1. 1 Dr. Scott Schaefer Geometric Modeling CSCE 645/VIZA 675

2. 2/55 Course Information Instructor  Dr. Scott Schaefer  HRBB 527B  Office Hours: MW 9:00am – 10:00am (or by appointment) Website: http://courses.cs.tamu.edu/schaefer/645_Spring2013

3. 3/55 Geometric Modeling Surface representations  Industrial design

4. 4/55 Geometric Modeling Surface representations  Industrial design  Movies and animation

5. 5/55 Geometric Modeling Surface representations  Industrial design  Movies and animation Surface reconstruction/Visualization

6. 6/55 Topics Covered Polynomial curves and surfaces  Lagrange interpolation  Bezier/B-spline/Catmull-Rom curves  Tensor Product Surfaces  Triangular Patches  Coons/Gregory Patches Differential Geometry Subdivision curves and surfaces Boundary representations Surface Simplification Solid Modeling Free-Form Deformations Barycentric Coordinates

7. 7/55 What you’re expected to know Programming Experience  Assignments in C/C++ Simple Mathematics Graphics is mathematics made visible

8. 8/55 How much math? General geometry/linear algebra Matrices  Multiplication, inversion, determinant, eigenvalues/vectors Vectors  Dot product, cross product, linear independence Proofs  Induction

9. 9/55 Required Textbook

10. 10/55 Grading 50% Homework 50% Class Project No exams!

11. 11/55 Class Project Topic: your choice  Integrate with research  Originality Reports  Proposal: 2/7  Update #1: 3/7  Update #2: 4/9  Final report/presentation: 4/25

12. 12/55 Class Project Grading 10% Originality 20% Reports (5% each) 5% Final Oral Presentation 65% Quality of Work http://courses.cs.tamu.edu/schaefer/645_Spring2013/assignments/project.html

13. Honor Code Your work is your own You may discuss concepts with others Do not look at other code.  You may use libraries not related to the main part of the assignment, but clear it with me first just to be safe. 13/55

14. 14/55 Questions?

15. 15/55 Vectors

16. 16/55 Vectors

17. 17/55 Vectors

18. 18/55 Vectors

19. 19/55 Vectors

20. 20/55 Vectors

21. 21/55 Vectors

22. 22/55 Points

23. 23/55 Points

24. 24/55 Points

25. 25/55 Points

26. 26/55 Points 1 p=p 0 p=0 (vector) c p=undefined where c 0,1 p – q = v (vector)

27. 27/55 Points

28. 28/55 Points

29. 29/55 Points

30. 30/55 Points

31. 31/55 Points

32. 32/55 Points

33. 33/55 Points

34. 34/55 Points

35. 35/55 Barycentric Coordinates

36. 36/55 Barycentric Coordinates

37. 37/55 Barycentric Coordinates

38. 38/55 Barycentric Coordinates

39. 39/55 Barycentric Coordinates

40. 40/55 Barycentric Coordinates

41. 41/55 Barycentric Coordinates

42. 42/55 Convex Sets If, then the form a convex combination

43. 43/55 Convex Hulls Smallest convex set containing all the

44. 44/55 Convex Hulls Smallest convex set containing all the

45. 45/55 Convex Hulls If p i and p j lie within the convex hull, then the line p i p j is also contained within the convex hull

46. 46/55 Convex Hulls If p i and p j lie within the convex hull, then the line p i p j is also contained within the convex hull

47. 47/55 Convex Hulls If p i and p j lie within the convex hull, then the line p i p j is also contained within the convex hull

48. 48/55 Convex Hulls If p i and p j lie within the convex hull, then the line p i p j is also contained within the convex hull

49. 49/55 Convex Hulls If p i and p j lie within the convex hull, then the line p i p j is also contained within the convex hull

50. 50/55 Convex Hulls If p i and p j lie within the convex hull, then the line p i p j is also contained within the convex hull

51. 51/55 Convex Hulls If p i and p j lie within the convex hull, then the line p i p j is also contained within the convex hull

52. 52/55 Convex Hulls If p i and p j lie within the convex hull, then the line p i p j is also contained within the convex hull

53. 53/55 Convex Hulls If p i and p j lie within the convex hull, then the line p i p j is also contained within the convex hull

54. 54/55 Affine Transformations Preserve barycentric combinations Examples: translation, rotation, uniform scaling, non-uniform scaling, shear

55. 55/55 Other Transformations Conformal  Preserve angles under transformation  Examples: translation, rotation, uniform scaling Rigid  Preserve angles and length under transformation  Examples: translation, rotation

56. 56/55 Vector Spaces A set of vectors v k are independent if The span of a set of vectors v k is A basis of a vector space is a set of independent vectors v k such that