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            <c val="#008080">VISUALIZATION AND SYNTHETIC IMAGES</c>
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              <c val="#800080">Ray Tracing and Mapping</c>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">It is possible to generate images by combining a "natural" image with a mathematically generated image, a technique frequently employed in digital art and advertising. Here we demonstrate the mathematics of the process. As an illustration, the Mona Lisa is reflected off the surface of a sphere. First, a square subimage of the original is loaded.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">For convenience, image coordinates are stored<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"> as complex numbers, and transformations are carried out by applying complex functions. </inlineAttr></f></p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Here the number of pixels in the synthetic image is set equal to the "natural" source image, though there's no reason these numbers need to be the same. The number of pixels is</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The pixel coordinates are stored in the array <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">I</inlineAttr></f>:</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The <b><u><link href="./Raytrace.xmcd" title="Links to section" popup="false">ray tracing model</link></u></b> has the following geometry: the viewer's eye is 2 units from the center of a sphere of radius 1. The image plane, .5 units on a side, lies halfway between the viewpoint and the surface of the sphere, 1.5 units from the center of the sphere. One unit <f family="Times New Roman" charset="0"><i><inlineAttr font-weight="normal" line-through="false">behind</inlineAttr></i></f> the viewpoint is a plane which contains the image of the Mona Lisa. The model will reflect this image off the sphere onto the image plane. </p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">If a ray misses the picture in this calculation, the pixel is set to black, otherwise, it follows the calculation. The reflected pixels are combined with the pixels that define the surface of the sphere.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The following two functions do the ray tracing:</p>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">d</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">root</ml:id>
            <ml:sequence>
              <ml:apply>
                <ml:minus />
                <ml:apply>
                  <ml:plus />
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:real>2</ml:real>
                    <ml:apply>
                      <ml:id xml:space="preserve">θ</ml:id>
                      <ml:id xml:space="preserve">u</ml:id>
                    </ml:apply>
                  </ml:apply>
                  <ml:apply>
                    <ml:id xml:space="preserve">α</ml:id>
                    <ml:id xml:space="preserve">u</ml:id>
                  </ml:apply>
                </ml:apply>
                <ml:apply>
                  <ml:div />
                  <ml:id xml:space="preserve">π</ml:id>
                  <ml:real>2</ml:real>
                </ml:apply>
              </ml:apply>
              <ml:id xml:space="preserve">u</ml:id>
            </ml:sequence>
          </ml:apply>
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      </math>
      <rendering item-idref="17" />
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    <region region-id="464" left="0" top="1038" width="6000" height="6" align-x="0" align-y="1038" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
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    <region region-id="465" left="6" top="1056.75" width="323.25" height="71.25" align-x="6" align-y="1068" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The sphere is given a realistic shape by making the brightness fall off from a point on the upper left. The function <b>sphere</b> takes care of this: The constan<inlineAttr font-weight="normal" line-through="false">t </inlineAttr><f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false">-</inlineAttr></f><inlineAttr font-weight="normal" line-through="false">0.1i + 0.1</inlineAttr><b><sp /></b>determines the lighting direction, and the random term dithers the gray levels. (If you delete this term, you'll see distinct rings marking the transitions between levels.)</p>
      </text>
      <indexes>
        <phrase>sphere function</phrase>
      </indexes>
    </region>
    <region region-id="466" left="36" top="1138.5" width="225.75" height="39" align-x="88.5" align-y="1158" 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">sphere</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">z</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:apply>
            <ml:plus />
            <ml:apply>
              <ml:div />
              <ml:real>235</ml:real>
              <ml:apply>
                <ml:plus />
                <ml:real>1</ml:real>
                <ml:apply>
                  <ml:pow />
                  <ml:parens>
                    <ml:apply>
                      <ml:absval />
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:parens>
                          <ml:apply>
                            <ml:plus />
                            <ml:apply>
                              <ml:minus />
                              <ml:id xml:space="preserve">z</ml:id>
