polySDF: Regular Polygon Distance Field

Parametric Regular Polygons with Variable Vertex Count

The polySDF function generates signed distance fields for regular polygons with arbitrary vertex count. This function utilizes angular subdivision and polar coordinate mathematics to create precise geometric shapes.

Mathematical Foundation

The polygon distance function operates through angular discretization:

$$d_{\text{poly}} = \cos\left(\lfloor\frac{a}{v} + 0.5\rfloor \cdot v - a\right) \cdot r$$

where:

• $a = \arctan(p_x, p_y) + \pi$ is the angle from point to origin
• $r = |p|$ is the radial distance
• $v = \frac{2\pi}{V}$ is the angular step for $V$ vertices

The floor operation discretizes the angle into $V$ equal segments, creating the polygon structure.

Function Parameters

Parameter	Type	Description
st	vec2	Sample point in normalized coordinates
V	int	Number of polygon vertices

Implementation Demonstrations

Live Editor

const fragment = () => Scope(() => {
      const center = vec2(0.5)
      const minDist = float(1).toVar()
      Loop(4, ({ i }) => {
              const sides = i.add(3)
              const offset = vec2(
                      i.mod(2).mul(0.5).add(0.25),
                      i.div(2).floor().mul(0.5).add(0.25)
              )
              const p = uv.sub(offset)
              const d = polySDFFloat(p.add(0.5), sides.toFloat()).sub(0.2).abs().sub(0.02)
              minDist.assign(minDist.min(d))
      })
      const brightness = minDist.step(0).mul(0.9).add(float(0.02).smoothstep(0, minDist.abs()).mul(0.3))
      const hue = uv.x.add(uv.y).mul(0.5)
      return vec4(brightness.mul(vec3(hue.mul(0.8).add(0.2), 0.6, 1)), 1)
})