Procedural Generation

Procedural shapes

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

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)

quad(P0, P1, P2, triangulate=False)

Generates a quad from three vertices

P1------- / / / / P0-------P2

Parameters:

Name	Type	Description	Default
P0,P1,P2	Vec	coordinates of corners. The fourth point is deduced as P2 + P1 - 2*P0	required
triangulate	bool	whether to output two triangles instead of a quad. Defaults to False.	False

Returns:

Name	Type	Description
SurfaceMesh		a quad

triangle(P0, P1, P2)

Generates a triangle from three vertices

Parameters:

Name	Type	Description	Default
P0,P1,P2	Vec	three points	required

Returns:

Name	Type	Description
SurfaceMesh		a triangle

ring(N, defect, open=False, n_cover=1)

Computes a ring of triangles with prescribed number of triangles and angle defect at the center.

Parameters:

Name	Type	Description	Default
N	int	number of triangles in the ring	required
defect	float	target angle defect to achieve. Position of the central point is adjusted via dichotomy to match this value.	required
open	bool)	whether to connect the last vertex to the first.	False
n_cover	int	Number of covering of the ring. Defaults to 1.	1

Raises:

Type	Description
Exception	Fails if N<3

Returns:

Name	Type	Description
SurfaceMesh	SurfaceMesh	the ring

AABB(p_min, p_max)

Axis Aligned Bounding Box in n dimensions.

Parameters:

Name	Type	Description	Default
p_min	Iterable	minimal values for each dimension	required
p_max	Iterable	maximal values for each dimension	required

Raises:

Type	Description
Exception	fails if p_min and p_max have different sizes (inconsistent dimension)

center: Vec property

Coordinates of the center point

Returns:

Name	Type	Description
Vec	Vec	center point

dim: int property

Dimension of the axis-aligned bounding box

Returns:

Name	Type	Description
int	int	dimension

maxi: Vec property

Maximum coordinates of any point inside the box. The box is an axis-aligned hexahedron which opposite corners are mini and maxi

Returns:

Name	Type	Description
Vec	Vec	maximum coordinates

mini: Vec property

Minimum coordinates of any point inside the box The box is an axis-aligned hexahedron which opposite corners are mini and maxi

Returns:

Name	Type	Description
Vec	Vec	minimum coordinates

span: Vec property

Dimensions of the box

Returns:

Name	Type	Description
Vec	Vec	dimensions of the box

contains_point(pt)

Point - bounding box intersection predicate. If the point is on the boundary of the box, the convention is as follows: inclusive if the point touches the min coordinates, exclusive for the max coordinates

Parameters:

Name	Type	Description	Default
pt	Vec	a query position	required

Raises:

Type	Description
IncompatibleDimensionError	fails if the point has a different dimension than the bounding box.

Returns:

Name	Type	Description
bool	bool	whether the point 'pt' is inside the bounding box.

distance(pt, which='l2')

Computes the distance from a point to the bounding box.

Parameters:

Name	Type	Description	Default
pt	Vec	coordinates of the point	required
which	str	which distance to consider. Choices are 'l2', 'l1' or 'linf'. Defaults to "l2".	'l2'

Raises:

Type	Description
IncompatibleDimensionError	fails if the point has a different dimension than the bounding box.

Returns:

Name	Type	Description
float	float	the distance from the point to the bounding box

do_intersect(b1, b2) staticmethod

Intersection test between two bounding boxes

Parameters:

Name	Type	Description	Default
b1	AABB	first bounding box	required
b2	AABB	second bounding box	required

Raises:

Type	Description
IncompatibleDimensionError	fails if b1 and b2 have different dimensions

Returns:

Name	Type	Description
bool	bool	whether the two BB intersect

infinite(dim) classmethod

Computes a bounding box with bounds at infinity, containing the whole space R^n

Parameters:

Name	Type	Description	Default
dim	int	dimension of the BB to build	required

Returns:

Name	Type	Description
AABB	AABB	A bounding box containing all of R^n

intersection(b1, b2) staticmethod

Computes the intersection bounding box of two bounding boxes

Parameters:

Name	Type	Description	Default
b1	AABB	first bounding box	required
b2	AABB	second bounding box	required

Raises:

Type	Description
IncompatibleDimensionError	fails if b1 and b2 have different dimensions

Returns:

Name	Type	Description
AABB	AABB	a bounding box representing the intersection (may be empty).

is_empty()

Tests if the bounding box encloses an empty domain

Returns:

Name	Type	Description
bool	bool	whether the bounding box is empty

of_mesh(mesh, padding=0.0) classmethod

Computes the 3D bounding box of all vertices of a mesh

Parameters:

Name	Type	Description	Default
mesh	Mesh	input mesh. Can be of any type (only the 'vertices' container is accessed)	required
padding	float	slack to be added between the mesh and the box. Defaults to 0 for a tight bounding box.	0.0

Returns:

Name	Type	Description
AABB	AABB	3D bounding box of the vertices of the mesh

of_points(points, padding=0.0) classmethod

Computes the bounding box of a set of nD points.

