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list-style-type="inherit" tabs="inherit"><f family="Arial" charset="0"><u><inlineAttr line-through="false"><c val="#808000"><link href="./nonparam.xmcd" title="Links to Section Contents" popup="false"><sp count="6" />NONPARAMETRIC FITTING</link></c></inlineAttr></u></f></p></text></region><region region-id="114" left="36" top="72.75" width="72.75" height="13.5" align-x="36" align-y="84" 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="Sous-titre" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Cubic Splines</p></text><indexes><phrase>cubic splines</phrase><phrase>nonparametric fitting, cubic splines</phrase><phrase>splines, cubic</phrase></indexes></region><region region-id="583" left="66" top="97.5" width="71.25" height="12.75" align-x="66" align-y="108" 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="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">pspline(Mx,My)</p></text><indexes><phrase>pspline</phrase><phrase>functions, pspline</phrase></indexes></region><region region-id="525" left="180" top="97.5" width="70.5" height="12.75" align-x="180" align-y="108" 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="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">cspline(Mx,My)</p></text><indexes><phrase>cspline</phrase><phrase>functions, cspline</phrase></indexes></region><region region-id="584" left="66" top="121.5" width="69" height="12.75" align-x="66" align-y="132" 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="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">lspline(Mx,My)</p></text><indexes><phrase>lspline</phrase><phrase>functions, lspline</phrase></indexes></region><region region-id="526" left="180" top="121.5" width="87.75" height="12.75" align-x="180" align-y="132" 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="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">interp(vs,Mx,My,x)</p></text><indexes><phrase>interp</phrase><phrase>functions, interp</phrase></indexes></region><region region-id="901" left="66" top="145.5" width="113.25" height="12.75" align-x="66" align-y="156" 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="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Bicubic2D(vx,vy,Z,p,q)</p></text><indexes><phrase>Bicubic2D</phrase><phrase>functions, Bicubic2D</phrase></indexes></region><region region-id="533" left="36" top="170.25" width="278.25" height="96" align-x="36" align-y="180" 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">These functions construct cubic splines (piecewise polynomials) of the specified endpoint form, l for linear, p for parabolic, and c for cubic.<sp count="2" />Columns of Mx must be in ascending order.The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">interp</c></inlineAttr></b></f> function is required to construct the interpolated points from the output of the spline functions. <f family="Verdana" charset="0" size="10">The </f><f family="Verdana" charset="0" size="10"><b>Bicubic2D</b></f><f family="Verdana" charset="0" size="10"> function performs bicubic interpolation in two dimensions at the requested point </f><f family="Verdana" charset="0" size="10"><b>(p,q)</b></f><f size="10">.</f></p></text></region><region region-id="597" left="66" top="273" width="104.25" height="46.5" align-x="84" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><component hide-arguments="false" clsid-buddy="577d93c8-ea65-4d05-b631-401168b146e4" item-idref="4" disable-calc="false"><inputs /><outputs><ml:id xml:space="preserve">Cu</ml:id></outputs></component><rendering item-idref="5" /></region><region region-id="598" left="66" top="333" width="72.75" height="12.75" align-x="83.25" align-y="342" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id xml:space="preserve">Cu</ml:id><ml:apply><ml:id xml:space="preserve">csort</ml:id><ml:sequence><ml:id xml:space="preserve">Cu</ml:id><ml:real>1</ml:real></ml:sequence></ml:apply></ml:define></math><rendering item-idref="6" /></region><region region-id="599" left="156" top="332.25" width="151.5" height="12" align-x="156" align-y="342" 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">sort on the second column (1)</p></text></region><region region-id="545" left="36" top="355.5" width="60.75" height="12.75" align-x="36" align-y="366" 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">Cubic splines</p></text><indexes><phrase>cubic splines</phrase><phrase>splines, cubic</phrase></indexes></region><region region-id="590" left="36" top="380.25" width="306" height="72" align-x="36" align-y="390" 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">You can choose to have the second derivative of your spline at both endpoints equal to 0 (<f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">lspline</c></inlineAttr></b></f>), equal to that of the next nearest point (<f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">pspline</c></inlineAttr></b></f>), or determined by the straight line through the nearest two points (<f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">cspline</c></inlineAttr></b></f>), meaning that the piecewise polynomial that passes through the last point on each end of your data is either linear, parabolic, or cubic. </p></text></region><region region-id="645" left="66" top="471.75" width="45" height="18" align-x="81" align-y="486" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">vx</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">Cu</ml:id><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="7" /></region><region region-id="646" left="144" top="471.75" width="46.5" height="18" align-x="160.5" align-y="486" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">vy</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">Cu</ml:id><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="8" /></region><region