Equation for the ray with parallel projection

Question

What will be the equation for the ray and ray origin when we are using parallel projection and how to derive that?

Answer 1

In traditional raytracing, you use a ray that starts at your eye point. For each pixel you calculate where it is on a virtual screen in front of the camera and shoot a ray through that pixel.

Let pO be the eye point, d be the direction of the camera, r to be a vector pointing to the right and u to be a vector pointing up. Let w be the number of pixels in the screen horizontally and h be the number of pixels vertically.

The parametric equation for a ray going through any pixel x, y is then:

ray = pO + t * normalize (d + (x - 0.5w)/0.5w * r + (y - 0.5h)/0.5h * u)

where t is the parameter.

For a parallel projection, move the virtual screen to the origin and calculate the x, y to be the origin of the ray then use the same direction d for each ray:

ray = (pO + (x - 0.5w)/0.5w * r + (y - 0.5h)/0.5h * u) + t*d

Answer 2

For a perspective projection, you have an eye origin, direction, right and up vectors. You then run a vector from the eye origin to each pixel in a virtual screen by scaling the right and up vectors.

In a parallel projection, you do the same calculation for the point on the screen, but your origin becomes that point and you use the same direction for each ray.