linkSDF: Chain Link Distance Field

Toroidal Ring Geometry with Cylindrical Extension

The linkSDF function generates a chain link geometry by combining cylindrical extension with toroidal curvature. This primitive is ideal for creating interlocked rings, chains, and linked mechanical components.

Mathematical Foundation

The link SDF operates through coordinate transformation and dual-radius distance calculation:

$$\vec{q} = (p_x, \max(|p_y| - l_e, 0), p_z)$$

The final distance combines radial and axial components:

$$d_{\text{link}} = |\vec{d}| - r_2$$

where $\vec{d} = (|\vec{q}_{xy}| - r_1, q_z)$ represents the intermediate distance vector.

The geometry effectively creates a torus that is extended along the Y-axis by the length parameter $l_e$.

Function Parameters

Parameter	Type	Description
p	vec3	Sample point position
le	float	Link extension length (half-height)
r1	float	Major radius (ring radius)
r2	float	Minor radius (tube thickness)

Implementation Demonstrations

Live Editor

const fragment = () => {
      const up = vec3(0, 1, 0)
      const eps = vec3(0.01, 0, 0)
      const eye = rotate3dY(iTime).mul(vec3(5))
      const args = [0.8, 0.6, 0.2]
      const march = Fn(([eye, dir]: [Vec3, Vec3]) => {
              const p = eye.toVar()
              const d = linkSDF(p, ...args).toVar()
              Loop(16, ({ i }) => {
                      If(d.lessThanEqual(eps.x), () => {
                              const dx = linkSDF(p.add(eps.xyy), ...args).sub(d)
                              const dy = linkSDF(p.add(eps.yxy), ...args).sub(d)
                              const dz = linkSDF(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(linkSDF(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)
}