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Date: Mon, 16 Dec 1996 23:40:16 GMTServer: NCSA/1.5Content-type: text/htmlLast-modified: Mon, 26 Jun 1995 15:24:13 GMTContent-length: 1689<html><head><title>Normals</title></head><body><h2>Computing the normal to a surface</h2><hr><b>There are several ways of computing a normal to a surface. </b><hr>If the surfaceis defined as a list of triangles then one way of computing the normalis to compute the normalized cross-product of two edges of the triangle.Three vertices of a triangle are shown below as <b>P1, P2</b> and <b>P3</b>.<p><!WA0><!WA0><!WA0><!WA0><img src="http://www.tc.cornell.edu/Visualization/Education/cs417/SECTIONS/normals.diagram.triangle.gif"><p>If we denote <br><!WA1><!WA1><!WA1><!WA1><img src="http://www.tc.cornell.edu/Visualization/Education/cs417/SECTIONS/normals.eqn.diff.gif"><br> then the normal is given by<p><!WA2><!WA2><!WA2><!WA2><img src="http://www.tc.cornell.edu/Visualization/Education/cs417/SECTIONS/normals.eqn.triangle.gif"><p><hr>If the surface is defined as a list of general polygons then the Newellmethod is a good way to calculate the normal. It produces an "average"normal if the polygon is not quite planar. It also is not confused byco-linear vertices.Calculate the following, where m is the number of vertices in thepolygon.The normal is then <i>norm([nx, ny, nz])</i>.<p><!WA3><!WA3><!WA3><!WA3><img src="http://www.tc.cornell.edu/Visualization/Education/cs417/SECTIONS/normals.eqn.newell.gif"><p><hr>If the surface is defined as a parametric equation then differential methods may be used to derive the normal.If the equation for a parametric surface is <br><!WA4><!WA4><!WA4><!WA4><img src="http://www.tc.cornell.edu/Visualization/Education/cs417/SECTIONS/normals.eqn.parametric.gif"><br>where <i> 0&lt;u&lt;1 and 0&lt;v&lt;1</i> are two parameters, then the surface normalis given by <br><!WA5><!WA5><!WA5><!WA5><img src="http://www.tc.cornell.edu/Visualization/Education/cs417/SECTIONS/normals.eqn.parametric.normal.gif"><p><hr>Comments about Theory Center online documents are welcome and may be sent to<i>doc-comments@tc.cornell.edu</i>. <p>Last modified, 6/26/95 B. Land. <! Revision history:	Original document: P.Maxfield, 10/94><p><!WA6><!WA6><!WA6><!WA6><iMG SRC="http://www.tc.cornell.edu/copyright.xbm"><!WA7><!WA7><!WA7><!WA7><A HREF="http://www.tc.cornell.edu/ctcCopyright.html"> <i>Copyright Statement </I></A></body> </html>

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