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

2. Course Information Instructor Dr. Scott Schaefer HRBB 527B
Office Hours: MW 10:15am – 11:15am (or by appointment)
Website:

3. Geometric Modeling
Surface representations
Industrial design

4. Geometric Modeling Surface representations Industrial design
Movies and animation

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

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

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

9. Required Textbook

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

11. Class Project Topic: your choice Integrate with research Originality
Reports
Proposal: 9/15
Update #1: 10/13
Update #2: 11/10
Final report/presentation: 12/13

12. Class Project Grading 10% Originality 20% Reports (5% each)
5% Final Oral Presentation
65% Quality of Work

13. Questions?

14. Vectors

15. Vectors

16. Vectors

17. Vectors

18. Vectors

19. Vectors

20. Vectors

21. Points

22. Points

23. Points

24. Points

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

26. Points

27. Points

28. Points

29. Points

30. Points

31. Points

32. Points

33. Points

34. Barycentric Coordinates

35. Barycentric Coordinates

36. Barycentric Coordinates

37. Barycentric Coordinates

38. Barycentric Coordinates

39. Barycentric Coordinates

40. Barycentric Coordinates

41. Convex Sets
If , then the form a convex combination

42. Convex Hulls
Smallest convex set containing all the

43. Convex Hulls
Smallest convex set containing all the

44. Convex Hulls If pi and pj lie within the convex hull,
then the line pipj is also contained within the convex hull

45. Convex Hulls If pi and pj lie within the convex hull,
then the line pipj is also contained within the convex hull

46. Convex Hulls If pi and pj lie within the convex hull,
then the line pipj is also contained within the convex hull

47. Convex Hulls If pi and pj lie within the convex hull,
then the line pipj is also contained within the convex hull

48. Convex Hulls If pi and pj lie within the convex hull,
then the line pipj is also contained within the convex hull

49. Convex Hulls If pi and pj lie within the convex hull,
then the line pipj is also contained within the convex hull

50. Convex Hulls If pi and pj lie within the convex hull,
then the line pipj is also contained within the convex hull

51. Convex Hulls If pi and pj lie within the convex hull,
then the line pipj is also contained within the convex hull

52. Convex Hulls If pi and pj lie within the convex hull,
then the line pipj is also contained within the convex hull

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

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

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