1 Dr. Scott Schaefer Intersecting Simple Surfaces

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

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

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

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

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

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

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

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

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?

11/66 Point Inside Convex Polygon

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

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

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

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

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

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

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

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

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

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

22/66 Point Inside Convex Polygon

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

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

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

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

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

28/66 A Better Way

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39/66 A Better Way One winding = inside

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51/66 A Better Way zero winding = outside

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

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

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

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?

56/66 Intersecting Spheres

57/66 Intersecting Spheres

58/66 Intersecting Spheres

59/66 Intersecting Spheres is quadratic in t

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

61/66 Normals of Spheres

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

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

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

65/66 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 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

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

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