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

2. 2/67 Course Information Instructor  Dr. Scott Schaefer  HRBB 527B  Office Hours: TR 10:00am – 11:00am (or by appointment) Website: http://courses.cs.tamu.edu/schaefer/645_Fall2015

3. 3/67 Geometric Modeling Surface representations  Industrial design

4. 4/67 Geometric Modeling Surface representations  Industrial design  Movies and animation

5. 5/67 Geometric Modeling Surface representations  Industrial design  Movies and animation Surface reconstruction/Visualization

6. 6/67 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 Surface Parameterization

7. 7/67 What you’re expected to know Programming Experience  Assignments in C/C++ Simple Mathematics Graphics is mathematics made visible

8. 8/67 How much math? General geometry/linear algebra Matrices  Multiplication, inversion, determinant, eigenvalues/vectors Vectors  Dot product, cross product, linear independence Proofs  Induction

9. 9/67 Required Textbook

10. 10/67 Grading 60% Homework 40% Class Project No exams!

11. 11/67 Class Project Topic: your choice  Integrate with research  Originality Reports  Proposal: 9/22  Update #1: 10/22  Update #2: 11/12  Final report/presentation: 12/8, 12/11

12. 12/67 Class Project Grading 10% Originality 20% Reports (5% each) 5% Final Oral Presentation 65% Quality of Work http://courses.cs.tamu.edu/schaefer/645_Fall2015/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. 13/67

14. 14/67 Questions?

15. 15/67 Vectors

16. 16/67 Vectors

17. 17/67 Vectors

18. 18/67 Vectors

19. 19/67 Vectors

20. 20/67 Vectors

21. 21/67 Vectors

22. 22/67 Points

23. 23/67 Points

24. 24/67 Points

25. 25/67 Points

26. 26/67 Points 1 p=p 0 p=0 (vector) c p=undefined where c 0,1 p – q = v (vector)

27. 27/67 Points

28. 28/67 Points

29. 29/67 Points

30. 30/67 Points

31. 31/67 Points

32. 32/67 Points

33. 33/67 Points

34. 34/67 Points

35. 35/67 Barycentric Coordinates

36. 36/67 Barycentric Coordinates

37. 37/67 Barycentric Coordinates

38. 38/67 Barycentric Coordinates

39. 39/67 Barycentric Coordinates

40. 40/67 Barycentric Coordinates

41. 41/67 Barycentric Coordinates

42. 42/67 Convex Sets If, then the form a convex combination

43. 43/67 Convex Hulls Smallest convex set containing all the

44. 44/67 Convex Hulls Smallest convex set containing all the

45. 45/67 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. 46/67 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. 47/67 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. 48/67 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. 49/67 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. 50/67 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. 51/67 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. 52/67 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. 53/67 Convex Hulls If p i and p j lie within the convex hull, then the line segment p i p j is also contained within the convex hull

54. Convex Hulls Inductive Proof Base Case: 1 point p 0 is its own convex hull 54/67

55. Convex Hulls Inductive Proof Inductive Step: 55/67

56. Convex Hulls Inductive Proof Inductive Step: 56/67

57. Convex Hulls Inductive Proof Inductive Step: Case 1: 57/67

58. Convex Hulls Inductive Proof Inductive Step: Case 1: 58/67

59. Convex Hulls Inductive Proof Inductive Step: Case 2: 59/67

60. Convex Hulls Inductive Proof Inductive Step: Case 2: 60/67

61. Convex Hulls Inductive Proof Inductive Step: Case 2: 61/67

62. Convex Hulls Inductive Proof Inductive Step: Case 2: 62/67

63. Convex Hulls Inductive Proof Inductive Step: 63/67

64. Convex Hulls Inductive Proof Inductive Step: 64/67

65. 65/67 Affine Transformations Preserve barycentric combinations Examples: translation, rotation, uniform scaling, non-uniform scaling, shear

66. 66/67 Other Transformations Conformal  Preserve angles under transformation  Examples: translation, rotation, uniform scaling Rigid  Preserve angles and length under transformation  Examples: translation, rotation

67. 67/67 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