                              <ml:imag symbol="i">.1</ml:imag>
                            </ml:apply>
                            <ml:real>.1</ml:real>
                          </ml:apply>
                        </ml:parens>
                        <ml:real>6</ml:real>
                      </ml:apply>
                    </ml:apply>
                  </ml:parens>
                  <ml:real>2</ml:real>
                </ml:apply>
              </ml:apply>
            </ml:apply>
            <ml:apply>
              <ml:id xml:space="preserve">rnd</ml:id>
              <ml:real>6</ml:real>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="18" />
    </region>
    <region region-id="468" left="6" top="1188.75" width="305.25" height="28.5" align-x="6" align-y="1200" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">A mask function applies <b>sphere</b> to just those pixels which belong to the image of the sphere.</p>
      </text>
    </region>
    <region region-id="469" left="36" top="1222.5" width="69" height="24" align-x="58.5" align-y="1242" 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">MI</ml:id>
          <ml:apply>
            <ml:vectorize />
            <ml:apply>
              <ml:lessThan />
              <ml:apply>
                <ml:absval />
                <ml:id xml:space="preserve">I</ml:id>
              </ml:apply>
              <ml:id xml:space="preserve">d</ml:id>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="19" />
    </region>
    <region region-id="470" left="132" top="1222.5" width="89.25" height="24" align-x="149.25" align-y="1242" 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="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">K</ml:id>
          <ml:apply>
            <ml:vectorize />
            <ml:apply>
              <ml:mult style="default" />
              <ml:id xml:space="preserve">MI</ml:id>
              <ml:apply>
                <ml:id xml:space="preserve">sphere</ml:id>
                <ml:id xml:space="preserve">I</ml:id>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="20" />
    </region>
    <region region-id="471" left="6" top="1254.75" width="316.5" height="28.5" align-x="6" align-y="1266" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">For viewing, a gray background may be used instead of black. Here's the completed sphere:</p>
      </text>
    </region>
    <region region-id="475" left="36" top="1290.75" width="137.25" height="15.75" align-x="76.5" align-y="1302" 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">Sphere</ml:id>
          <ml:apply>
            <ml:plus />
            <ml:id xml:space="preserve">K</ml:id>
            <ml:apply>
              <ml:mult style="default" />
              <ml:real>150</ml:real>
              <ml:parens>
                <ml:apply>
                  <ml:minus />
                  <ml:real>1</ml:real>
                  <ml:id xml:space="preserve">MI</ml:id>
                </ml:apply>
              </ml:parens>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="21" />
    </region>
    <region region-id="534" left="36" top="1314" width="148.5" height="165" align-x="109.5" align-y="1314" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="22" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">Sphere</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="23" />
    </region>
    <region region-id="477" left="6" top="1488.75" width="321" height="57" align-x="6" align-y="1500" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The reflected image is constructed using the function <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">P, which gives the intersection point for a light ray and z with the plane containing the Mona Lisa picture. The real part of the value of P is the horizontal coordinate and the imaginary pa</inlineAttr></f>rt is the vertical coordinate.</p>
      </text>
    </region>
    <region region-id="478" left="6" top="1552.5" width="328.5" height="33.75" align-x="63" align-y="1572" 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">P</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">z</ml:id>
              <ml:id xml:space="preserve">theta</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:apply>
            <ml:mult style="default" />
            <ml:parens>
              <ml:apply>
                <ml:plus />
                <ml:apply>
                  <ml:id xml:space="preserve">sin</ml:id>
                  <ml:id xml:space="preserve">theta</ml:id>
                </ml:apply>
                <ml:apply>
                  <ml:mult style="default" />
                  <ml:parens>
                    <ml:apply>
                      <ml:minus />
                      <ml:real>3</ml:real>
                      <ml:apply>
                        <ml:id xml:space="preserve">cos</ml:id>
                        <ml:id xml:space="preserve">theta</ml:id>
                      </ml:apply>
                    </ml:apply>
                  </ml:parens>
                  <ml:apply>
                    <ml:id xml:space="preserve">tan</ml:id>
                    <ml:apply>
                      <ml:plus />
                      <ml:apply>
                        <ml:id xml:space="preserve">α</ml:id>
                        <ml:id xml:space="preserve">z</ml:id>
                      </ml:apply>
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:real>2</ml:real>
                        <ml:id xml:space="preserve">theta</ml:id>
                      </ml:apply>
                    </ml:apply>
