1. Intersecting Simple Surfaces
Dr. Scott Schaefer

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

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

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

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

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

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

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

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

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

11. Point Inside Convex Polygon

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

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

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

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

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

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

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

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

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

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

22. Point Inside Convex Polygon

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

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

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

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

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

28. A Better Way

29. A Better Way

30. A Better Way

31. A Better Way

32. A Better Way

33. A Better Way

34. A Better Way

35. A Better Way

36. A Better Way

37. A Better Way

38. A Better Way

39. A Better Way
One winding = inside

40. A Better Way

41. A Better Way

42. A Better Way

43. A Better Way

44. A Better Way

45. A Better Way

46. A Better Way

47. A Better Way

48. A Better Way

49. A Better Way

50. A Better Way

51. A Better Way
zero winding = outside

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

53. Computing Winding Number
Given unit normal n =0
For each edge (p1, p2)
If , then inside

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

55. 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. Intersecting Spheres

57. Intersecting Spheres

58. Intersecting Spheres

59. Intersecting Spheres
is quadratic in t

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

61. Normals of Spheres

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

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

64. 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. 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. 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

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

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