How does ray tracing work?

How does ray tracing work?

Ray tracing makes an image by sending one straight-line ray from the camera through every pixel and asking what it hits first. At that point, the laws of geometric optics — Lambert's cosine law, the inverse-square law, shadows and the law of reflection — decide how much light travels back along the ray, and that becomes the pixel's colour.

Why can light be modelled as rays?

Light is an electromagnetic wave, but when every object in the scene is much larger than its wavelength (about $400$–$700\ \text{nm}$ for visible light) diffraction is negligible and light travels in straight lines. This is geometric optics: energy flows along rays, and rays only change direction where they meet a surface. Every step of a ray tracer is a statement from geometric optics.

Real lamps emit rays in every direction and only a tiny fraction ever reaches a camera. Simulating them forwards would waste nearly all the work, so a ray tracer starts at the camera instead. This is allowed by the Helmholtz reciprocity principle: a light path from A to B behaves the same as the path from B to A. The method of tracing backwards with shadow and mirror rays is called Whitted-style ray tracing, after Turner Whitted (1980).

How is a ray sent through a pixel?

The camera sits at $\vec{O}$ and looks along $-\hat{w}$, with $\hat{u}$ pointing right and $\hat{v}$ up. A pixel is a point $(s_x, s_y)$ on an image plane one unit in front of the camera, spread across $\pm\tan(\text{fov}/2)$. The ray through it is:

Because $\hat{D}$ is a unit vector, $t$ is simply the distance travelled in metres. Notice that nothing here depends on the objects or the light — move them and this step does not change.

How do you find where a ray hits a sphere?

A sphere of centre $\vec{C}$ and radius $r$ is every point at distance $r$ from $\vec{C}$. Substituting the ray into that condition gives a quadratic equation in $t$:

The visible surface is the smallest positive root, $t = (-b - \sqrt{\Delta})/2a$. A negative root means the intersection is behind the camera. The flat floor is even simpler: the plane $y = 0$ is hit when $O_y + tD_y = 0$, so $t = -O_y/D_y$. Whichever object gives the smaller positive $t$ is the one the pixel shows.

At the hit point $\vec{P} = \vec{O} + t\hat{D}$ the ray tracer needs the surface normal — the unit vector sticking straight out of the surface. For a sphere it points from the centre through $\vec{P}$: $\hat{N} = (\vec{P}-\vec{C})/r$. Every lighting law below measures angles from $\hat{N}$.

How bright is the surface where the ray lands?

The inverse-square law

A point light of radiant intensity $I_0$ (watts per steradian) spreads its power over spheres of area $4\pi d^2$, so the irradiance it delivers at distance $d$ falls as the square of the distance. Doubling the distance to the light cuts the light arriving to a quarter.

Lambert's cosine law

A beam arriving at angle $\theta_i$ from the normal is spread over a patch $1/\cos\theta_i$ times larger than when it arrives head-on, so the energy per unit area falls by $\cos\theta_i$. With unit vectors, $\cos\theta_i = \hat{N}\cdot\hat{L}$, and a matte (diffuse) surface of reflectance $k_d$ looks this bright:

The $\max(0,\cdot)$ removes surfaces facing away from the light. This single dot product is why a sphere looks round: its brightness fades smoothly from the point facing the lamp to the terminator, where $\hat{N}\cdot\hat{L} = 0$. It is also why winter sunlight, arriving at a slant, heats the ground less.

Shiny highlights

Glossy surfaces also reflect light mostly in the mirror direction. The Phong model mirrors $\hat{L}$ about the normal to get $\hat{R} = 2(\hat{N}\cdot\hat{L})\hat{N} - \hat{L}$, then checks how closely it points at the viewer $\hat{V} = -\hat{D}$. The exponent $n$ (shininess) sets how tight the highlight is:

How does a ray tracer make shadows?

A point is in shadow when something blocks the straight line between it and the light. So the tracer casts a second ray, the shadow ray, from $\vec{P}$ toward the light. If it hits an object at a distance $s$ smaller than $d$, the direct light is removed and only the ambient term remains.

With one point light, shadows have perfectly sharp edges. Real lamps have a size, so points near the edge see part of the lamp: that partial shadow is the penumbra, which a point-light tracer cannot produce.

How are reflections traced?

At a mirror, the angle of incidence equals the angle of reflection, and the incoming ray, the normal and the outgoing ray lie in one plane. Subtracting twice the component of $\hat{D}$ along the normal flips it and gives exactly that direction:

The reflected ray is then traced with exactly the same steps — intersection, lighting, shadows — which makes ray tracing recursive. The pixel mixes the surface's own colour with what the mirror ray sees, weighted by the reflectivity $\rho$: $\text{colour} = (1-\rho)\,\text{local} + \rho\,\text{reflected}$. Recursion stops after a fixed number of bounces.

What does simple ray tracing leave out?

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