1. Surfaces
Dr. Scott Schaefer

2. Types of Surfaces Implicit Surfaces Parametric Surfaces
Deformed Surfaces

3. Implicit Surfaces

4. Implicit Surfaces

5. Implicit Surfaces
Examples
Spheres
Planes
Cylinders
Cones
Tori

6. Intersecting Implicit Surfaces

7. Intersecting Implicit Surfaces

8. Intersecting Implicit Surfaces
Example

9. Intersecting Implicit Surfaces
Example

10. Intersecting Implicit Surfaces
Example

11. Intersecting Implicit Surfaces
Example

12. Normals of Implicit Surfaces
Given F(x,y,z)=0, find the normal at a point (x,y,z)

13. Normals of Implicit Surfaces
Given F(x,y,z)=0, find the normal at a point (x,y,z)
Assume we have a parametric curve (x(t),y(t),z(t)) on the surface of F(x,y,z)

14. Normals of Implicit Surfaces
Given F(x,y,z)=0, find the normal at a point (x,y,z)
Assume we have a parametric curve (x(t),y(t),z(t)) on the surface of F(x,y,z)
F(x(t),y(t),z(t))=0

15. Normals of Implicit Surfaces
Given F(x,y,z)=0, find the normal at a point (x,y,z)
Assume we have a parametric curve (x(t),y(t),z(t)) on the surface of F(x,y,z)
F(x(t),y(t),z(t))=0

16. Normals of Implicit Surfaces
Given F(x,y,z)=0, find the normal at a point (x,y,z)
Assume we have a parametric curve (x(t),y(t),z(t)) on the surface of F(x,y,z)
F(x(t),y(t),z(t))=0

17. Normals of Implicit Surfaces
Given F(x,y,z)=0, find the normal at a point (x,y,z)
Assume we have a parametric curve (x(t),y(t),z(t)) on the surface of F(x,y,z)
F(x(t),y(t),z(t))=0

18. Normals of Implicit Surfaces
Given F(x,y,z)=0, find the normal at a point (x,y,z)
Assume we have a parametric curve (x(t),y(t),z(t)) on the surface of F(x,y,z)
F(x(t),y(t),z(t))=0
Tangent of curve!!!

19. Normals of Implicit Surfaces
Given F(x,y,z)=0, find the normal at a point (x,y,z)
Assume we have a parametric curve (x(t),y(t),z(t)) on the surface of F(x,y,z)
F(x(t),y(t),z(t))=0
Normal of surface!!!

20. Normals of Implicit Surfaces
Example

21. Normals of Implicit Surfaces
Example

22. Normals of Implicit Surfaces
Example

23. Implicit Surfaces Advantages Easy to determine inside/outside
Easy to determine if a point is on the surface
Disadvantages
Hard to generate points on the surface

24. Parametric Surfaces
Geri’s Game Copyright Pixar

25. Parametric Surfaces

26. Parametric Surfaces

27. Intersecting Parametric Surfaces
Solve three equations (one for each of x, y, z) for the parameters t, u, v
Plug parameters back into equation to find actual intersection

28. Intersecting Parametric Surfaces

29. Intersecting Parametric Surfaces

30. Intersecting Parametric Surfaces

31. Intersecting Parametric Surfaces

32. Normals of Parametric Surfaces
Assume t is fixed

33. Normals of Parametric Surfaces
Assume t is fixed
Curve on surface

34. Normals of Parametric Surfaces
Assume t is fixed
Tangent at s

35. Normals of Parametric Surfaces
Assume s is fixed

36. Normals of Parametric Surfaces
Assume s is fixed
Curve on surface

37. Normals of Parametric Surfaces
Assume s is fixed
Tangent at t

38. Normals of Parametric Surfaces
Normal at s, t is

39. Parametric Surfaces Advantages Easy to generate points on the surface
Disadvantages
Hard to determine inside/outside
Hard to determine if a point is on the surface

40. Deformed Surfaces
Assume we have some surface S and a deformation function D(x,y,z)
D(S) is deformed surface
Useful for creating complicated
shapes from simple objects

41. Intersecting Deformed Surfaces
Assume D(x,y,z) is simple… a matrix
First deform line L(t) by inverse of D
Calculate intersection with undeformed surface S
Transform intersection point and normal by D

42. Intersecting Deformed Surfaces
Example

43. Intersecting Deformed Surfaces
Example

44. Intersecting Deformed Surfaces
Example

45. Intersecting Deformed Surfaces
Example

46. Intersecting Deformed Surfaces
Example

47. Intersecting Deformed Surfaces
Example

48. Normals of Deformed Surfaces
Define how tangents transform first
Assume curve C(t) on surface

49. Normals of Deformed Surfaces
Define how tangents transform first
Assume curve C(t) on surface

50. Normals of Deformed Surfaces
Define how tangents transform first
Assume curve C(t) on surface
Tangents transform by just applying the deformation D!!!

51. Normals of Deformed Surfaces
Normals and tangents are orthogonal before and after deformation

52. Normals of Deformed Surfaces
Normals and tangents are orthogonal before and after deformation

53. Normals of Deformed Surfaces
Normals and tangents are orthogonal before and after deformation

54. Normals of Deformed Surfaces
Normals and tangents are orthogonal before and after deformation

55. Normals of Deformed Surfaces
Normals and tangents are orthogonal before and after deformation

56. Normals of Deformed Surfaces
Example

57. Normals of Deformed Surfaces
Example

58. Normals of Deformed Surfaces
Example

59. Deformed Surfaces Advantages
Simple surfaces can represent complex shapes
Affine transformations yield simple calculations
Disadvantages
Complicated deformation functions can be difficult to use (inverse may not exist!!!)