                  </ml:apply>
                </ml:apply>
              </ml:apply>
            </ml:parens>
            <ml:apply>
              <ml:div />
              <ml:id xml:space="preserve">z</ml:id>
              <ml:apply>
                <ml:absval />
                <ml:id xml:space="preserve">z</ml:id>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="24" />
    </region>
    <region region-id="487" 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="479" left="6" top="1608.75" width="309" height="85.5" align-x="6" align-y="1620" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
          <f family="Times New Roman" charset="0">
            <inlineAttr font-weight="normal" line-through="false">P is applied to every pixel, generating a matrix R</inlineAttr>
          </f>
          <b>
            <sp />
          </b>of corresponding locations on the plane behind the view point. For pixels outside the sphere, the results are nonsense, but these will be masked. Ray-tracing calculations are typically very time-consuming, and even this simple example takes quite a while, since we're following <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">32,400</inlineAttr></f> rays. </p>
      </text>
    </region>
    <region region-id="480" left="36" top="1701.75" width="74.25" height="24.75" align-x="51.75" align-y="1722" 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="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">R</ml:id>
          <ml:apply>
            <ml:vectorize />
            <ml:apply>
              <ml:id xml:space="preserve">P</ml:id>
              <ml:sequence>
                <ml:id xml:space="preserve">I</ml:id>
                <ml:apply>
                  <ml:id xml:space="preserve">θ</ml:id>
                  <ml:id xml:space="preserve">I</ml:id>
                </ml:apply>
              </ml:sequence>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="25" />
    </region>
    <region region-id="481" left="6" top="1734.75" width="328.5" height="71.25" align-x="6" align-y="1746" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The size and location on the plane of the Mona Lisa picture are specified by the constants <b>size</b>, <b>up,</b> and <b>right</b>. Increasing <b>size</b> corresponds to making the painting larger, and positive values for <b>up</b> and <b>right</b> move it in those directions. You can experiment by changing these values after following through with the original calculation.</p>
      </text>
      <indexes>
        <phrase>right constant, ray tracing</phrase>
        <phrase>size constant, ray tracing</phrase>
        <phrase>up constant, ray tracing</phrase>
      </indexes>
    </region>
    <region region-id="482" left="36" top="1812.75" width="41.25" height="15.75" align-x="60.75" align-y="1824" 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">size</ml:id>
          <ml:real>1</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="26" />
    </region>
    <region region-id="483" left="96" top="1812.75" width="42" height="15.75" align-x="115.5" align-y="1824" 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">up</ml:id>
          <ml:real>20</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="27" />
    </region>
    <region region-id="484" left="156" top="1812.75" width="50.25" height="15.75" align-x="183.75" align-y="1824" 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">right</ml:id>
          <ml:real>50</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="28" />
    </region>
    <region region-id="485" left="36" top="1840.5" width="180.75" height="33.75" align-x="51.75" align-y="1860" 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">R</ml:id>
          <ml:apply>
            <ml:plus />
            <ml:apply>
              <ml:plus />
              <ml:apply>
                <ml:plus />
                <ml:apply>
                  <ml:mult style="default" />
                  <ml:apply>
                    <ml:div />
                    <ml:real>20</ml:real>
                    <ml:id xml:space="preserve">size</ml:id>
                  </ml:apply>
                  <ml:id xml:space="preserve">R</ml:id>
                </ml:apply>
                <ml:real>90</ml:real>
              </ml:apply>
              <ml:id xml:space="preserve">up</ml:id>
            </ml:apply>
            <ml:apply>
              <ml:mult style="default" />
              <ml:parens>
                <ml:apply>
                  <ml:plus />
                  <ml:real>90</ml:real>
                  <ml:id xml:space="preserve">right</ml:id>
                </ml:apply>
              </ml:parens>
              <ml:imag symbol="i">1</ml:imag>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="29" />
    </region>
    <region region-id="428" left="6" top="1884.75" width="317.25" height="57" align-x="6" align-y="1896" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The intersection of each ray with the plane containing the picture is determined. If the ray falls within the picture, the image is given the value of the closest picture coordinate, otherwise the image value is set to 0. This step also takes a while to compute.</p>
      </text>
    </region>
    <region region-id="429" left="36" top="1950.75" width="102" height="15.75" align-x="69" align-y="1962" 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">rd</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">x</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:apply>
