tetrahedronSDF: Regular Tetrahedron Distance Field

Four-Faced Platonic Solid Geometry

The tetrahedronSDF function computes the signed distance from any 3D point to a regular tetrahedron. This primitive represents the simplest of the Platonic solids, featuring four triangular faces meeting at each vertex with perfect symmetry.

Mathematical Foundation

The tetrahedron SDF calculation involves distance computation from multiple planes:

$$d_1 = |q_z| - \max(0, y)$$ $$d_2 = \max(0.5|q_x| + 0.5y, 0) - \min(h, h + y)$$ $$d_{\text{tetrahedron}} = \text{length}(\max(\vec{d}, 0.005)) + \min(\max(d_1, d_2), 0)$$

where $\vec{q} = |p|$ represents the absolute coordinates and $y = p_y$ is the vertical component.

Function Signature

Parameter	Type	Description
p	vec3	Sample point position
h	float	Tetrahedron height parameter

Implementation Demonstrations

ライブエディター

const fragment = () => {
      const up = vec3(0, 1, 0)
      const eps = vec3(0.01, 0, 0)
      const eye = rotate3dX(iTime).mul(vec3(4))
      const args = [0.5]
      const march = Fn(([eye, dir]: [Vec3, Vec3]) => {
              const p = eye.toVar()
              const d = tetrahedronSDF(p, ...args).toVar()
              Loop(128, ({ i }) => {
                      If(d.lessThanEqual(eps.x), () => {
                              const dx = tetrahedronSDF(p.add(eps.xyy), ...args).sub(d)
                              const dy = tetrahedronSDF(p.add(eps.yxy), ...args).sub(d)
                              const dz = tetrahedronSDF(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(tetrahedronSDF(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)
}

結果

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