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lecnotes20_fixed - 13.472J/1.128J/2.158J/16.940J...

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Unformatted text preview: 13.472J/1.128J/2.158J/16.940J COMPUTATIONAL GEOMETRY Lecture 20 Dr. K. H. Ko Prof. N. M. Patrikalakis Copyright c 2003 Massachusetts Institute of Technology Contents 20 Advanced topics in differential geometry 2 20.1 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 20.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 20.1.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 20.1.3 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 20.1.4 Two-point boundary value problem . . . . . . . . . . . . . . . . . . . . . . . . . 5 20.1.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 20.2 Developable surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 20.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 20.2.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 20.2.3 Developable surface in terms of B ezier surface . . . . . . . . . . . . . . . . . . . 12 20.2.4 Development of developable surface (attening) . . . . . . . . . . . . . . . . . . 13 20.3 Umbilics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 20.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 20.3.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 20.3.3 Computation of umbilical points . . . . . . . . . . . . . . . . . . . . . . . . . . 15 20.3.4 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 20.3.5 Characteristic lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 20.4 Parabolic, ridge and sub-parabolic points . . . . . . . . . . . . . . . . . . . . . . . . . 21 20.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 20.4.2 Focal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 20.4.3 Parabolic points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 20.4.4 Ridge points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 20.4.5 Sub-parabolic points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Bibliography 25 Reading in the Textbook Geodesics : Chapter 10, pp.265- 291 Umbilics : Chapter 9, pp.231- 264 1 Lecture 20 Advanced topics in differential geometry 20.1 Geodesics In this section we study the computation of shortest path between two points on free-form surfaces [14, 11]....
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