CSE 681 RayObject Intersections: Planes and Polygons - PowerPoint PPT Presentation
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CSE 681 RayObject Intersections: Planes and Polygons
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x. y. z. Ray-Plane Intersection. If P = (x, y, z) then. Given: ax by cz = -d. (a, b, c)T (x, y, z) N P = -d. How shall we intersect a ray with a plane? ... – PowerPoint PPT presentation
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Title: CSE 681 RayObject Intersections: Planes and Polygons
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CSE 681Ray-Object IntersectionsPlanes and
Polygons
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Mathematical Toolbox
- Objects are represented by mathematical
descriptions - Implicit form
- F (x, y, z) 0
- Explicit form
- Analytic x F (y, z)
- Parametric x Px(t)
- y Py(t)
- z Pz(t)
Testing
Generating
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Ray-Object Intersection
- Implicit Form F(x, y, z) 0
- Ray P(t) (x, y, z)
- source t Direction
- R t D
- Intersection F(P(t)) 0
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Ray-Sphere Intersection Recap
- Solve at2 bt c 0, where
- Discriminant
- Solutions
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Ray-Sphere Test
- b2 4ac lt 0 Þ No intersection
- b2 4ac gt 0 Þ Two solutions (enter and exit)
- b2 4ac 0 Þ One solution (ray grazes sphere)
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Ray-Plane Intersection
- Infinite Plane
- Contains at least three points (not collinear)
- A line drawn through any two points in it lies
wholly in the plane - Plane Equation
- ax by cz d 0, where a, b, c, d are
constants
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Plane Normal
- N (a, b, c)T
- d is the distance to the plane
d
N
y
x
z
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Ray-Plane Intersection
- If P (x, y, z) then
- Given ax by cz -d
- Þ (a, b, c)T (x, y, z)
- Þ N P -d
- How shall we intersect a ray with a plane?
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Ray-Plane Intersection
V2
V1
- Let P(t) R t D
- Solve for t
- N P -d
- Þ N (R t D) -d
- Þ N R t (N D) -d
- Þ t -(d N R)/(N D)
D
R
V4
V3
What can we say about this equation?
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Ray-Plane Intersection
V2
V1
- Let P(t) R t D
- Solve for t
- N P -d
- Þ N (R t D) -d
- Þ N R t (N D) -d
- Þ t -(d N R)/(N D)
- N D 0 ray parallel to the plane
- N D gt 0 ray points away from the plane
D
R
V4
V3
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Side Note On Plane Handedness
- N (V3 - V2) x (V1 - V2)
- Front face vs back face depends on
- Left-hand vs right-handedness
- Vertex order (clockwise vs counter-clockwise)
V2
V1
Take left-handedness
V3
V5
V4
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Ray-Polygon IntersectionConcavity and
Collinearity
- Watch for special cases
- V1, V2, and V3 are collinear
- Angle gt 180o
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Ray-Polygon Intersection
- Perform the ray-plane intersection
- Test if intersection is in the polygon
- Solve on the 2D plane
- How do we do test this?
R
D
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Polygon Projection
- Project the plane to an axial plane
- Project to the two smallest coords of the normal
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Point Inside/Outside Test
- Ray intersection test
- Cast a ray in any direction What direction?
- Axis-aligned is easy
- Count the number of intersection
- If an odd number, point is inside else outside
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Point Inside/Outside Test
- Take this polygon
- Do you for see any problems?
- How do we count ray-vertex intersections?
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Special Case Vertices
- How should we count this?
- Just count vertices? No odd number of vertices
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Special Case Vertices
N
N
Y
N
Y
Y
Y
Y
Y
- If we encounter a vertex, just count the bottom
one on an edge
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Perform Check
- Check
- ( ( (y lt y2) (y gt y1) ) ( (y lt y1) (y
gt y2))) - All Cases
y1
y2
y2
y1
y
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Special Case Vertices
N
N
Y
N
Y
Y
Y
Y
Y
- Try the check in this example
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Winding Number Test
- Compute the sum of the angles created by point p
and all the polygon edges - How do we compute the angles?
1
1
2
5
2
5
p
p
3
4
3
4
Inside ??i 3600
Outside ??i 0
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