opSubtraction: Boolean Subtraction Operation

Geometric Carving for Distance Field Composition

The opSubtraction operations perform boolean subtraction by removing one SDF shape from another. This creates carved or cut-out effects, enabling complex geometric compositions through subtractive modeling.

Mathematical Foundation

Basic subtraction uses the maximum of the negated first distance and the second distance:

$$d_{\text{subtraction}} = \max(-d_1, d_2)$$

Smooth subtraction blends the transition using a smoothstep interpolation:

$$h = \text{clamp}\left(\frac{1}{2} - \frac{d_2 + d_1}{2k}, 0, 1\right)$$ $$d_{\text{smooth}} = \text{mix}(d_2, -d_1, h) + kh(1-h)$$

Function Variants

Function	Parameters	Description
opSubtraction	d1, d2	Sharp boolean subtraction
opSubtractionVec4	d1, d2	Vec4 distance with material info
opSubtractionSmooth	d1, d2, k	Smoothed subtraction with blending factor
opSubtractionSmoothVec4	d1, d2, k	Smooth vec4 subtraction

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 sphere = sphereSDFRadius(p, 1)
              const cylinder = cylinderSDFHeightRadius(p, 0.5, 2)
              return opSubtraction(cylinder, sphere)
      })
      const march = Fn(([eye, dir]: [Vec3, Vec3]) => {
              const p = eye.toVar()
              const d = sdf(p).toVar()
              Loop(16, ({ 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)
}