Parameters:

Name	Type	Description	Default
points	ndarray	input points	required
padding	float	slack to be added between the mesh and the box. Defaults to 0 for a tight bounding box.	0.0

Returns:

Name	Type	Description
AABB	AABB	nD bounding box of the points

pad(pad)

Enlarges the bounding box by adding pad on each dimension on each side. Does nothing if the padding values are negative.

Parameters:

Name	Type	Description	Default
pad	float | iterable	Additional dimensions to be added. If a float is provided, will assume that padding is the same for each dimensions. If an array is provided, its size should match the dimension of the box. Values are clamped to be >=0.	required

project(pt)

Computes the closest point from point 'pt' in the bounding box in Euclidean distance

Parameters:

Name	Type	Description	Default
pt	Vec	the point to project	required

Raises:

Type	Description
IncompatibleDimensionError	fails if the point has a different dimension than the bounding box.

Returns:

Name	Type	Description
Vec	Vec	the projected point

union(b1, b2) staticmethod

Computes the union bounding box of two bounding boxes

Parameters:

Name	Type	Description	Default
b1	AABB	first bounding box	required
b2	AABB	second bounding box	required

Raises:

Type	Description
IncompatibleDimensionError	fails if b1 and b2 have different dimensions

Returns:

Name	Type	Description
AABB	AABB	a bounding box representing the union

unit_cube(dim, centered=False) classmethod

Computes the unit bounding box [0,1]^n or [-0.5, 0.5]^n

Parameters:

Name	Type	Description	Default
dim	int	dimension of the BB to build	required
centered	bool	whether to generate [0,1]^n (False) or [-0.5;0.5]^n (True). Defaults to False.	False

Returns:

Name	Type	Description
AABB	AABB	a hypercube of side length 1

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)

angle_2vec2D(V1, V2)

Angle between two 2D vectors

Parameters:

Name	Type	Description	Default
V1	Vec	first vector	required
V2	Vec	second vector	required

Returns:

Name	Type	Description
float	float	the angle

angle_2vec3D(V1, V2)

Angle between two 3D vectors

Parameters:

Name	Type	Description	Default
V1	Vec	first vector	required
V2	Vec	second vector	required

Returns:

Name	Type	Description
float	float	the angle

angle_3pts(A, B, C)

Angle ABC between three points

Parameters:

Name	Type	Description	Default
A	Vec	first point	required
B	Vec	central point	required
C	Vec	second point	required

Returns:

Name	Type	Description
float	float	the angle

aspect_ratio(A, B, C)

Computes the aspect ratio of triangle ABC, defined as the ratio between the circumradius to twice the inradius. This ratio equals 1 for equilateral triangle and goes to zero as the triangle gets close to degenerate.

Parameters:

Name	Type	Description	Default
A	Vec	first point of the triangle	required
B	Vec	second point of the triangle	required
C	Vec	third point of the triangle	required

Returns:

Name	Type	Description
float	float	the aspect ratio of triangle ABC

Note

https://stackoverflow.com/a/10290011

axis_aligned_cube(colored=False, triangulate=False)

generated an axis aligned cube as 6 quad faces.

7--------6 /| /| / | / | 4--------5 | | | | | | 3-----|--2 | / | / |/ |/ 0--------1

Parameters:

Name	Type	Description	Default
colored	bool	if set to true, will add a colo rattribute on faces to determine. Defaults to False.	False
triangulate	bool	if set to true, will triangulate the faces. Defaults to False.	False

Returns:

Name	Type	Description
SurfaceMesh	SurfaceMesh	a cube

axis_rot_from_z(v)

The (smallest) rotation to align the z axis (0,0,1) with the vector v

check_argument(argname, argvalue, expected_type, expected_values=None)

check input sanity on the argument of a function

Parameters:

