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<HTML><HEAD><TITLE>CIS 510 Handout 1</TITLE><BODY><! BODY BGCOLOR = "#000000" TEXT = "#FFFFFF"><BODY bgcolor="#FFEFDB"> <! AntiqueWhite1><H1><CENTER>CIS 510, Spring 96 </CENTER> <P><center>COMPUTER AIDED GEOMETRIC DESIGN</CENTER><CENTER>Course Information</CENTER><CENTER>January 9</CENTER></H1><P><H4><H2>Coordinates:</H2> Moore 224, MW 12-1:30<P><H2>Instructor:</H2> <!WA0><A HREF="mailto:jean@saul.cis.upenn.edu">Jean H. Gallier</A>, MRE 176, 8-4405, jean@saul <P><H2>Office Hours:</H2> 2:00-3:00 Tuesday, Thursday, 2:00-3:00, Friday<P><H2>Teaching Assistant:</H2>TBA <p><H2>Office Hours:</H2> TBA<P><h2> Prerequesites:</h2>Basic knowledge of linear algebra, calculus,and elementary geometry <br>(CIS560 NOT required).<H2>Textbooks (not required):</H2> <I>Computer Aided Geometric Design </I>Hoschek, J. and Lasser, D., AK Peters, 1993<br><BR>Also recommended:<BR><BR><I>Curves and Surfaces for Computer Aided Geometric Design</I>,D. Wood, Wiley<br><P><H2>Grades:</H2><P> Problem Sets (3 or 4 of them) and a project<ul><li><!WA1><a href="http://www.cis.upenn.edu/~jean/geom96hm1.dvi.Z"> Homework1 </a><li><!WA2><a href="http://www.cis.upenn.edu/~jean/geom96hm2.dvi.Z"> Homework2 </a><li><!WA3><a href="http://www.cis.upenn.edu/~jean/geom96hm3.dvi.Z"> Homework3 </a></ul><P><H2>Brief description:</H2>A more appropriate name for the course would be <h2><center>Curves and Surfaces for Computer Aided Geometric Design</center></h2>or perhaps<h2><center>Mathematical Foundations of Computer Graphics</center></h2>(as CS348a is called at Stanford University).The course should be of interest to anyone who likes geometry (with an algebraic twist)!<p>Basically, the course will be about mathematical techniques used for geometric design in computer graphics (but also in robotics, vision, and computational geometry).Such techniques are used in 2D and 3D drawing and plot, object silhouettes,animating positions, product design (cars, planes, buildings),topographic data, medical imagery, active surfaces of proteins,attribute maps (color, texture, roughness), weather data, art(!), ... .Three broad classes of problems will be considered: <p><ul><li> <em> Approximating</em> curved shapes, using smooth curves or surfaces.<li> <em> Interpolating</em> curved shapes, using smooth curves or surfaces.<li> <em> Rendering</em> smooth curves or surfaces. <p></ul> <p> Specific topics include: basic geometric materialon affine spaces and affine maps.Be'zier curves will be introduced ``gently'', in terms ofmultiaffine symmetric polar forms, also known as ``blossoms''.We will begin with degree 2, move up to degree 3,giving lots of examples, and derive the fundamental``de Casteljau algorithm'', and show where the Bernsteinpolynomials come from.Then, we will consider polynomial curvesof arbitrary degree. It will be shown how a constructionembedding an affine space into a vector space, where pointsand vectors can be treated uniformly, together with polar forms, yield a very elegant and effective treatment of tangents andosculating flats. The conditions for joining polynomial curveswill be derived using polar forms, and this will lead toa treatment of B-splines in terms of polar forms. In particular,the de Boor algorithm will be derived as a natural extension ofthe de Casteljau algorithm. Rectangular (tensor product) Be'zier surfaces, and triangularBe'zier surfaces will also be introduced using polar forms,and the de Casteljau algorithm will be derived.Subdivision algorithms and their application torendering will be discussed extensively.Joining conditions will be derivedusing polar forms. <p>Using the embeddingof an affine space into a vector space, we will contruct theprojective completion of an affine space, and show howrational curves can be dealt with as central projectionsof polynomial curves, with appropriate generalizationsof the de Casteljau algorithm. <p>Rational surfaces will be obtained as central projectionsof polynomial surfaces. If time permits, NURBS andgeometric continuity will be discussed.This will require a little bit of differential geometry. <p>A class-room treatment of curves and surfaces in terms of polar formsis rather new (althoughused at Stanford by Leo Guibas and Lyle Ramshaw), but should be illuminating and exciting.Since books (even recent) do not follow such an approach,I have written extensive course notes, which will be available. <p>I will mix assignments not involving programming, andsmall programming projects.There are plenty of opportunities for trying outthe algorithms presented in the course. In particular,it is fairly easy to program many of these algorithms in Mathematica(I have done so, and I'm not such a great programmer!). <p>At the end of the course, you will know how to write your <em> own </em>algorithms to display the half Klein bottle shown below.<p><!WA4><a href="http://www.cis.upenn.edu/~jean/klein5.ps.Z"> Half Klein bottle</a><p> <P></UL><P><I>published by:<H2><!WA5><A HREF="mailto:jean@saul.cis.upenn.edu">Jean Gallier</A></H2></H4><BODY><HTML>⌨️ 快捷键说明
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