region-id="642" left="66" top="501" width="106.5" height="12.75" align-x="105" align-y="510" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">paramsc</ml:id><ml:apply><ml:id xml:space="preserve">cspline</ml:id><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">vy</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="9" /></region><region region-id="643" left="66" top="525" width="102" height="12.75" align-x="102.75" align-y="534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">paramsl</ml:id><ml:apply><ml:id xml:space="preserve">lspline</ml:id><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">vy</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="10" /></region><region region-id="602" left="36" top="548.25" width="293.25" height="48" align-x="36" align-y="558" 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">These functions return a vector which contains the values of the second derivative of the desired spline at each point of <b>M</b><b>x</b>.<sp count="2" />Pass the output to the interp function to generate the interpolated values.</p></text></region><region region-id="603" left="66" top="603" width="138" height="12.75" align-x="96" align-y="612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">fitc</ml:id><ml:boundVars><ml:id 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align-y="696" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="14" /><rendering item-idref="15" /></region><region region-id="914" left="36" top="895.5" width="52.5" height="12.75" align-x="36" align-y="906" 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">Derivatives</p></text><indexes><phrase>derivatives</phrase></indexes></region><region region-id="915" left="36" top="914.25" width="300.75" height="48" align-x="36" 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">You may also wish to know the derivatives of the spline fit, to find maxima and minima, slope, or other features.<sp count="2" />Mathcad's numerical derivative operators work with the fitted curves, e.g.</p></text></region><region region-id="916" left="60" top="967.5" width="79.5" height="29.25" align-x="97.5" align-y="984" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">deriv</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:derivative /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars><ml:apply><ml:id xml:space="preserve">fitc</ml:id><ml:id 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z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:derivative /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">x1</ml:id></ml:boundVars><ml:apply><ml:id xml:space="preserve">fitc</ml:id><ml:id xml:space="preserve">x0</ml:id></ml:apply></ml:lambda><ml:degree><ml:real>2</ml:real></ml:degree></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0</ml:real></result></ml:eval></math><rendering item-idref="23" /></region><region region-id="944" left="36" top="1112.25" width="303.75" height="24" align-x="36" align-y="1122" 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">You can see an example of <b><u><link href="./integrate.xmcd" title="Links to section" popup="false">integrating using splines</link></u></b> in the Application section.</p></text></region><region region-id="807" left="36" top="1147.5" width="146.25" height="12.75" align-x="36" align-y="1158" 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">Interpolation in two dimensions</p></text><indexes><phrase>interpolation in two dimensions</phrase></indexes></region><region region-id="853" left="36" top="1172.25" width="310.5" height="36" align-x="36" align-y="1182" 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 you are dealing with a square matrix of values, that is, the same number of x points as y points in a grid, then you can use the cubic spline functions in two dimensions as well:</p></text></region><region region-id="839" left="66" top="1221" width="285" height="46.5" align-x="85.5" align-y="1230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><component hide-arguments="false" clsid-buddy="a6cfc4d6-3a88-437d-94de-30d1a591cf29" item-idref="24" disable-calc="false"><inputs /><outputs><ml:id xml:space="preserve">Mz</ml:id></outputs></component><rendering item-idref="25" /></region><region region-id="840" left="66" top="1275" width="58.5" height="12.75" align-x="77.25" align-y="1284" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" 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height="6" align-x="0" align-y="1320" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="864" left="66" top="1332.75" width="142.5" height="29.25" align-x="101.25" align-y="1350" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">fit</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">y</ml:id></ml:boundVars></ml:function><ml:apply><ml:id xml:space="preserve">interp</ml:id><ml:sequence><ml:id xml:space="preserve">S</ml:id><ml:id xml:space="preserve">Mxy</ml:id><ml:id xml:space="preserve">Mz</ml:id><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:sequence></ml:apply></ml:define></math><rendering item-idref="32" /></region><region 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/><ml:real>4</ml:real><ml:id xml:space="preserve">n</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:real>4</ml:real><ml:id xml:space="preserve">n</ml:id></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="34" /></region><region region-id="894" left="66" top="1398" width="142.5" height="164.25" align-x="136.5" align-y="1398" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="35" disable-calc="false"><inputs><ml:id xml:space="preserve">FIT</ml:id></inputs><outputs /></component><rendering item-idref="36" /></region><region