Ray-Object Intersections

Intersection Testing

Finding where a ray hits geometric objects is fundamental to ray tracing.

Ray-Sphere Intersection

Given:

• Ray: $\mathbf{P}(s) = \mathbf{O} + s\mathbf{D}$
• Sphere: $(\mathbf{P} - \mathbf{C}) \cdot (\mathbf{P} - \mathbf{C}) = r^2$

Substituting the ray equation: $(\mathbf{O} + s\mathbf{D} - \mathbf{C}) \cdot (\mathbf{O} + s\mathbf{D} - \mathbf{C}) = r^2$

This is a quadratic in $s$: $as^2 + bs + c = 0$

where:

• $a = \mathbf{D} \cdot \mathbf{D}$
• $b = 2\mathbf{D} \cdot (\mathbf{O} - \mathbf{C})$
• $c = (\mathbf{O} - \mathbf{C}) \cdot (\mathbf{O} - \mathbf{C}) - r^2$

Discriminant

The discriminant $d = b^2 - 4ac$ determines intersection type:

• $d < 0$: No intersection (miss)
• $d = 0$: Tangent (one intersection)
• $d > 0$: Two intersections (enter and exit)

Solutions

$s_1 = \frac{-b + \sqrt{d}}{2a}, \quad s_2 = \frac{-b - \sqrt{d}}{2a}$

The closest valid intersection is the smallest positive $s$.

Ray-Plane Intersection

Given:

• Ray: $\mathbf{P}(s) = \mathbf{O} + s\mathbf{D}$
• Plane: $\mathbf{P} \cdot \mathbf{N} + d = 0$

where $\mathbf{N}$ is the plane normal and $d$ is the distance offset.

Substituting: $s = \frac{-d - \mathbf{N} \cdot \mathbf{O}}{\mathbf{N} \cdot \mathbf{D}}$

Valid if:

• $s \geq 0$ (intersection in front of origin)
• $\mathbf{N} \cdot \mathbf{D} \neq 0$ (ray not parallel to plane)

Ray-Polygon Intersection

For a polygon:

1. Intersect the ray with the plane containing the polygon
2. Check if the intersection point lies inside the polygon (2D geometry test)

For a disc: Check if distance from centre is less than radius.

Ray-Cone Intersection (2022 Paper 3 Question 3)

For a cone with apex at origin, base radius $r$, height $h$, axis along $y$:

Implicit equation: $x^2 + z^2 - \left(\frac{r}{h}\right)^2 y^2 = 0$

Intersection Method

1. Base: Intersect ray with plane $y = h$; accept if $x^2 + z^2 \leq r^2$
2. Side: Substitute ray into implicit equation, solve quadratic in $s$

Normal Calculation

For a point $(x, y, z)$ on the cone side, the normal is proportional to: $\mathbf{N} \propto \left(2x, -2\left(\frac{r}{h}\right)^2 y, 2z\right)$

The normal is the gradient of the implicit function $f(x, y, z) = 0$.

Transformed Objects

For an object transformed by matrix $\mathbf{M}$:

1. Transform the ray by $\mathbf{M}^{-1}$
2. Intersect with the canonical (untransformed) object
3. Transform the intersection point back by $\mathbf{M}$
4. Transform the normal by $(\mathbf{M}^{-1})^\top$

Summary

• Ray-sphere intersection reduces to a quadratic equation
• Ray-plane intersection is a simple division
• Polygon intersections require inside/outside tests
• Cone normals are computed from the implicit gradient
• Transformed objects require transforming the ray by the inverse matrix

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Past Paper Questions

2022 Paper 3 Question 3: Derive the ray-cone intersection. Find the normal at an intersection point.