            <ml:id xml:space="preserve">floor</ml:id>
            <ml:apply>
              <ml:plus />
              <ml:id xml:space="preserve">x</ml:id>
              <ml:real>.5</ml:real>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="30" />
    </region>
    <region region-id="430" left="36" top="1974.75" width="325.5" height="15.75" align-x="65.25" align-y="1986" 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="WarnRedefinedBIUnit">
          <ml:function>
            <ml:id xml:space="preserve">T</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">z</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:apply>
            <ml:mult style="default" />
            <ml:apply>
              <ml:mult style="default" />
              <ml:apply>
                <ml:mult style="default" />
                <ml:parens>
                  <ml:apply>
                    <ml:greaterOrEqual />
                    <ml:apply>
                      <ml:id xml:space="preserve">Re</ml:id>
                      <ml:id xml:space="preserve">z</ml:id>
                    </ml:apply>
                    <ml:real>0</ml:real>
                  </ml:apply>
                </ml:parens>
                <ml:parens>
                  <ml:apply>
                    <ml:lessOrEqual />
                    <ml:apply>
                      <ml:id xml:space="preserve">Re</ml:id>
                      <ml:id xml:space="preserve">z</ml:id>
                    </ml:apply>
                    <ml:apply>
                      <ml:minus />
                      <ml:id xml:space="preserve">D</ml:id>
                      <ml:real>1</ml:real>
                    </ml:apply>
                  </ml:apply>
                </ml:parens>
              </ml:apply>
              <ml:parens>
                <ml:apply>
                  <ml:greaterOrEqual />
                  <ml:apply>
                    <ml:id xml:space="preserve">Im</ml:id>
                    <ml:id xml:space="preserve">z</ml:id>
                  </ml:apply>
                  <ml:real>0</ml:real>
                </ml:apply>
              </ml:parens>
            </ml:apply>
            <ml:parens>
              <ml:apply>
                <ml:lessOrEqual />
                <ml:apply>
                  <ml:id xml:space="preserve">Im</ml:id>
                  <ml:id xml:space="preserve">z</ml:id>
                </ml:apply>
                <ml:apply>
                  <ml:minus />
                  <ml:id xml:space="preserve">D</ml:id>
                  <ml:real>1</ml:real>
                </ml:apply>
              </ml:apply>
            </ml:parens>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="31" />
    </region>
    <region region-id="431" left="36" top="1998.75" width="259.5" height="22.5" align-x="70.5" align-y="2010" 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="WarnRedefinedBIUnit">
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">F</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">m</ml:id>
              <ml:id xml:space="preserve">n</ml:id>
            </ml:sequence>
          </ml:apply>
          <ml:apply>
            <ml:id xml:space="preserve">if</ml:id>
            <ml:sequence>
              <ml:apply>
                <ml:id xml:space="preserve">T</ml:id>
                <ml:apply>
                  <ml:indexer />
                  <ml:id xml:space="preserve">R</ml:id>
                  <ml:sequence>
                    <ml:id xml:space="preserve">n</ml:id>
                    <ml:id xml:space="preserve">m</ml:id>
                  </ml:sequence>
                </ml:apply>
              </ml:apply>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">S</ml:id>
                <ml:sequence>
                  <ml:apply>
                    <ml:id xml:space="preserve">rd</ml:id>
                    <ml:apply>
                      <ml:id xml:space="preserve">Re</ml:id>
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">R</ml:id>
                        <ml:sequence>
                          <ml:id xml:space="preserve">n</ml:id>
                          <ml:id xml:space="preserve">m</ml:id>
                        </ml:sequence>
                      </ml:apply>
                    </ml:apply>
                  </ml:apply>
                  <ml:apply>
                    <ml:id xml:space="preserve">rd</ml:id>
                    <ml:apply>
                      <ml:id xml:space="preserve">Im</ml:id>
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">R</ml:id>
                        <ml:sequence>
                          <ml:id xml:space="preserve">n</ml:id>
                          <ml:id xml:space="preserve">m</ml:id>
                        </ml:sequence>
                      </ml:apply>
                    </ml:apply>
                  </ml:apply>
                </ml:sequence>
              </ml:apply>
              <ml:real>0</ml:real>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">These image values are combined with the values defining the sphere. The values can simply be <b>OR</b>ed together, while setting the background to an intermediate gray as was done before. Here's the result.</p>
      </text>
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        <ml:define>
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                    <ml:id xml:space="preserve">F</ml:id>
                    <ml:id xml:space="preserve">MI</ml:id>
                  </ml:apply>
                </ml:apply>
                <ml:id xml:space="preserve">K</ml:id>
