Boundary First Flattening

Example

import mouette as M

mesh = M.mesh.load("path/to/mesh")
bff = M.parametrization.BoundaryFirstFlattening(mesh, bnd_scale_fctr=scale, verbose=True)
mesh = bff.run() # /!\ mesh is not modified in place

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

ArrayAttribute(elem_type, n_elem, elem_size=1, default_value=None)

Bases: _BaseAttribute

init method and the whole Attribute class are not supposed to be manipulated outside of the DataContainer class. An ArrayAttribute stores its values in a numpy array. This is a less flexible but safer approach than Attribute.

Parameters:

Name	Type	Description	Default
elem_type		the type of the attribute (bool, int, float, complex, string)	required
n_elem	int	Total number of elements in the container. Should match the size of the DataContainer the attribute is stored in.	required
elem_size	int	Number of elem_type objects to be stored per element. Defaults to 1.	1
dense	bool	description. Defaults to False.	required
default_value	optional)	the default value of the attribute is n_elem is not specified. If it is not specified either, it will correspond to the default value of the type provided in elem_type.	None

__iter__()

Sparse attributes allow to iterate only over non-default elements

clear()

Empties the attribute. Frees the memory and ensures that all access return default value

empty()

Check if an attribute is empty. For an attribute to be empty, its number of elements should not be fixed, and the dictionnary should be empty

Attribute(elem_type, elem_size=1, default_value=None)

Bases: _BaseAttribute

init method and the whole Attribute class are not supposed to be manipulated outside of the DataContainer class

Parameters:

Name	Type	Description	Default
elem_type		the type of the attribute (bool, int, float, complex, string)	required
elem_size	int	Number of elem_type objects to be stored per element. Defaults to 1.	1
default_value	optional)	the default value of the attribute is n_elem is not specified. If it is not specified either, it will correspond to the default value of the type provided in elem_type.	None

__iter__()

Sparse attributes allow to iterate only over non-default elements

clear()

Empties the attribute. Frees the memory and ensures that all access return default value

empty()

Check if an attribute is empty. For an attribute to be empty, its number of elements should not be fixed, and the dictionnary should be empty

BoundaryFirstFlattening(mesh, bnd_scale_fctr=None, bnd_curvature=None, verbose=False, **kwargs)

Bases: BaseParametrization

Boundary First Flattening: A conformal flattening algorithm with control over the boundary conditions, either in terms of scale factor or curvature

References

• [1] Boundary First Flattening, Rohan Sawhney and Keenan Crane, ACM ToG, 2017

Warning

This algorithm reorders the vertices so that boundary vertices are labeled [0, N-1] in order. Therefore, the uvs coordinates are not computed on the original mesh but on a reordered copy. Access the final result with self.mesh

Parameters:

Name	Type	Description	Default
mesh	SurfaceMesh	the input surface. Should be a triangulation of a topological disk.	required
bnd_scale_fctr	Attribute	Scale factor on the boundary. Ignores values for interior vertices. Defaults to None. If provided, will automatically compute boundary curvature and ignore the bnd_curvaturea argument.	None
bnd_curvature	Attribute	Geodesic curvature on the boundary. Ignores values for interior vertices. Defaults to None. If provided (and no bnd_scale_fctr is provided), will automatically compute boundary scale factors.	None
verbose	bool	verbose mode. Defaults to False.	False

Other Parameters:

Name	Type	Description
save_on_corners	bool	if True, stores the results on face corners instead of vertices. Defaults to True
use_cotan	bool	if True, uses the cotan Laplacian instead of the connectivity Laplacian. Defaults to True.
hilbert_transform	bool	if True, extends the boundary values with the Hilbert transform

Raises:

Type	Description
Exception	if the mesh is not the triangulation of a topological disk

Note

If neither of bnd_scale_fctr and bnd_curvature are provided, the algorithm will run in default mode with scale factors = 0 on the boundary. Otherwise, provided scale factors have priority over provided curvatures.

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

run()

Run the algorithm

Raises:

Type	Description
Exception	If the mesh is not a triangulation of a topological disk

Returns:

Name	Type	Description
SurfaceMesh	SurfaceMesh	a copy of the original mesh with reordered vertices and computed uv-coordinates

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

Vec

Bases: ndarray

A simple class to manipulate vectors in mouette. Basically, it inherits from a numpy array and implements some quality of life features for 2D and 3D vectors especially.

x: float property writable

First coordinate of the vector

Returns:

Name	Type	Description
float	float	vec[0]

xy property

First two coordinates of the vector

Returns:

Name	Type	Description
float		vec[:2]

y property writable

Second coordinate of the vector

Returns:

Name	Type	Description
float		vec[1]

z property writable

Third coordinate of the vector

Returns:

Name	Type	Description
float		vec[3]

X() classmethod

The [1,0,0] vector

Returns:

Name	Type	Description
Vec		[1,0,0]

Y() classmethod

The [0,1,0] vector

Returns:

Name	Type	Description
Vec		[0,1,0]

Z() classmethod

The [0,0,1] vector

Returns:

Name	Type	Description
Vec		[0,0,1]

dot(other)

Dot product between two vectors:

$ a \cdot b = \sum_i a[i]b[i]$

Parameters:

Name	Type	Description	Default
other	Vec	other vector to dot with	required

Returns:

Name	Type	Description
float	float	the dot product

from_complex(c) classmethod

2D vector from complex number

Parameters:

Name	Type	Description	Default
c	complex		required

Returns:

Name	Type	Description
Vec		Vec(c.real, c.imag)

norm(which='l2')

Vector norm. Three norms are implemented: the Euclidean l2 norm, the l1 norm or the l-infinite norm:

l2 : $ \sqrt{ \sum_i v[i]^2 } $

l1 : $ \sum_i |v[i]| $

linf : $ \max_i |v[i]| $

Parameters:

Name	Type	Description	Default
which	str	which norm to compute. Choices are "l2", "l1" and "linf". Defaults to "l2".	'l2'

Returns:

Name	Type	Description
float	float	the vector's norm

normalize(which='l2')

Normalizes the vector to have unit norm.

Parameters:

Name	Type	Description	Default
which	str	which norm to compute. Choices are "l2", "l1" and "linf". Defaults to "l2".	'l2'

normalized(vec, which='l2') staticmethod

Computes and returns a normalized vector of the input vector vec.

Parameters:

Name	Type	Description	Default
vec	Vec	the input vector	required
which	str	which norm to compute. Choices are "l2", "l1" and "linf". Defaults to "l2".	'l2'

Returns:

Name	Type	Description
Vec		the normalized vector

outer(other)

Outer product between two vectors:

\(a \otimes b = c \in \mathbb{R}^{n \times n}\) such that \(c[i,j] = a[i]b[j]\).

Parameters:

Name	Type	Description	Default
other	Vec	the second vector	required

Returns:

Type	Description
ndarray	np.array: an array of shape (n,n)

random(n) classmethod

Generates a random vector of size n with coefficients sampled uniformly and independently in [0;1)

Parameters:

Name	Type	Description	Default
n	int	size	required

Returns:

Name	Type	Description
Vec

zeros(n) classmethod

Generates a vector of size n full of zeros.

Parameters:

Name	Type	Description	Default
n	int	size	required

Returns:

Name	Type	Description
Vec

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)