Intersection Testing

Ray vs Sphere:
- A ray is represented as $r (t) = o + t d$ , where $o$ is the origin and $d$ is the direction.
- A sphere is defined by its center $c$ and radius $r$ .
- Substitute the ray equation into the sphere’s equation $∣∣ p - c ∣ ∣^{2} = r^{2}$ to check for intersections.
- Solve the resulting quadratic equation for $t$ (parameter along the ray) to find intersection points.
Ray vs Triangle:
- A triangle is defined by three vertices $v_{0}, v_{1}, v_{2}$ .
- A point inside the triangle can be represented as $t (u, v) = v_{0} + u (v_{1} - v_{0}) + v (v_{2} - v_{0})$ where $u, v \geq 0$ and $u + v \leq 1$ .
- Set $t (u, v) = r (t)$ and solve for $t, u, v$ using methods like Cramer’s rule.
Ray vs Plane:
- A plane is defined by the equation $n \cdot x + d = 0$ , where $n$ is the normal vector.
- To find where a ray intersects the plane, solve for $t$ using: $x = r (t)$ $n \cdot (o + t d) + d = 0$ $t = \frac{- ( d + n \cdot o )}{n \cdot d}$ .

Slab Test (Ray/Box Intersection):
- A slab is the region between two parallel planes.
- For a box (defined as the intersection of three slabs), compute the intersection of the ray with each pair of planes in each dimension (x, y, z).
- Keep the maximum $t_{min}$ (entry time) and minimum $t_{max}$ (exit time) across all dimensions:
  - If $t_{min} < t_{max}$ , the ray intersects the box.
- Special case: Handle parallel rays (direction vector perpendicular to slab normals).
Separating Axis Theorem (SAT):
- Two convex polyhedra are disjoint if there exists an axis where their projections do not overlap.
- For collision detection, test the following axes:
  - Normals of each face.
  - Cross products of edges from both objects.
- If projections overlap on all axes, the objects intersect.

Used to handle moving objects over time (e.g., between animation frames).
Example: Sphere/Plane:
- Compute the signed distances $sc$ (current) and $se$ (expected) between the sphere’s center and the plane.
- If the sphere moves to $∣ sc - r ∣$ , calculate the time of collision: $t = \frac{∣ sc ∣ - r}{∣ sc ∣ - ∣ se ∣}$ .
- Adjust the sphere’s movement to account for the collision response (reflection vector).

View Frustum Culling:
- Used to determine if objects fall within the camera’s view.
- A frustum consists of six planes (near, far, top, bottom, left, right).
- Test objects (e.g., spheres or AABB) against these planes:
  - If outside any plane, reject the object.
  - Otherwise, consider the object inside or intersecting.
Sphere vs AABB Intersection:
- Compute the squared distance from the sphere’s center to the closest point on the box.
- Compare this to the sphere’s radius squared to check for collision.

Jonas Notes