boxSDF: Three-Dimensional Rectangular Distance Field

Fundamental Cubic and Rectangular Primitive Geometry

The boxSDF function computes the signed distance from any point to a rectangular box. This fundamental SDF primitive supports both unit cubes and custom-sized boxes, forming the foundation for many complex geometric constructions.

Mathematical Foundation

The box SDF operates through coordinate space analysis and distance decomposition:

$$d_{\text{box}}(p, b) = \|\max(\mathbf{0}, |p| - b)\| + \min(0, \max_i(|p_i| - b_i))$$

where $|p|$ represents the absolute coordinates and $b$ defines the box half-extents.

For a unit box (no size parameter), the formula simplifies to:

$$d_{\text{box}}(p) = \|\max(\mathbf{0}, |p|)\| + \min(0, \max_i(|p_i|))$$

Function Signatures

boxSDF (Unit Box)

Parameter	Type	Description
p	vec3	Sample point position

boxSDFSize (Custom Box)

Parameter	Type	Description
p	vec3	Sample point position
b	vec3	Box half-dimensions

Implementation Demonstrations

Live Editor

const fragment = () => {
      const up = vec3(0, 1, 0)
      const eps = vec3(0.01, 0, 0)
      const eye = rotate3dY(iTime.mul(0.5)).mul(vec3(6))
      const args = [vec3(0.8, 1.2, 0.6)]
      const march = Fn(([eye, dir]: [Vec3, Vec3]) => {
              const p = eye.toVar()
              const d = boxSDFSize(p, ...args).toVar()
              Loop(16, ({ i }) => {
                      If(d.lessThanEqual(eps.x), () => {
                              const dx = boxSDFSize(p.add(eps.xyy), ...args).sub(d)
                              const dy = boxSDFSize(p.add(eps.yxy), ...args).sub(d)
                              const dz = boxSDFSize(p.add(eps.yyx), ...args).sub(d)
                              return vec4(vec3(dx, dy, dz).normalize().mul(0.5).add(0.5), 1)
                      })
                      p.addAssign(d.mul(dir))
                      d.assign(boxSDFSize(p, ...args))
              })
              return vec4(0)
      })
      const z = eye.negate().normalize()
      const x = z.cross(up)
      const y = x.cross(z)
      const scr = vec3(uv.sub(0.5), 2)
      const dir = mat3(x, y, z).mul(scr).normalize()
      return march(eye, dir)
}