opRevolve: Radial Revolution Operation

2D to 3D Transformation Through Axis Revolution

The opRevolve operation transforms 3D coordinates for evaluating 2D cross-sections revolved around the Y-axis. This technique creates radially symmetric shapes from 2D profiles, effectively implementing a computational lathe operation.

Mathematical Foundation

The revolution operation maps 3D coordinates to 2D cross-section space:

$$\text{opRevolve}(\vec{p}, w) = (|\vec{p}_{xz}| - w, p_y)$$

where $|\vec{p}_{xz}| = \sqrt{p_x^2 + p_z^2}$ represents the radial distance from the Y-axis, and $w$ defines the revolution radius offset.

This transformation allows any 2D SDF function $f(x, y)$ to be evaluated as:

$$d_{\text{revolved}} = f(\text{opRevolve}(\vec{p}, w))$$

Function Signature

Parameter	Type	Description
p	vec3	3D sample point
w	float	Revolution radius offset

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 revolvedPos = opRevolve(p, 1.5)
              return circleSDF(revolvedPos, vec2(0.8))
      })
      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)
}