1. 1 Dr. Scott Schaefer Intersecting Simple Surfaces

2. 2/66 Types of Surfaces Infinite Planes Polygons  Convex  Ray Shooting  Winding Number Spheres Cylinders

3. 3/66 Infinite Planes Defined by a unit normal n and a point o

4. 4/66 Infinite Planes Defined by a unit normal n and a point o

5. 5/66 Infinite Planes Defined by a unit normal n and a point o

6. 6/66 Infinite Planes Defined by a unit normal n and a point o

7. 7/66 Infinite Planes Defined by a unit normal n and a point o

8. 8/66 Infinite Planes Defined by a unit normal n and a point o

9. 9/66 Polygons Intersect infinite plane containing polygon Determine if point is inside polygon

10. 10/66 Polygons Intersect infinite plane containing polygon Determine if point is inside polygon How do we know if a point is inside a polygon?

11. 11/66 Point Inside Convex Polygon

12. 12/66 Check if point on same side of all edges Point Inside Convex Polygon

13. 13/66 Check if point on same side of all edges Point Inside Convex Polygon

14. 14/66 Check if point on same side of all edges Point Inside Convex Polygon

15. 15/66 Check if point on same side of all edges Point Inside Convex Polygon

16. 16/66 Check if point on same side of all edges Point Inside Convex Polygon

17. 17/66 Check if point on same side of all edges Point Inside Convex Polygon

18. 18/66 Check if point on same side of all edges Point Inside Convex Polygon

19. 19/66 Check if point on same side of all edges Point Inside Convex Polygon

20. 20/66 Check if point on same side of all edges Point Inside Convex Polygon

21. 21/66 Check if point on same side of all edges Point Inside Convex Polygon

22. 22/66 Point Inside Convex Polygon

23. 23/66 Point Inside Polygon Test Given a point, determine if it lies inside a polygon or not

24. 24/66 Ray Test Fire ray from point Count intersections  Odd = inside polygon  Even = outside polygon

25. 25/66 Problems With Rays Fire ray from point Count intersections  Odd = inside polygon  Even = outside polygon Problems  Ray through vertex

26. 26/66 Problems With Rays Fire ray from point Count intersections  Odd = inside polygon  Even = outside polygon Problems  Ray through vertex

27. 27/66 Problems With Rays Fire ray from point Count intersections  Odd = inside polygon  Even = outside polygon Problems  Ray through vertex  Ray parallel to edge

28. 28/66 A Better Way

29. 29/66 A Better Way

30. 30/66 A Better Way

31. 31/66 A Better Way

32. 32/66 A Better Way

33. 33/66 A Better Way

34. 34/66 A Better Way

35. 35/66 A Better Way

36. 36/66 A Better Way

37. 37/66 A Better Way

38. 38/66 A Better Way

39. 39/66 A Better Way One winding = inside

40. 40/66 A Better Way

41. 41/66 A Better Way

42. 42/66 A Better Way

43. 43/66 A Better Way

44. 44/66 A Better Way

45. 45/66 A Better Way

46. 46/66 A Better Way

47. 47/66 A Better Way

48. 48/66 A Better Way

49. 49/66 A Better Way

50. 50/66 A Better Way

51. 51/66 A Better Way zero winding = outside

52. 52/66 Requirements Oriented edges Edges can be processed in any order

53. 53/66 Computing Winding Number Given unit normal n =0 For each edge (p 1, p 2 ) If, then inside

54. 54/66 Advantages Extends to 3D! Numerically stable Even works on models with holes (sort of) No ray casting

55. 55/66 Intersecting Spheres Three possible cases  Zero intersections: miss the sphere  One intersection: hit tangent to sphere  Two intersections: hit sphere on front and back side How do we distinguish these cases?

56. 56/66 Intersecting Spheres

57. 57/66 Intersecting Spheres

58. 58/66 Intersecting Spheres

59. 59/66 Intersecting Spheres is quadratic in t

60. 60/66 Intersecting Spheres is quadratic in t Solve for t using quadratic equation If, no intersection If, one intersection Otherwise, two intersections

61. 61/66 Normals of Spheres

62. 62/66 Infinite Cylinders Defined by a center point C, a unit axis direction A and a radius r

63. 63/66 Infinite Cylinders Defined by a center point C, a unit axis direction A and a radius r

64. 64/66 Infinite Cylinders Defined by a center point C, a unit axis direction A and a radius r 1.Perform an orthogonal projection to the plane defined by C, A on the line L(t) and intersect with circle in 2D

65. 65/66 Infinite Cylinders Defined by a center point C, a unit axis direction A and a radius r 2.Substitute t parameters from 2D intersection to 3D line equation

66. 66/66 Infinite Cylinders Defined by a center point C, a unit axis direction A and a radius r 3.Normal of 2D circle is the same normal of cylinder at point of intersection

67. 67/66

68. 68/66 Infinite Cylinders Defined by a center point C, a unit axis direction A and a radius r

69. 69/66 Infinite Cylinders Defined by a center point C, a unit axis direction A and a radius r