Ray Tracing

Rasterization asks “for each triangle, which pixels does it cover?” Ray tracing flips the question: “for each pixel, what does the camera see through it?” It traces a ray from the eye out into the scene and reports what it hits. Render the image below row by row, or all at once, and move the light to relight the sphere.

Casting a ray per pixel

Picture the image as a window floating in front of the camera. For every pixel we build a ray — a starting point (the camera) and a direction pointing through that pixel into the scene. A point along the ray is

$\mathbf{p}(t) = \mathbf{o} + t\,\mathbf{d}$

where $\mathbf{o}$ is the origin, $\mathbf{d}$ the direction, and $t$ the distance travelled. Finding what the pixel sees means finding the smallest positive $t$ where the ray meets an object.

Ray-sphere intersection

A sphere is every point at distance $r$ from its center $\mathbf{c}$. Substituting the ray into that condition gives a quadratic in $t$:

$t^2 + 2\,(\mathbf{d}\cdot(\mathbf{o}-\mathbf{c}))\,t + |\mathbf{o}-\mathbf{c}|^2 - r^2 = 0$

The discriminant decides everything: negative means the ray misses the sphere entirely (those pixels stay background); zero grazes the edge; positive gives two hits, and we keep the nearer one — the front of the sphere. This closed-form solve is why spheres are the traditional first object in every ray tracer.

Diffuse shading

A hit point alone is just a silhouette; shading gives it form. At the hit we compute the surface normal $\mathbf{n}$, the unit vector pointing straight out of the sphere, which is simply $(\mathbf{p}-\mathbf{c})/r$. Lambert’s cosine law says a matte surface looks brightest when it faces the light head-on and fades as it turns away. That cosine is exactly a dot product of unit vectors:

$\text{brightness} = \max(0,\ \mathbf{n}\cdot\mathbf{l})$

where $\mathbf{l}$ points toward the light. Facing the light gives 1 (bright), a right angle gives 0 (dark), and the $\max$ clamps away the back side. That single dot product per pixel produces the smooth rounded gradient you see — move the light and the bright spot follows.

Beyond one bounce

Real ray tracers do not stop at the first hit. They spawn new rays from it: toward lights to test for shadows, and in mirror or refracted directions for reflections and glass. Following light across many bounces is what makes ray tracing produce such realistic images — at the cost of casting far more rays.

Takeaways

• Ray tracing casts a ray from the camera through each pixel and finds the nearest hit.
• Ray-sphere intersection is a quadratic; its discriminant tells you miss, graze, or hit, and you keep the nearest positive $t$.
• Diffuse brightness is $\max(0, \mathbf{n}\cdot\mathbf{l})$ — the cosine of the angle between the surface normal and the light.
• Secondary rays for shadows, reflections, and refraction extend the same idea to photorealistic images.

References

• Scratchapixel — A minimal ray tracer
• Peter Shirley — Ray Tracing in One Weekend
• Akenine-Moller et al., “Real-Time Rendering” — ray tracing chapter