Name	Type	Description	Default
argname	str	Name of the argument (for exhaustivity of error message)	required
argvalue	any	value of the argument	required
expected_type	type	expected type of the argument. If type(argvalue) does not match, will raise an InvalidArgumentTypeError	required
expected_values	list	List of possible values the argument is allowed to take. If argvalue is not in the list, will raise an InvalidArgumentValueError. Defaults to None.	None

circumcenter(v1, v2, v3)

Circumcenter of the triangle formed by three points in space

Parameters:

Name	Type	Description	Default
v1	Vec	first point	required
v2	Vec	second point	required
v3	Vec	third point	required

Warning

Circumcenter of triangle (v1,v2,v3) may not lay inside the triangle

Returns:

Name	Type	Description
Vec	Vec	coordinates of the circumcenter

cotan(A, B, C)

cotangent of the angle \(\hat{ABC}\) Parameters: A (Vec): point A B (Vec): point B C (Vec): point C

Returns:

Name	Type	Description
float	float	the cotangent

cross(A, B)

Cross product of vectors A and B

Parameters:

Name	Type	Description	Default
A	Vec	Size 3 vector	required
B	Vec	Size 3 vector	required

Returns:

Name	Type	Description
Vec	Vec	cross product AxB

cylinder(P1, P2, radius=1.0, N=50, fill_caps=True)

Generates a cylinder around the segment defined by P1 and P2

Parameters:

Name	Type	Description	Default
P1	Vec	start of the cylinder (center of bottom face)	required
P2	Vec	end of the cylinder (center of top face)	required
radius	float	radius. Defaults to 1..	1.0
N	int	number of segments. Defaults to 50.	50
fill_caps	bool	whether to also generate faces at caps. Defaults to True.	True

Returns:

Name	Type	Description
SurfaceMesh	SurfaceMesh	a cylinder

det_2x2(A, B)

Computes a 2x2 determinant

Parameters:

Name	Type	Description	Default
A	Union[complex, ndarray]	first column vector. Can also be a complex number	required
B	Union[complex, ndarray]	second column vector. Can also be a complex number	required

Returns:

Name	Type	Description
float	float	the determinant

det_3x3(*args)

Computes a 3x3 using the rule of Sarrus

Parameters:

Name	Type	Description	Default
args		Either a 3x3 numpy array representing a matrix, or 3 3x1 numpy array representing three column vectors	()

Returns:

Name	Type	Description
float	float	the determinant

distance(A, B, which='l2')

Distance between A and B. Three distances are implemented, the Euclidean distance l2, the Manhattan l1 distance of the l-infinity distance:

l2: \(\sqrt{(B-A)^T(B-A)} = \sqrt{\sum_i (A_i - B_i)^2}\)

l1: \(\sum_i |A_i - B_i|\)

linf: \(\max_i |A_i - B_i|\)

Parameters:

Name	Type	Description	Default
A	Vec	first point	required
B	Vec	second point	required
which	str	which norm to compute. Choices are "l2", "l1" and "linf". Defaults to "l2".	'l2'

Returns:

Name	Type	Description
float	float	distance

distance_to_segment2D(P, A, B)

Computes, in the plane, the distance of point P to the segment [A;B]

Parameters:

Name	Type	Description	Default
P	Vec	Query point in 2D	required
A	Vec	First segment extremity	required
B	Vec	Second segment extremity	required

Returns:

Name	Type	Description
float	float	the euclidean distance from P to [A;B]

dodecahedron()

Generate a unit dodecahedron as the dual mesh of an icosahedron

Returns:

Name	Type	Description
SurfaceMesh	SurfaceMesh	a dodecahedron

dot(A, B)

dot product \(A^TB\) between two vectors

Parameters:

Name	Type	Description	Default
A	ndarray	first vector	required
B	ndarray	second vector	required

Returns:

Name	Type	Description
float	float	dot product

face_basis(*f)

Orthonormal basis of face Given three points A,B,C, returns a basis such that the first vector is along direction AB and third vector is normal to the plane ABC

Parameters:

Name	Type	Description	Default
A,B,C		three points (in a list or not)	required

Returns:

Type	Description
	Vec,Vec,Vec: an orthonormal 3D basis of the face

hexahedron(P1, P2, P3, P4, P5, P6, P7, P8, colored=False, triangulate=False, volume=False)

Generate an hexahedron in arbitrary configuration given 8 points. Order and connectivity of points is:

7--------6 /| /| / | / | 4--------5 | | | | | | 3-----|--2 | / | / |/ |/ 0--------1

Parameters:

Name	Type	Description	Default
P1	to P8 (Vec	coordinates of eight vertices	required
colored	bool	if set to true, will add a color attribute on faces. Defaults to False.	False
triangulate	bool	if set to true, will triangulate the faces. Defaults to False.	False
volume	bool	if set to true, will also generate three tetrahedra to fill the volume. Defaults to False.	False

Returns:

Name	Type	Description
SurfaceMesh	SurfaceMesh	a cube

hexahedron_4pts(P1, P2, P3, P4, colored=False, volume=False)

Generate an hexahedron given by an absolute position and three points building a basis.

