torusSDF: Three-Dimensional Torus Distance Field

Donut-Shaped Geometry with Parametric Variations

The torusSDF function family generates signed distance fields for torus (donut) geometries with both standard circular torus and sector-based variants. These primitives enable the creation of ring-shaped objects with precise radial control.

Mathematical Foundation

The standard torus distance function operates through radial projection:

$$d_{\text{torus}} = |\vec{q}| - t_y$$

where $\vec{q} = (|\vec{p}_{xz}| - t_x, p_y)$ represents the distance from the torus center line.

For sector torus with angular constraints:

$$k = \begin{cases} \text{dot}(\vec{p}_{xy}, \vec{sc}) & \text{if } sc_y \cdot |p_x| > sc_x \cdot p_y \\ |\vec{p}_{xy}| & \text{otherwise} \end{cases}$$ $$d_{\text{sector}} = \sqrt{|\vec{p}|^2 + r_a^2 - 2r_a k} - r_b$$

Function Variants

Function	Parameters	Description
torusSDF	p, t	Standard torus with major/minor radii
torusSDFSector	p, sc, ra, rb	Angular sector torus with sine/cosine constraints

Implementation Demonstrations

Live Editor

const fragment = () => {
      const up = vec3(0, 1, 0)
      const eye = vec3(5)
      const eps = vec3(0.01, 0, 0)
      const args = [1, 0.25]
      const march = Fn(([eye, dir]: [Vec3, Vec3]) => {
              const p = eye.toVar()
              const d = torusSDF(p, vec2(...args)).toVar()
              Loop(16, ({ i }) => {
                      If(d.lessThanEqual(eps.x), () => {
                              const dx = torusSDF(p.add(eps.xyy), vec2(...args)).sub(d)
                              const dy = torusSDF(p.add(eps.yxy), vec2(...args)).sub(d)
                              const dz = torusSDF(p.add(eps.yyx), vec2(...args)).sub(d)
                              return vec4(vec3(dx, dy, dz).normalize().mul(0.5).add(0.5), 1)
                      })
                      p.addAssign(d.mul(dir))
                      d.assign(torusSDF(p, vec2(...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)
}