>
							</ml:function>
							<ml:program>
								<ml:localDefine>
									<ml:id xml:space="preserve">LS</ml:id>
									<ml:apply>
										<ml:id xml:space="preserve">loess</ml:id>
										<ml:sequence>
											<ml:apply>
												<ml:div/>
												<ml:apply>
													<ml:id xml:space="preserve">augment</ml:id>
													<ml:sequence>
														<ml:apply>
															<ml:matcol/>
															<ml:id xml:space="preserve">Data</ml:id>
															<ml:real>1</ml:real>
														</ml:apply>
														<ml:apply>
															<ml:matcol/>
															<ml:id xml:space="preserve">Data</ml:id>
															<ml:real>2</ml:real>
														</ml:apply>
													</ml:sequence>
												</ml:apply>
												<ml:id xml:space="preserve">mm</ml:id>
											</ml:apply>
											<ml:apply>
												<ml:div/>
												<ml:apply>
													<ml:matcol/>
													<ml:id xml:space="preserve">Data</ml:id>
													<ml:real>0</ml:real>
												</ml:apply>
												<ml:id xml:space="preserve">μm</ml:id>
											</ml:apply>
											<ml:id xml:space="preserve">lp</ml:id>
										</ml:sequence>
									</ml:apply>
								</ml:localDefine>
								<ml:str xml:space="preserve">Number of points over full circle</ml:str>
								<ml:localDefine>
									<ml:id xml:space="preserve">N</ml:id>
									<ml:apply>
										<ml:id xml:space="preserve">round</ml:id>
										<ml:apply>
											<ml:div/>
											<ml:apply>
												<ml:mult style="default"/>
												<ml:id xml:space="preserve">π</ml:id>
												<ml:id xml:space="preserve">φ</ml:id>
											</ml:apply>
											<ml:id xml:space="preserve">dp</ml:id>
										</ml:apply>
									</ml:apply>
								</ml:localDefine>
								<ml:localDefine>
									<ml:id xml:space="preserve">dp</ml:id>
									<ml:apply>
										<ml:div/>
										<ml:apply>
											<ml:mult style="default"/>
											<ml:id xml:space="preserve">π</ml:id>
											<ml:id xml:space="preserve">φ</ml:id>
										</ml:apply>
										<ml:id xml:space="preserve">N</ml:id>
									</ml:apply>
								</ml:localDefine>
								<ml:localDefine>
									<ml:id xml:space="preserve" subscript="0">r</ml:id>
									<ml:apply>
										<ml:div/>
										<ml:id xml:space="preserve">φ</ml:id>
										<ml:real>2</ml:real>
									</ml:apply>
								</ml:localDefine>
								<ml:localDefine>
									<ml:id xml:space="preserve">id</ml:id>
									<ml:real>0</ml:real>
								</ml:localDefine>
								<ml:for>
									<ml:id xml:space="preserve">i</ml:id>
									<ml:range>
										<ml:real>0</ml:real>
										<ml:apply>
											<ml:minus/>
											<ml:id xml:space="preserve">N</ml:id>
											<ml:real>1</ml:real>
										</ml:apply>
									</ml:range>
									<ml:program>
										<ml:str xml:space="preserve">p is the perimeter location</ml:str>
										<ml:localDefine>
											<ml:id xml:space="preserve">p</ml:id>
											<ml:apply>
												<ml:mult style="default"/>
												<ml:parens>
													<ml:apply>
														<ml:plus/>
														<ml:id xml:space="preserve">i</ml:id>
														<ml:real>0.5</ml:real>
													</ml:apply>
												</ml:parens>
												<ml:id xml:space="preserve">dp</ml:id>
											</ml:apply>
										</ml:localDefine>
										<ml:localDefine>
											<ml:id xml:space="preserve">ω</ml:id>
											<ml:apply>
												<ml:div/>
												<ml:id xml:space="preserve">p</ml:id>
												<ml:id xml:space="preserve" subscript="0">r</ml:id>
											</ml:apply>
										</ml:localDefine>
										<ml:localDefine>
											<ml:id xml:space="preserve">x</ml:id>
											<ml:apply>
												<ml:mult style="default"/>
												<ml:id xml:space="preserve" subscript="0">r</ml:id>
												<ml:apply>
													<ml:id xml:space="preserve">cos</ml:id>
													<ml:id xml:space="preserve">ω</ml:id>
												</ml:apply>
											</ml:apply>
										</ml:localDefine>
										<ml:localDefine>
											<ml:id xml:space="preserve">y</ml:id>
											<ml:apply>
												<ml:mult style="default"/>
												<ml:id xml:space="preserve" subscript="0">r</ml:id>
												<ml:apply>
													<ml:id xml:space="preserve">sin</ml:id>
													<ml:id xml:space="preserve">ω</ml:id>
												</ml:apply>
											</ml:apply>
										</ml:localDefine>
										<ml:tryCatch>
											<ml:localDefine>
												<ml:id xml:space="preserve">dz</ml:id>
												<ml:apply>
													<ml:id xml:space="preserve">interp</ml:id>
													<ml:sequence>
														<ml:id xml:space="preserve">LS</ml:id>
														<ml:apply>
															<ml:div/>
															<ml:apply>
																<ml:id xml:space="preserve">augment</ml:id>
																<ml:sequence>
																	<ml:apply>
																		<ml:matcol/>
																		<ml:id xml:space="preserve">Data</ml:id>
																		<ml:real>1</ml:real>
																	</ml:apply>
																	<ml:apply>
																		<ml:matcol/>
																		<ml:id xml:space="preserve">Data</ml:id>
																		<ml:real>2</ml:real>
																	</ml:apply>
																</ml:sequence>
															</ml:apply>
															<ml:id xml:space="preserve">mm</ml:id>