4 | | 3 | / |/ 1--------2

Parameters:

Name	Type	Description	Default
P1	to P4 (Vec	coordinates of vertices	required
colored	bool	if set to true, will add a color attribute on faces to determine. Defaults to False.	False
volume	bool	if set to true, will also generate three tetrahedra to fill the volume. Defaults to False.	False

Returns:

Name	Type	Description
SurfaceMesh	SurfaceMesh	[description]

icosahedron(center=Vec(0, 0, 0), radius=1.0, uv=False)

Generate a unit icosahedron

Parameters:

Name	Type	Description	Default
center	Vec	center position. Defaults to Vec(0,0,0).	Vec(0, 0, 0)
uv	bool	whether to generate uv coordinates. Defaults to False.	False

Returns:

Name	Type	Description
_type_		description

icosphere(n_refine=3, center=Vec(0.0, 0.0, 0.0), radius=1.0)

Generates an icosphere, that is a subdivision of the icosahedron

Parameters:

Name	Type	Description	Default
n_refine	int	number of subdivisions. Defaults to 3.	3
center	Vec	center of the sphere. Defaults to Vec(0.,0.,0.).	Vec(0.0, 0.0, 0.0)
radius	float	radius of the sphere. Defaults to 1..	1.0

Returns:

Name	Type	Description
SurfaceMesh	SurfaceMesh	description

intersect_2lines2D(p1, d1, p2, d2)

Computes the intersection of two lines in the plane

Parameters:

Name	Type	Description	Default
p1	Vec	point on the first line	required
d1	Vec	direction vector of the first line	required
p2	Vec	point on the second line	required
d2	Vec	direction vector of the second line	required

Returns:

Name	Type	Description
Vec	Vec	the intersection point. None if lines are parallel

match_rotation(Ra, Rb, symgroup=Rotation.create_group('O'), threshold=math.pi / 4)

Given two rotation matrices, compute the matching between the two: the minimal rotation going from Ra to s(Rb) with s in some symmetry group (most often the octahedral group for cube symmetries)

Parameters:

Name	Type	Description	Default
symgroup	Rotation)	a rotation group (given by the scipy.spatial.transform.Rotation.create_group method). Defaults to the octahedral group (24 direct symmetries of the cube)	create_group('O')

Returns:

Name	Type	Description
Rotation		the minimal rotation going from Ra to Rb

norm(x, 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
x	ndarray	vector from which we compute the norm. Will be flattened if it has more than one dimension.	required
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

octahedron()

Generate a unit octahedron as the dual mesh of the unit hexahedron

Reference

https://danielsieger.com/blog/2021/01/03/generating-platonic-solids.html

project_to_plane(P, N, orig)

Projects vector P onto the plane of normal N and passing through point 'orig'

Parameters:

Name	Type	Description	Default
P	Vec	query position to project	required
N	Vec	normal vector of the plane	required

Returns:

Name	Type	Description
Vec	Vec	P projected onto the plane

quad_area(A, B, C, D)

Area of the quad ABCD

Parameters:

Name	Type	Description	Default
A	Vec	first point	required
B	Vec	second point	required
C	Vec	third point	required
D	Vec	fourth point	required

Returns:

Name	Type	Description
float	float	area

rotate_2d(v, angle)

Rotates a 2D vector in the plane

Parameters:

Name	Type	Description	Default
v	Vec	the vector to rotate	required
angle	float	angle of rotation in radiants	required

Returns:

Name	Type	Description
Vec	Vec	the rotated vector

sign(x)

\[\text{sign}(x)=\begin{cases} 0 \text{ if } x=0 \\ 1 \text{ if } x>0 \\ -1 \text{ if } x<0 \end{cases}\]

Parameters:

Name	Type	Description	Default
x	float	input number	required

Returns:

Name	Type	Description
float		x/|x| and 0 for x=0

sign0(x)

\[\text{sign0}(x)=\begin{cases} 1 \text{ if } x\geqslant0 \\ -1 \text{ if } x<0 \end{cases}\]

Parameters:

Name	Type	Description	Default
x	float	input number	required

Returns:

