opRepeat: Infinite Pattern Repetition Operation

Spatial Tiling Transform for Distance Fields

The opRepeat operations transform coordinate space to create infinite repetitions of SDF primitives. These functions enable the creation of regular patterns, grids, and tiled arrangements by folding space into repeating cells.

Mathematical Foundation

The basic repetition operation uses modular arithmetic to fold space:

$$\text{opRepeat}(\vec{p}, s) = \text{mod}(\vec{p} + \frac{s}{2}, s) - \frac{s}{2}$$

For multi-dimensional repetition:

$$\text{opRepeat}(\vec{p}, \vec{c}) = \text{mod}(\vec{p} + \frac{\vec{c}}{2}, \vec{c}) - \frac{\vec{c}}{2}$$

Limited repetition constrains the pattern to a finite region:

$$\text{opRepeatLimited}(\vec{p}, \vec{a}, \vec{b}, s) = \vec{p} - s \cdot \text{clamp}(\lfloor\frac{\vec{p}}{s}\rfloor, \vec{a}, \vec{b})$$

Function Variants

Function	Parameters	Description
opRepeatVec2	p, s	2D uniform repetition with spacing s
opRepeatVec3	p, c	3D repetition with per-axis spacing
opRepeatVec2Limited	p, lima, limb, s	2D repetition within bounds
opRepeatVec3Limited	p, lima, limb, s	3D repetition within bounds

Implementation Demonstration

Live Editor

const fragment = () => {
      const up = vec3(0, 1, 0)
      const eps = vec3(0.01, 0, 0)
      const eye = rotate3dY(iTime).mul(vec3(4))
      const sdf = Fn(([p]: [Vec3]) => {
              const repeatedPos = opRepeatVec3(p, vec3(5))
              return sphereSDFRadius(repeatedPos, 0.2)
      })
      const march = Fn(([eye, dir]: [Vec3, Vec3]) => {
              const p = eye.toVar()
              const d = sdf(p).toVar()
              Loop(128, ({ i }) => {
                      If(d.lessThanEqual(eps.x), () => {
                              const dx = sdf(p.add(eps.xyy)).sub(d)
                              const dy = sdf(p.add(eps.yxy)).sub(d)
                              const dz = sdf(p.add(eps.yyx)).sub(d)
                              const normal = vec3(dx, dy, dz).normalize()
                              const light = vec3(2, 4, 3).sub(p).normalize()
                              const diffuse = normal.dot(light).max(0.1)
                              return vec4(vec3(diffuse.mul(0.8).add(0.2)), 1)
                      })
                      p.addAssign(d.mul(dir))
                      d.assign(sdf(p))
              })
              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)
}