														</ml:apply>
														<ml:apply>
															<ml:div/>
															<ml:apply>
																<ml:matcol/>
																<ml:id xml:space="preserve">Data</ml:id>
																<ml:real>0</ml:real>
															</ml:apply>
															<ml:id xml:space="preserve">μm</ml:id>
														</ml:apply>
														<ml:apply>
															<ml:mult style="default"/>
															<ml:apply>
																<ml:pow/>
																<ml:id xml:space="preserve">mm</ml:id>
																<ml:real>-1</ml:real>
															</ml:apply>
															<ml:matrix rows="2" cols="1">
																<ml:id xml:space="preserve">x</ml:id>
																<ml:id xml:space="preserve">y</ml:id>
															</ml:matrix>
														</ml:apply>
													</ml:sequence>
												</ml:apply>
											</ml:localDefine>
											<ml:continue/>
										</ml:tryCatch>
										<ml:localDefine>
											<ml:apply>
												<ml:indexer/>
												<ml:id xml:space="preserve">out</ml:id>
												<ml:sequence>
													<ml:id xml:space="preserve">id</ml:id>
													<ml:real>0</ml:real>
												</ml:sequence>
											</ml:apply>
											<ml:apply>
												<ml:mult style="default"/>
												<ml:id xml:space="preserve">dz</ml:id>
												<ml:id xml:space="preserve">μm</ml:id>
											</ml:apply>
										</ml:localDefine>
										<ml:localDefine>
											<ml:apply>
												<ml:indexer/>
												<ml:id xml:space="preserve">out</ml:id>
												<ml:sequence>
													<ml:id xml:space="preserve">id</ml:id>
													<ml:real>1</ml:real>
												</ml:sequence>
											</ml:apply>
											<ml:id xml:space="preserve">x</ml:id>
										</ml:localDefine>
										<ml:localDefine>
											<ml:apply>
												<ml:indexer/>
												<ml:id xml:space="preserve">out</ml:id>
												<ml:sequence>
													<ml:id xml:space="preserve">id</ml:id>
													<ml:real>2</ml:real>
												</ml:sequence>
											</ml:apply>
											<ml:id xml:space="preserve">y</ml:id>
										</ml:localDefine>
										<ml:localDefine>
											<ml:apply>
												<ml:indexer/>
												<ml:id xml:space="preserve">out</ml:id>
												<ml:sequence>
													<ml:id xml:space="preserve">id</ml:id>
													<ml:real>3</ml:real>
												</ml:sequence>
											</ml:apply>
											<ml:id xml:space="preserve">p</ml:id>
										</ml:localDefine>
										<ml:localDefine>
											<ml:id xml:space="preserve">id</ml:id>
											<ml:apply>
												<ml:plus/>
												<ml:id xml:space="preserve">id</ml:id>
												<ml:real>1</ml:real>
											</ml:apply>
										</ml:localDefine>
									</ml:program>
								</ml:for>
								<ml:id xml:space="preserve">out</ml:id>
							</ml:program>
						</ml:define>
					</math>
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			</area>
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		</region>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">We first select the data points that fall within a certain distance of the circle of diameter <f family="Symbol" charset="2" size="10">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">f</inlineAttr>
					</f>. The last parameter of the function call below defines this maximum distance.</p>
			</text>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">subData2</ml:id>
					<ml:apply>
						<ml:id xml:space="preserve">SelectDataRd</ml:id>
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							<ml:id xml:space="preserve">φ</ml:id>
							<ml:id xml:space="preserve">Residuals</ml:id>
							<ml:apply>
								<ml:mult style="default"/>
								<ml:real>10</ml:real>
								<ml:id xml:space="preserve">μm</ml:id>
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				</ml:define>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">subX</ml:id>
					<ml:apply>
						<ml:matcol/>
						<ml:id xml:space="preserve">subData2</ml:id>
						<ml:real>1</ml:real>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="104"/>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">subY</ml:id>
					<ml:apply>
						<ml:matcol/>
						<ml:id xml:space="preserve">subData2</ml:id>
						<ml:real>2</ml:real>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="105"/>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">subZ</ml:id>
					<ml:apply>
						<ml:matcol/>
						<ml:id xml:space="preserve">subData2</ml:id>
						<ml:real>0</ml:real>
					</ml:apply>
				</ml:define>
			</math>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">subρ</ml:id>
					<ml:apply>
						<ml:matcol/>
						<ml:id xml:space="preserve">subData2</ml:id>
						<ml:real>3</ml:real>
					</ml:apply>
				</ml:define>
			</math>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">subω</ml:id>
					<ml:apply>
						<ml:div/>
						<ml:apply>
							<ml:matcol/>
							<ml:id xml:space="preserve">subData2</ml:id>
							<ml:real>4</ml:real>
						</ml:apply>