Name	Type	Description
float		the sign of x

signed_angle_2vec3D(V1, V2, N)

Signed angle between two vectors with orientation given by normal N

Parameters:

Name	Type	Description	Default
V1	Vec	First vector	required
V2	Vec	Second vector	required
N	Vec	reference normal direction	required

Returns:

Name	Type	Description
float	float	the angle

signed_angle_3pts(A, B, C, N)

Signed angle between three points ABC with orientation givne by normal N

Parameters:

Name	Type	Description	Default
A	Vec	first point	required
B	Vec	central point	required
C	Vec	second point	required
N	Vec	reference normal direction	required

Returns:

Name	Type	Description
float	float	the angle

sphere_fibonacci(n_pts, radius=1.0, build_surface=True)

Generates a point cloud or a surface mesh using fibonacci sampling of a sphere.

Parameters:

Name	Type	Description	Default
n_pts	int	total number of vertices	required
radius	float	Radius of the sphere. Defaults to 1.	1.0
build_surface	bool	If specified to True, the function will also compute a triangulation of the vertices. This is obtained through a convex hull algorithm (since points lay on a convex shape, the convex hull and the Delaunay triangulation are equivalent). Defaults to True.	True

Returns:

Type	Description
SurfaceMesh	[SurfaceMesh | PointCloud]: the generated mesh

sphere_uv(n_lat, n_long, center=Vec(0.0, 0.0, 0.0), radius=1.0)

Generates a sphere using classical spherical uv-coordinates

Parameters:

Name	Type	Description	Default
n_lat	int	number of different latitudes for points	required
n_long	int	number of different longitudes for points	required
center	Vec	Center position of the sphere. Defaults to Vec(0.,0.,0.).	Vec(0.0, 0.0, 0.0)
radius	float	Radius of the sphere. Defaults to 1.	1.0

Returns:

Name	Type	Description
SurfaceMesh	SurfaceMesh	the sphere

tetrahedron(P1, P2, P3, P4, volume=False)

Simple tetrahedron from four points

Parameters:

Name	Type	Description	Default
P1	Vec	first point	required
P2	Vec	second point	required
P3	Vec	third point	required
P4	Vec	fourth point	required
volume	bool	whether to generate a cell or just a surface. Defaults to False.	False

Returns:

Name	Type	Description
SurfaceMesh	SurfaceMesh	description

torus(major_segments, minor_segments, major_radius, minor_radius, triangulate=False)

Generates a torus From https://danielsieger.com/blog/2021/05/03/generating-primitive-shapes.html

Parameters:

Name	Type	Description	Default
major_segments	int	number of major segments	required
minor_segments	int	number of minor segments	required
major_radius	float	global radius of the torus	required
minor_radius	float	thickness of the torus	required
triangulate	bool	whether to output a triangular or quadmesh. Defaults to False.	False

Returns:

Name	Type	Description
SurfaceMesh	SurfaceMesh	a torus

triangle_area(A, B, C)

Area of 3D triangle ABC

Parameters:

Name	Type	Description	Default
A	Vec	first point	required
B	Vec	second point	required
C	Vec	third point	required

Returns:

Name	Type	Description
float	float	area

triangle_area_2D(A, B, C)

Area of 2D triangle ABC

Parameters:

Name	Type	Description	Default
A	Vec	first point	required
B	Vec	second point	required
C	Vec	third point	required

Returns:

Name	Type	Description
float	float	area

Transformations

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

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)

cylindrify_edges(mesh, radius=0.05, N=50)

Transforms edges of a polyline as cylinder surface

Parameters:

Name	Type	Description	Default
mesh	PolyLine	the input mesh	required
radius	float	Radius of the output cylinders. Defaults to 5e-2.	0.05
N	int	Number of points inside each circle of the cylinders. Defaults to 50.	50

Returns:

Type	Description
SurfaceMesh	SurfaceMesh

spherify_vertices(points, radius=0.01, n_subdiv=1)

Transforms vertices of a point cloud as icospheres

Parameters:

Name	Type	Description	Default
points	PointCloud	the input point cloud	required
radius	float	radius of each sphere. Defaults to 1e-2.	0.01
n_subdiv	int	number of subdivisions of the icospheres. Defaults to 1.	1

Returns:

Type	Description
SurfaceMesh	SurfaceMesh

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

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)

dual_mesh(mesh)

Computes the dual mesh of a mesh

Parameters:

Name	Type	Description	Default
mesh	SurfaceMesh	input surface	required

Returns:

Type	Description
SurfaceMesh	SurfaceMesh