1. Ray Tracing Forward Ray Tracing
Modeling interaction of light with the objects/surfaces
Problem:
Many rays will not contribute
to the image!

2. Ray Tracing Backward Ray Tracing
Rays from camera (viewer) through each pixel to the scene
Backward Ray Tracing = Ray Tracing

3. Primary and Secondary Rays
Ray Tracing
Backward Ray Tracing
Primary and Secondary Rays D B
Viewer E C A
View Plane F

4. Ray Tracing Backward Ray Tracing Shadow Rays
Visibility check with respect
to the light source

5. Ray Tracing
Ray Casting D B
Viewer E C A
View Plane F

6. Ray Tracing Two Issues Ray-object intersection
Visibility test: Closest to the viewer
Pixel color determination (shading)
Illumination model

7. Ray Tracing
Ray Object Intersection
Sphere Rd Ro

8. Ray Tracing Ray Object Intersection Sphere Implicit Form
Center Sc = [ Xc Yc Zc ]
Radius Sr
Surface Point [ Xs Ys Zs ]

9. Ray Tracing Ray Object Intersection Sphere
To solve the intersection problem the ray equation is substituted into the sphere equation and the result is
solved for t
That is

10. Ray Tracing
Ray Object Intersection
Sphere

11. [ Xi Yi Zi ] = [ Xo + Xdt , Yo + Ydt , Zo + Zdt ]
Ray Tracing
Ray Object Intersection
Sphere
Smaller positive among t0 and t1 gives the closest intersection point
[ Xi Yi Zi ] = [ Xo + Xdt , Yo + Ydt , Zo + Zdt ]

12. Ray Tracing
Ray Object Intersection
Sphere
Normal

13. Ray Tracing Ray Sphere Intersection Sum up Calculate A B C
Compute the discriminant
Calculate min (t0 , t1)
Compute the intersection point
Compute the normal

14. Ray Tracing Ray Object Intersection Sphere Geometric Approach tca thcRd d r R0 L O

15. Ray Tracing Ray Object Intersection Sphere Geometric Approach tca thcRd d r R0 L O

16. Ray Tracing Ray Object Intersection Sphere Geometric Approach tca thcRd d r R0 L O

17. Ray Tracing Ray Object Intersection Sphere Geometric Approach tca thcRd d r R0 L O

18. Ray Tracing Ray Plane Intersection Ray Ro = [ Xo Yo Zo ] (ray origin)
Rd = [ Xd Yd Zd ] (ray direction)
Xd2 + Yd2 + Zd2 =1 (normalized)
R(t) = Ro + Rdt t > 0
Plane
P : Ax + By + Cz + D = 0
A2 + B2 +C2 = 1
Pnormal = Pn = [ A B C ]
D : Distance from origin

19. Ray Tracing Ray Plane Intersection
Substituting ray equation in plane’s equation
A ( Xo + Xdt ) + B (Yo + Ydt ) + C ( Zo + Zdt ) + D = 0
Solving for t

20. Ray Tracing Ray Plane Intersection Let
If Vd = 0 then the ray is parallel to the plane
(no intersection)
Vd > 0 normal is pointing away from the ray
(may be used for back-face culling)
Normal
Ray

21. Ray Tracing Ray Plane Intersection Let
If t < 0 then plane is behind ray’s origin
else compute intersection
ri = [ Xi Yi Zi ] = [Xo + Xdt , Yo + Ydt , Zo + Zdt ]
rnormal = Pn
Let

22. Ray Tracing Polygon Intersection Containment Test
Parity Test: If the number of intersection is odd then point
is inside (special case for vertices)

23. Triangle: Barycentric Coordinates
Ray Tracing
Triangle Intersection
Containment Test
Triangle: Barycentric Coordinates V3 P V1 V2

24. Triangle: Barycentric Coordinates
Ray Tracing
Triangle Intersection
Containment Test
Triangle: Barycentric Coordinates V3 A2 P A1
Total A V1 V2 A3
Area Coordinates

25. Triangle: Barycentric Coordinates
Ray Tracing
Triangle Intersection
Containment Test
Triangle: Barycentric Coordinates V3 A2 A1 P V1 V2 A3

26. Ray Tracing Ray Quadric Intersection Quadrics:
Cylinders, Cone, Sphere, Ellipsoids,
Paraboloids, Hyperboloids, etc.
Implict form f(X,Y,Z) = 0
Ax2 + 2Bxy + 2Cxz + 2Dx + Ey2 +2Fyz +26y + Hz2 + 2Iz +J = 0
Ray: Parametric form
Ro = [ Xo Yo Zo ] (ray origin)
Rd = [ Xd Yd Zd ] (ray direction)
Xd2 + Yd2 + Zd2 =1 (normalized)
R(t) = Ro + Rdt t > 0

27. Ray Tracing
Ray Quadric Intersection
Matrix Form
f(X,Y,Z) = 0

28. Ray Tracing
Ray Quadric Intersection
Substituting

29. Ray Tracing
Ray Quadric Intersection
Normal

30. 3D Clipping: Cyrus Beck/Liang Barsky
Ray Tracing
Ray Box Intersection
3D Clipping: Cyrus Beck/Liang Barsky