Signed distance fields and ray marching

Level AdvancedDifficulty ★★★★★Application⌖ Open in the map

What is it?

Represent a shape by the signed distance $d(x)$ to its surface. Rays march forward by $d(x)$ (it is always safe), the surface is $d = 0$ , and the normal is $\nabla d$ . Neural SDFs learn $d$ with a network.

Formulas

$\norm{\nabla d} = 1, \qquad n(x) = \nabla d(x) \approx \frac{1}{2h}\big(d(x + h e_i) - d(x - h e_i)\big)_{i=1}^{3}$
eikonal property; normal by central differences

The mathematics behind it

Surfaces and level sets★★★★★fundamental
An SDF represents a shape as the zero level set of a distance function.
Gradient★★★★★fundamental
For a signed distance $d$ , $\norm{\nabla d} = 1$ and $\nabla d$ is the surface normal; ray marchers estimate it by finite differences.
Scalar fields★★★★★frequent
An SDF is a scalar field whose value is the signed distance to a shape.

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.