pyramidSDF: Square Pyramid Distance Field

Geometric Apex Structure with Symmetric Base

The pyramidSDF function generates a signed distance field for a square pyramid geometry. This primitive combines linear tapering from a square base to an apex point, creating architectural and geometric visualizations with precise mathematical control over height and proportions.

Mathematical Foundation

The pyramid distance calculation involves coordinate transformation and multi-dimensional optimization:

$$m^2 = h^2 + 0.25$$

The algorithm performs symmetry folding and coordinate remapping:

$$\vec{q} = (p_z, h \cdot p_y - 0.5 \cdot p_x, h \cdot p_x + 0.5 \cdot p_y)$$

Distance optimization uses two candidate calculations:

$$a = m^2(q_x + s)^2 + q_y^2$$ $$b = m^2(q_x + 0.5t)^2 + (q_y - m^2t)^2$$

where $s = \max(-q_x, 0)$ and $t$ is the clamped projection parameter.

Function Signature

Parameter	Type	Description
p	vec3	Sample point position
h	float	Pyramid height (apex distance from base)

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 = rotate3dY(iTime).mul(vec3(2))
      const args = [0.7]
      const march = Fn(([eye, dir]: [Vec3, Vec3]) => {
              const p = eye.toVar()
              const d = pyramidSDF(p, ...args).toVar()
              Loop(16, ({ i }) => {
                      If(d.lessThanEqual(eps.x), () => {
                              const dx = pyramidSDF(p.add(eps.xyy), ...args).sub(d)
                              const dy = pyramidSDF(p.add(eps.yxy), ...args).sub(d)
                              const dz = pyramidSDF(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(pyramidSDF(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)
}