Module fair 

Fairing: move chosen vertices to the smoothest surface their fixed surroundings allow.

Smoothing (crate::smooth) relaxes vertices a step at a time and never arrives anywhere in particular. Fairing solves for the arrival: the free vertices are placed where the discrete Laplacian (harmonic, a membrane) or bi-Laplacian (biharmonic, a thin plate) of the surface vanishes, with every other vertex held fixed. The biharmonic solution also matches the fixed surface’s first derivatives across the border, which is what makes a filled hole continue its surroundings tangentially instead of meeting them at a crease. This is the fairing step of Liepa (2003) and of CGAL’s Polygon_mesh_processing::fair.

The system

With symmetric edge weights w_ij the Laplacian L (L_ij = w_ij, L_ii = -sum_j w_ij) is symmetric, and with a positive diagonal mass M the biharmonic conditions (L M^-1 L x)_i = 0 for the free vertices F read

(L_{:,F})^T M^-1 L_{:,F} x_F = -(L_{:,F})^T M^-1 L_{:,C} x_C

whose matrix is a Gram matrix: symmetric positive definite whenever each connected group of free vertices touches a fixed one. It is solved for each coordinate by axiolid_numeric::conjugate_gradient; the harmonic case solves -L_FF x_F = L_FC x_C the same way. Only rows of L at the free vertices and their neighbours enter, so the cost follows the faired region, not the mesh.

Weights are computed once, on the positions before fairing. FairingWeights::Cotangent uses (cot a + cot b) / 2 with the barycentric (one-third) vertex area as mass, the standard linear finite element discretisation; FairingWeights::Uniform uses unit weights and the valence as mass (Kobbelt’s umbrella operator), so it ignores the shape of the triangles and also works on polygonal faces.

Exactness

Rows of L sum to zero, so every affine function of position is reproduced: fixed vertices that all lie in one plane keep the free ones in it, up to the solver tolerance, and in an axis-aligned plane exactly.

Structs

FairOptions

Options of fair.

FairReport

What fair did.

Enums

FairError

Why fair refused. On every refusal the mesh is unchanged.

FairingOrder

Which derivative vanishes on the faired region.

FairingWeights

Edge weights of the discrete Laplacian.

Functions

fair

Fair the given vertices of a mesh in place; every other vertex is fixed.