              </ml:sequence>
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            <ml:apply>
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              <ml:real>150</ml:real>
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                </ml:apply>
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      <pageBreak />
    </region>
    <region region-id="536" left="24" top="2136" width="149.25" height="165" align-x="97.5" align-y="2136" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="34" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">View1</ml:id>
        </inputs>
        <outputs />
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      <rendering item-idref="35" />
    </region>
    <region region-id="499" left="6" top="2316.75" width="330.75" height="28.5" align-x="6" align-y="2328" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">An alternative approach is to boost the reflected values where the sphere values are high, using a function like<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"> g:</inlineAttr></f></p>
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    </region>
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        <ml:define warning="WarnRedefinedBIUnit">
          <ml:function>
            <ml:id xml:space="preserve">g</ml:id>
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              <ml:id xml:space="preserve">a</ml:id>
              <ml:id xml:space="preserve">b</ml:id>
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              </ml:parens>
              <ml:apply>
                <ml:pow />
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                    <ml:apply>
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                      <ml:id xml:space="preserve">a</ml:id>
                    </ml:apply>
                    <ml:real>255</ml:real>
                  </ml:apply>
                </ml:parens>
                <ml:apply>
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                  <ml:real>2</ml:real>
                  <ml:real>3</ml:real>
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            <ml:id xml:space="preserve">if</ml:id>
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                    <ml:id xml:space="preserve">m</ml:id>
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                </ml:apply>
              </ml:apply>
              <ml:apply>
                <ml:id xml:space="preserve">g</ml:id>
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                  <ml:apply>
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                    <ml:id xml:space="preserve">S</ml:id>
                    <ml:sequence>
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                              <ml:id xml:space="preserve">m</ml:id>
                            </ml:sequence>
                          </ml:apply>
                        </ml:apply>
                      </ml:apply>
                      <ml:apply>
                        <ml:id xml:space="preserve">rd</ml:id>
                        <ml:apply>
                          <ml:id xml:space="preserve">Im</ml:id>
                          <ml:apply>
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                </ml:sequence>
              </ml:apply>
              <ml:apply>
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                </ml:sequence>
              </ml:apply>
            </ml:sequence>
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      </math>
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    </region>
    <region region-id="508" left="36" top="2446.5" width="152.25" height="24" align-x="74.25" align-y="2466" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
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            </ml:apply>
            <ml:apply>
              <ml:mult style="default" />
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              </ml:parens>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="38" />
    </region>
    <region region-id="538" left="42" top="2484" width="147.75" height="165" align-x="114.75" align-y="2484" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="39" disable-calc="false">
        <inputs>
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        </inputs>
        <outputs />
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      <rendering item-idref="40" />
    </region>
    <region region-id="513" left="0" top="2670" width="6000" height="6" align-x="0" align-y="2670" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="511" left="6" top="2682.75" width="321" height="85.5" align-x="6" align-y="2694" 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="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Neither of these approaches is particularly realistic - this sphere has a matte surface, whereas a reflected image assumes a mirror-like surface, so the combination looks a bit strange. But the 3D illusion is still fairly strong and could be embellished by better specifying the quality and quantity of light reflected from the sphere, as described in the<inlineAttr font-weight="normal" underline="false"> section on </inlineAttr><b><u><link href="./Raytrace.xmcd" title="Links to section" popup="false">ray tracing</link></u></b>.</p>
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        <phrase>3-D illusion</phrase>
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