Ray Equation

Ray Representation

A ray is represented parametrically as: $\mathbf{P}(s) = \mathbf{O} + s\mathbf{D}, \quad s \geq 0$

where:

• $\mathbf{O}$ is the ray origin (starting point)
• $\mathbf{D}$ is the direction vector (typically normalised)
• $s$ is the parameter measuring distance along the ray

Properties

• For $s = 0$: $\mathbf{P}(0) = \mathbf{O}$ (origin)
• For $s > 0$: Points along the ray in direction $\mathbf{D}$
• Negative $s$ values are invalid (behind the origin)

Constructing Rays from Camera

For a pixel at screen coordinates $(x, y)$:

1. Compute the world-space position of the pixel on the image plane
2. Direction: $\mathbf{D} = \frac{\mathbf{P}_{\text{pixel}} - \mathbf{O}_{\text{camera}}}{|\mathbf{P}_{\text{pixel}} - \mathbf{O}_{\text{camera}}|}$

Why Parametric?

The parametric form is ideal for intersection testing:

• Substitute $\mathbf{P}(s)$ into the surface equation
• Solve for $s$ (the distance to intersection)
• IF $s \geq 0$, the intersection is in front of the ray origin

Summary

• A ray is defined by origin $\mathbf{O}$ and direction $\mathbf{D}$
• The parameter $s$ gives distance along the ray
• Parametric form enables algebraic intersection calculations