Class TubeMapDensity

Class Documentation

class TubeMapDensity

Closed-form boundary density of the tube maps of Nogina and Sellán 2026 [NS26].

The boundary density at a point with signed distance d to the surface is computed by a 1D integral along the normal direction t in [-h, h] of the cubic spline kernel projected onto the tangent plane. The integrand is weighted by

  • the linear fall-off Gamma(s) = 1 - factor * s / h, s = d + t, which smoothly cuts off the density at s = h / factor, and

  • an area correction which accounts for the curvature of the surface. The order of the tube map determines the correction:

    • order 0: planar boundary, A(s) = 1

    • order 1: mean curvature H, A(s) = 1 + 0.5 H d - 0.5 H s

    • order 2: mean curvature H and Gaussian curvature K, A(s) = 1 + 0.5 H d - 0.5 H s + (2K + 0.5 H^2) s^2, together with a second kernel moment term. The curvatures are clamped to the range in which the expansion is valid (principal curvatures |k| <= 2/h, saddle-type Gaussian curvature K >= -1/(4h^2)) since discrete curvatures of fine or noisy meshes can be orders of magnitude larger.

All polynomial integrals are evaluated exactly.

References:

  • [NS26] Daria Nogina and Silvia Sellán. Tube Maps: Fast SPH Boundary Handling in Tubular Coordinates. In SIGGRAPH 2026 Conference Papers. 2026.

Public Static Functions

static double boundaryDensity(const double d, const double H, const double K, const double h, const double factor, const unsigned int order)

Compute the (unnormalized) boundary density.

Parameters

  • d – signed distance to the surface

  • H – mean curvature at the closest surface point, H = -(k1 + k2), i.e., negative for convex surfaces (sphere of radius R: H = -2/R)

  • K – Gaussian curvature at the closest surface point

  • h – support radius of the cubic spline kernel

  • factor – fall-off factor of Gamma

  • order – order of the tube map (0, 1 or 2)