region-id="895" left="234" top="1398" width="144" height="164.25" align-x="305.25" align-y="1398" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="37" disable-calc="false"><inputs><ml:id xml:space="preserve">Mz</ml:id></inputs><outputs /></component><rendering item-idref="38" /></region><region region-id="898" left="36" top="1579.5" width="210.75" height="12.75" align-x="36" align-y="1590" 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">2D Bicubic Interpolation on a non-square grid</p></text><indexes><phrase>2D bicubic interpolation on a non-square grid</phrase><phrase>interpolation, 2D bicubic on a non-square grid</phrase></indexes></region><region region-id="851" left="36" top="1604.25" width="300.75" height="84" align-x="42" align-y="1614" 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<f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000"> Bicubic2D </c></inlineAttr></b></f>function is equivalent to using <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000"><link href="./splines_1.xmcd" title="Links to section" popup="false">lspline</link></c></inlineAttr></b></f> in two dimensions, with no restrictions on the number of rows and columns of the input values (they need not be equal). To obtain a value for a point in a table that does not previously exist, we first interpolate down each column of the table, creating a new row; then we interpolate across the new row.</p></text></region><region region-id="809" left="114" top="1701.75" width="77.25" height="18" align-x="123.75" align-y="1716" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id xml:space="preserve">x</ml:id><ml:apply><ml:transpose /><ml:matrix rows="1" cols="3"><ml:real>0</ml:real><ml:real>200</ml:real><ml:real>300</ml:real></ml:matrix></ml:apply></ml:define></math><rendering item-idref="39" /></region><region region-id="913" left="66" top="1728" width="135" height="78.75" align-x="105" align-y="1770" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Altitude</ml:id><ml:matrix rows="5" cols="3"><ml:real>125</ml:real><ml:real>124.8</ml:real><ml:real>114.4</ml:real><ml:real>121.1</ml:real><ml:real>123.2</ml:real><ml:real>139.8</ml:real><ml:real>135.5</ml:real><ml:real>139.2</ml:real><ml:real>137.7</ml:real><ml:real>135</ml:real><ml:real>119.8</ml:real><ml:real>120</ml:real><ml:real>122.3</ml:real><ml:real>121.1</ml:real><ml:real>123.3</ml:real></ml:matrix></ml:define></math><rendering item-idref="40" /></region><region region-id="811" left="222" top="1728" width="46.5" height="78.75" align-x="233.25" align-y="1770" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">y</ml:id><ml:matrix rows="5" cols="1"><ml:real>0</ml:real><ml:real>10</ml:real><ml:real>50</ml:real><ml:real>120</ml:real><ml:real>190</ml:real></ml:matrix></ml:define></math><rendering item-idref="41" /></region><region region-id="816" left="36" top="1814.25" width="276.75" height="48" align-x="36" align-y="1824" 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 x and y values must be in sorted order. If we do this for enough points, we can produce a smooth interpolation of the sketchy information in the table, and plot a representative map.</p></text></region><region region-id="945" left="66" top="1869" width="36.75" height="12.75" align-x="74.25" align-y="1878" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id xml:space="preserve">i</ml:id><ml:range><ml:real>0</ml:real><ml:real>19</ml:real></ml:range></ml:define></math><rendering item-idref="42" /></region><region region-id="946" left="132" top="1869" width="37.5" height="12.75" align-x="141" align-y="1878" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id 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xml:space="preserve">q</ml:id><ml:id xml:space="preserve">j</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:real>10</ml:real><ml:id xml:space="preserve">j</ml:id></ml:apply></ml:define></math><rendering item-idref="45" /></region><region region-id="943" left="0" top="1896" width="6000" height="6" align-x="0" align-y="1896" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="949" left="66" top="1905" width="177" height="18" align-x="111" align-y="1914" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id xml:space="preserve">Zinterp</ml:id><ml:sequence><ml:id xml:space="preserve">i</ml:id><ml:id xml:space="preserve">j</ml:id></ml:sequence></ml:apply><ml:apply><ml:id xml:space="preserve">Bicubic2D</ml:id><ml:sequence><ml:id xml:space="preserve">y</ml:id><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">Altitude</ml:id><ml:apply><ml:indexer /><ml:id xml:space="preserve">p</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id xml:space="preserve">q</ml:id><ml:id xml:space="preserve">j</ml:id></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="46" /></region><region region-id="837" left="36" top="1938" width="140.25" height="163.5" align-x="105" align-y="1938" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="47" disable-calc="false"><inputs><ml:id xml:space="preserve">Altitude</ml:id></inputs><outputs /></component><rendering item-idref="48" /></region><region region-id="838" left="198" top="1938" width="151.5" height="165" align-x="273" align-y="1938" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="49" disable-calc="false"><inputs><ml:id xml:space="preserve">Zinterp</ml:id></inputs><outputs /></component><rendering item-idref="50" /></region><region region-id="900" left="36" top="2120.25" width="302.25" height="60" align-x="36" align-y="2130" 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<f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000"> Bicubic2D</c></inlineAttr></b></f> function is based on SPLIN2 (p.128) from the book "Numerical Recipes in C, The Art of Scientific Computing" (Cambridge 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