Curvature

Mean curvature of a mesh's interfaces, from the cotangent Laplacian, as the Laplace pressure methods read it.

Curvatures are in inverse mesh length units. An interface all of whose vertices lie on junction lines has no curvature (nan); see foambryo.pressure_inference.UndeterminedPressureWarning.

Sacha Ichbiah, 2021. Matthieu Perez, 2024.

compute_laplacian_cotan

compute_laplacian_cotan(points: NDArray[float64], triangles: NDArray[int64]) -> tuple[NDArray[np.float64], NDArray[np.float64]]

Laplacian computation.

Consider a mesh M = (V, F), with verts of shape Nx3 and faces of shape Mx3. The Laplacian matrix L is a NxN matrix such that LV gives a matrix of vectors: LV[i] gives the normal scaled by the discrete mean curvature. For vertex i, assume S[i] is the set of neighboring vertices to i, a_ij and b_ij are the "outside" angles in the two triangles connecting vertex v_i and its neighboring vertex v_j for j in S[i], as seen in the diagram below. .. code-block:: python a_ij /\ / \ / \ / \ v_i /________\ v_j \ / \ / \ / \ / \/ b_ij The definition of the Laplacian is LV[i] = sum_j w_ij (v_j - v_i) For the uniform variant, w_ij = 1 / |S[i]| For the cotangent variant, w_ij = (cot a_ij + cot b_ij) / (sum_k cot a_ik + cot b_ik) For the cotangent curvature, w_ij = (cot a_ij + cot b_ij) / (4 A[i]) where A[i] is the sum of the areas of all triangles containing vertex v_i. There is a nice trigonometry identity to compute cotangents. Consider a triangle with side lengths A, B, C and angles a, b, c. .. code-block:: python c /|\ / | \ / | \ B / H| \ A / | \ / | \ /a_____|_____b\ C Then cot a = (B^2 + C^2 - A^2) / 4 * area We know that area = CH/2, and by the law of cosines we have A^2 = B^2 + C^2 - 2BC cos a => B^2 + C^2 - A^2 = 2BC cos a Putting these together, we get: B^2 + C^2 - A^2 2BC cos a _______________ = _________ = (B/H) cos a = cos a / sin a = cot a 4 * area 2CH [1] Desbrun et al, "Implicit fairing of irregular meshes using diffusion and curvature flow", SIGGRAPH 1999. [2] Nealan et al, "Laplacian Mesh Optimization", Graphite 2006.

compute_gaussian_curvature_vertices

compute_gaussian_curvature_vertices(mesh: DcelData) -> NDArray[np.float64]

Return the discrete gaussian curvarture at every vertex of the mesh.

Parameters:

Name	Type	Description	Default
mesh	DcelData	Mesh.	required

Returns:

Type	Description
NDArray[float64]	NDArray[np.float64]: Discrete gaussian curvature at every vertex of the mesh.

compute_curvature_vertices_cotan

compute_curvature_vertices_cotan(mesh: DcelData) -> tuple[NDArray[np.float64], NDArray[np.float64], NDArray[np.float64]]

Return the mean curvature at every vertex along with other unidentified data.

compute_curvature_interfaces

compute_curvature_interfaces(mesh: DcelData, weighted: bool = True) -> dict[tuple[int, int], float]

Compute the mean curvature on the interfaces of the mesh.

The vertex mean curvature of the cotangent Laplacian, signed by the vertex normal, averaged over each interface's vertices that are not on a junction line.

Parameters:

Name	Type	Description	Default
mesh	DcelData	the mesh.	required
weighted	bool	weight each vertex by its area (PressureComputationMethod. WeightedLaplace) rather than equally (Laplace). Defaults to True.	True

Returns:

Type	Description
dict[tuple[int, int], float]	dict[tuple[int, int], float]: interface (label_1, label_2) -> mean curvature, in inverse mesh length units; nan for an interface all of whose vertices lie on junction lines.

laplacian_cot

laplacian_cot(points: NDArray[float64], triangles: NDArray[int64]) -> tuple[NDArray[np.float64], NDArray[np.float64]]

Returns the Laplacian matrix with cotangent weights and the inverse of the face areas.

Parameters:

Name	Type	Description	Default
points	NDArray[float64]	vertices of the mesh.	required
triangles	NDArray[int64]	triangles of the mesh (topology).	required

Returns:

Type	Description
NDArray[float64]	tuple[NDArray[np.float64], NDArray[np.float64]]: the Laplacian matrix L, of shape
NDArray[float64]	(V, V), with L[i, j] = cot a_ij + cot b_ij if (i, j) is an edge of the mesh; and the
tuple[NDArray[float64], NDArray[float64]]	inverse of the sum of the areas of the faces around each vertex, of shape (V,).

sphere_fit_residue

sphere_fit_residue(points: NDArray[float64]) -> float

Get residue from the fit of a sphere from a sample of points in 3D using least squares method.

From https://jekel.me/2015/Least-Squares-Sphere-Fit/ .

Parameters:

Name	Type	Description	Default
points	NDArray[float64]	sample of points in 3D	required

Returns:

Name	Type	Description
float	float	reside from least squares method.

compute_sphere_fit_residues_dict

compute_sphere_fit_residues_dict(graph: Graph) -> dict

Compute a map edge of the graph -> residue from sphere fitting.

Parameters:

Name	Type	Description	Default
graph	Graph	Graph constructed from the DcelData.	required

Returns:

Name	Type	Description
dict	dict	map an edge of the graph to a fit residue.

compute_areas_interfaces

compute_areas_interfaces(mesh: DcelData) -> dict[tuple[int, int], float]

Compute the area of each interface.

Parameters:

Name	Type	Description	Default
mesh	DcelData	Mesh	required

Returns:

Type	Description
dict[tuple[int, int], float]	dict[tuple[int, int], float]: map interface defined by two labels -> area.

compute_triangles_areas

compute_triangles_areas(points: NDArray[float64], triangles: NDArray[int64]) -> NDArray[np.float64]

Compute the area of every triangles in a mesh.

Parameters:

Name	Type	Description	Default
points	NDArray[float64]	points of a mesh.	required
triangles	NDArray[float64]	triangles (topology) of the mesh.	required

Returns:

Type	Description
NDArray[float64]	the areas for each triangle.