Ray Tracing / Ray Casting

Tracing / Ray Casting

- Provides a related , powerful approach to render scenes

Overview of Ray Tracing Process

Pseudo code of a ray tracer:

  for(int r = 0; r <nRows ; r++)

  for(int c = 0; c <nCols ; c++)

            {

1.                   build the rc-th ray

2.                   find all intersections of the rc-th ray with objects in the scene

3.                  Identify the intersection that lies closest to and in front of the eye

4.                  Compute the hit point where the ray hits this object and the normal vector at that point

5.                  Find the color of the light returning to the eye along the ray from the point of intersection

6.                   place the color in the rc-th pixel

}                                 

Intersection of a Ray with an Object

Common shapes used in ray tracing

§     Sphere

§     Cylinder

§     Cone cube

§     hex cone

If ‘S’ is the starting point of a ray and ‘c’ is its direction then the ray that intersects with a shape is given by

  r(t) = S + ct         // implicit form of shape is F(P)

Condition for r(t) to coincide with a point of the surface is

  F(r(t)) = 0

The hit time t_hit  can be found by solving

  F(S+ c t_hit ) = 0

Intersection of a Ray with the Generic Plane

§      generic plane – xy plane or z = 0

§     Implicit form is F(x ,y , z) = z

§     The ray S+ct intersects the generic plane when

            S_z +c_zt_h = 0 , where t_h = - ( S_z  /  c_z )

If c_z = 0 , the ray is moving parallel to the plane & there is no intersection

Otherwise , the ray hits the plane at the point P_hit = S – c(S_z / c_z )

Intersection of a Ray with the Generic Shape

§     Consider a generic shape whose implicit form is F(P) = |P|² – 1

§     The point of intersection of the ray is given by |S + ct |²   - 1 = 0

  |c|²T² + 2.(S.C)t + (|S|² - 1) = 0   which is of the form At²  +2Bt + c = 0

by solving   t_h = -(B/A) ±(√B² – AC)/ A

ü    If B²  - Ac is ‘-’ ve , the ray misses the sphere

ü    If B²  - Ac  is zero , the ray grazes the sphere at one point

ü    If B²  - Ac is ‘+’ ve , there are 2 hit times t₁& t₂