						<ml:id xml:space="preserve">m</ml:id>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="108"/>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Nominal interpolation sampling step along the perimeter:</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Exact sampling step:</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Use 4-point linear interpolation:</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Because the data wrap around in this case we pad both ends of the data sent to the interpolation function.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The roundness error at diameter <f family="Symbol" charset="2" size="10">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">f</inlineAttr>
					</f>
					<f family="Arial" charset="0" size="10">
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					</f>is equal to:</p>
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					<f family="Arial" charset="0">
						<b>
							<inlineAttr font-style="normal" line-through="false">Note</inlineAttr>
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					</f>: A typical roundness tolerance is on the order of 1 micrometer.</p>
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				<p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Profile filtering</p>
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			<indexes>
				<phrase>profile filtering</phrase>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Roundness profiles are usually lowpass, band-pass and highpass filtered to extract form, waviness and roughness information. We illustrate here how to extract form with a Gaussian filter. A typical cutoff frequency (50% transfer function) for this type of data is "50 upr," which corresponds to 50 undulations per revolution.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Since there may be missing data points in the interpolated roundness profile, we use linear interpolation to fill in the missing data and then use a Fourier transform to apply the Gaussian filter. Note that since the data is effectively periodic for a roundness profile there are no edge discontinuities issues for the Fourier transformation.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Generate Gaussian filter for the frequency domain:</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Filtered profile:</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The roundness from error at diameter <f family="Symbol" charset="2" size="10">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">f</inlineAttr>
					</f>
					<f family="Arial" charset="0" size="10">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
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					</f>is equal to:</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Roundness profiles are frequently presented as polar plots.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Offset radius:</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Plot range:</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The number of grid points on the polar axis should be set manually to <region region-id="5541" left="50.25" top="6777" width="51" height="27.75" align-x="79.5" align-y="6792.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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					</region> to get a scale of <region region-id="5542" left="192.75" top="6783.75" width="27" height="12.75" align-x="204.75" align-y="6792.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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								<ml:real>0.1</ml:real>
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					</region> per radial division.</p>
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				<p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">3D plot detail</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">As a final step, let's display the subregion of the data where the roundness and straightness profiles intersect. Rotate the figure to observe how the profiles follow the original surface deviation.</p>
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						<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Straightness data:</p>
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						<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Surface deviation data:</p>
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						<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Find the corresponding section of the roundness profile.</p>
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				<p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">References</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Metrology data courtesy of Zygo Corporation, all rights reserved, used by permission. </p>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"/>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">For more information on surface metrology: D. J. Whitehouse (1994), <f family="Verdana" charset="0">
						<i>
							<inlineAttr line-through="false">Handbook of Surface Metrology</inlineAttr>
						</i>
					</f>, Institute of Physics Publishing, ISBN 0-7503-0039-6.</p>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"/>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">For more information on optical profilers: Peter de Groot, Xavier Colonna de Lega (May 2003), "Valve cone measurement using white light interference microscopy in a spherical measurement geometry," <f family="Verdana" charset="0">
						<i>
							<inlineAttr line-through="false">Optical Engineering</inlineAttr>
						</i>
					</f>, Vol. 42, No. 5, pp. 1232-1237.</p>
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