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

2/67 Course Information Instructor  Dr. Scott Schaefer  HRBB 527B  Office Hours: TR 10:00am – 11:00am (or by appointment) Website:

3/67 Geometric Modeling Surface representations  Industrial design

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

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

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

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

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

9/67 Required Textbook

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

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

12/67 Class Project Grading 10% Originality 20% Reports (5% each) 5% Final Oral Presentation 65% Quality of Work

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

14/67 Questions?

15/67 Vectors

16/67 Vectors

17/67 Vectors

18/67 Vectors

19/67 Vectors

20/67 Vectors

21/67 Vectors

22/67 Points

23/67 Points

24/67 Points

25/67 Points

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

27/67 Points

28/67 Points

29/67 Points

30/67 Points

31/67 Points

32/67 Points

33/67 Points

34/67 Points

35/67 Barycentric Coordinates

36/67 Barycentric Coordinates

37/67 Barycentric Coordinates

38/67 Barycentric Coordinates

39/67 Barycentric Coordinates

40/67 Barycentric Coordinates

41/67 Barycentric Coordinates

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

43/67 Convex Hulls Smallest convex set containing all the

44/67 Convex Hulls Smallest convex set containing all the

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

46/67 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

47/67 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

48/67 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

49/67 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

50/67 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

51/67 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

52/67 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

53/67 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

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

Convex Hulls Inductive Proof Inductive Step: 55/67

Convex Hulls Inductive Proof Inductive Step: 56/67

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

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

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

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

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

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

Convex Hulls Inductive Proof Inductive Step: 63/67

Convex Hulls Inductive Proof Inductive Step: 64/67

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

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 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