Tutte's Embedding

Example

import mouette as M
mesh = M.mesh.load(args.model)
tutte = M.parametrization.TutteEmbedding(mesh, [boundary_mode], use_cotan=True, verbose=True)
tutte.run()
M.mesh.save(mesh, "tutte_model.obj")
M.mesh.save(tutte.flat_mesh, "tutte_flat.obj")

See https://github.com/GCoiffier/mouette/blob/main/examples/parametrization/tutte.py

PointCloud(data=None)

Bases: Mesh

A data structure for representing point clouds

Attributes:

Name	Type	Description
vertices	DataContainer	the container for all vertices
__str__	str	Representation of the object and its elements as a string.

id_vertices property

Shortcut for range(len(self.vertices))

append(x)

Shortcut for self.vertices.append(x), since we can only append elements in the 'vertices' container

PolyLine(data=None)

Bases: Mesh

A data structure for representing polylines.

Attributes:

Name	Type	Description
vertices	DataContainer	the container for all vertices
edges	DataContainer	the container for all edges
__str__		Representation of the object and its elements as a string.

id_edges property

Shortcut for range(len(self.edges))

id_vertices property

Shortcut for range(len(self.vertices))

SurfaceMesh(data=None)

Bases: Mesh

A data structure for representing polygonal surfaces.

Attributes:

Name	Type	Description
vertices	DataContainer	the container for all vertices
edges	DataContainer	the container for all edges
faces	DataContainer	the container for all faces
face_corners	DataContainer	the container for all corner of faces
boundary_edges	list	list of all edge indices on the boundary
interior_edges	list	list of all interior edge indices (all edges \ boundary_edges)
boundary_vertices	list	list of all vertex indices on the boundary
interior_vertices	list	list of all interior verticex indices (all vertices \ boundary_vertices)
connectivity	_SurfaceConnectivity	the connectivity utility class

id_corners property

Shortcut for range(len(self.face_corners))

id_edges property

Shortcut for range(len(self.edges))

id_faces property

Shortcut for range(len(self.faces))

id_vertices property

Shortcut for range(len(self.vertices))

clear_boundary_data()

Clear all boundary data. Next call to a boundary/interior container or method will recompute everything

is_edge_on_border(u, v)

whether edge (u,v) is a boundary edge or not

Parameters:

Name	Type	Description	Default
u	int	vertex id	required
v	int	vertex id	required

Returns:

Name	Type	Description
bool	bool	whether edge (u,v) is a boundary edge or not. Returns False if (u,v) is not a valid edge.

is_quad()

Returns:

Name	Type	Description
bool	bool	True if the mesh is quadrangular (all faces are quad)

is_triangular()

Returns:

Name	Type	Description
bool	bool	True if the mesh is triangular (all faces are triangles)

is_vertex_on_border(u)

whether vertex u is a boundary vertex or not.

Parameters:

Name	Type	Description	Default
u	int	vertex id	required

Returns:

Name	Type	Description
bool	bool	whether vertex u is a boundary vertex or not.

ith_vertex_of_face(fid, i)

helper function to get the i-th vertex of a face, i.e. self.faces[fid][i]

Parameters:

Name	Type	Description	Default
fid	int	face id	required
i	int	vertex id in face. Should be 0 <= vid < len(face)	required

Returns:

Name	Type	Description
int	int	the id of the i-th vertex in face fid (self.faces[fid][i])

pt_of_face(fid)

point coordinates of vertices of face fid

Parameters:

Name	Type	Description	Default
fid	int	face id	required

Returns:

Name	Type	Description
Iterable		iterator of Vec objects representing point coordinates of vertices

TutteEmbedding(mesh, boundary_mode='circle', use_cotan=False, verbose=False, **kwargs)

Bases: BaseParametrization

Tutte's embedding parametrization method for a disk inside a fixed boundary. The parametrization is locally injective (Floater, 1997) provided the boundary is convex.

References

• [1] How to draw a graph, Tutte W.T., 1963

• [2] Parametrization and smooth approximation of surface triangulations, Floater M.S., 1997

Parameters:

Name	Type	Description	Default
mesh	SurfaceMesh	the mesh to embed. Should be a surface with disk topology.	required
boundary_mode	str	Shape of the boundary. Possible choices are ["square", "circle"]. Defaults to "circle".	'circle'
use_cotan	bool	If True, uses cotangents as weights in the Laplacian matrix [2]. Otherwise, use Tutte's original barycentric embedding [1]. Defaults to False.	False
verbose	bool	verbose mode. Defaults to False.	False

Other Parameters:

Name	Type	Description
save_on_corners	bool	if True, the resulting uv-coordinates will be stored in an attribute on face corners. Otherwise, they are stored in an attribute on vertices. Defaults to True
custom_boundary	(ndarray, optionnal)	a Nx2 array containing custom coordinates for the boundary vertices (N being the number of boundary vertices).

Raises:

Type	Description
InvalidArgumentValueError	if 'boundary_mode' is not "square" or "circle".

flat_mesh: SurfaceMesh property

A flat representation of the mesh where uv-coordinates are copied to xy.

Returns:

Name	Type	Description
SurfaceMesh	SurfaceMesh	the flat mesh

BoundaryMode

Bases: Enum

Enum that represents the shape of the boundary curve.

VolumeMesh(data=None)

Bases: Mesh

id_cells property

Shortcut for range(len(self.cells))

id_corners property

Shortcut for range(len(self.face_corners))

id_edges property

Shortcut for range(len(self.edges))

id_faces property

Shortcut for range(len(self.faces))

id_vertices property

Shortcut for range(len(self.vertices))

is_edge_on_border(*args)

Simple test to determine if a given edge is on the boundary of the mesh.

Returns:

Name	Type	Description
bool	bool	Returns True if the given edge is on the boundary of the mesh.

is_face_on_border(*args)

Simple test to determine if a given face is on the boundary of the mesh.

Returns:

Name	Type	Description
bool	bool	Returns True is the given face exists and is on the boundary of the mesh

is_tetrahedral()

Returns:

Name	Type	Description
bool	bool	True if the mesh is tetrahedral (all cells are tetrahedra)