"""This module provides functions to create quad and triangle grids."""
import numpy as np
def _quad_faces2D(array):
"""
Create quad faces of a grid from a 2D array of grid vertex indices.
Parameters
----------
array : np.ndarray of shape (n, m)
The vertex indices of the grid.
Returns
-------
faces : np.ndarray of shape (k, 4)
Faces of the quad grid, where k is the number of faces.
Examples
--------
>>> import numpy as np
>>> from ddg.math.grids import _quad_faces2D
>>> a = np.arange(3 * 4).reshape(3, 4)
>>> a
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11]])
>>> _quad_faces2D(a)
array([[ 0, 1, 5, 4],
[ 1, 2, 6, 5],
[ 2, 3, 7, 6],
[ 4, 5, 9, 8],
[ 5, 6, 10, 9],
[ 6, 7, 11, 10]])
"""
faces = np.column_stack(
(
array[:-1, :-1].ravel(),
array[:-1, 1:].ravel(),
array[1:, 1:].ravel(),
array[1:, :-1].ravel(),
)
)
return faces
def _triangle_faces2D(array):
"""
Create triangle faces of a grid from a 2D array of grid vertex indices.
Parameters
----------
array : np.ndarray of shape (n, m)
The vertex indices of the grid.
Returns
-------
faces : np.ndarray of shape (k, 4)
Faces of the quad grid, where k is the number of faces.
Examples
--------
>>> import numpy as np
>>> from ddg.math.grids import _triangle_faces2D
>>> a = np.arange(3 * 4).reshape(3, 4)
>>> a
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11]])
>>> _triangle_faces2D(a)
array([[ 0, 1, 4],
[ 1, 2, 5],
[ 2, 3, 6],
[ 4, 5, 8],
[ 5, 6, 9],
[ 6, 7, 10],
[ 1, 5, 4],
[ 2, 6, 5],
[ 3, 7, 6],
[ 5, 9, 8],
[ 6, 10, 9],
[ 7, 11, 10]])
"""
faces = np.concatenate(
(
np.column_stack(
(
array[:-1, :-1].ravel(),
array[:-1, 1:].ravel(),
array[1:, :-1].ravel(),
)
),
np.column_stack(
(
array[:-1, 1:].ravel(),
array[1:, 1:].ravel(),
array[1:, :-1].ravel(),
)
),
)
)
return faces
def _grid3D_along_axis(array, axis):
"""
Create quad faces from a 3D index array along a given axis.
This is a helper function to create faces of a 3D grid
in different axes of the index array.
Parameters
----------
array : np.ndarray of shape (n, m, o)
The 3D vertex index array.
axis : {0, 1, 2}
Axis along which the 2D grid faces of the index array is to be created.
Returns
-------
np.ndarray of shape (k, 4)
Faces of the grid, where k is the number of faces.
Examples
--------
>>> import numpy as np
>>> from ddg.math.grids import _grid3D_along_axis
>>> index_array = np.arange(2 * 3 * 2).reshape(2, 3, 2)
>>> _grid3D_along_axis(index_array, 0)
array([[ 0, 1, 3, 2],
[ 2, 3, 5, 4],
[ 6, 7, 9, 8],
[ 8, 9, 11, 10]])
This gives you the faces parallel to the x-y-plane (if default
coordinates are used).
>>> _grid3D_along_axis(index_array, 1)
array([[ 0, 1, 7, 6],
[ 2, 3, 9, 8],
[ 4, 5, 11, 10]])
This gives you the faces parallel to the x-z-plane (if default
coordinates are used).
>>> _grid3D_along_axis(index_array, 2)
array([[ 0, 2, 8, 6],
[ 2, 4, 10, 8],
[ 1, 3, 9, 7],
[ 3, 5, 11, 9]])
This gives you the faces parallel to the y-z-plane (if default
coordinates are used).
"""
return np.vstack(
[_quad_faces2D(array.take(i, axis)) for i in range(array.shape[axis])]
)
def _quad_faces3D(array):
"""
Create the face array of a 3D grid.
Parameters
----------
array : np.ndarray of shape (n, m, o)
The vertex index array.
Returns
-------
np.ndarray of shape (k, 4)
Faces of the grid, where k is the number of faces.
Examples
--------
>>> import numpy as np
>>> from ddg.math.grids import _quad_faces3D
>>> index_array = np.arange(2 * 3 * 2).reshape(2, 3, 2)
>>> index_array
array([[[ 0, 1],
[ 2, 3],
[ 4, 5]],
<BLANKLINE>
[[ 6, 7],
[ 8, 9],
[10, 11]]])
>>> _quad_faces3D(index_array)
array([[ 0, 1, 3, 2],
[ 2, 3, 5, 4],
[ 6, 7, 9, 8],
[ 8, 9, 11, 10],
[ 0, 1, 7, 6],
[ 2, 3, 9, 8],
[ 4, 5, 11, 10],
[ 0, 2, 8, 6],
[ 2, 4, 10, 8],
[ 1, 3, 9, 7],
[ 3, 5, 11, 9]])
"""
return np.vstack([_grid3D_along_axis(array, axis) for axis in range(3)])
[docs]def quad_grid(shape):
"""
Create a rectangular grid of the input shape.
Returns the quad faces and points of a box grid in Z^2 or Z^3.
Parameters
----------
shape : tuple of length 2 or 3
Shape of the grid. The first entry defines the number of vertices
in the first coordinate axis, the second the number of vertices
in the second coordinate axid and the third (if given) the
number of vertices in the third coordinate axis.
Returns
-------
faces : np.ndarray of shape (k, 4)
The quad faces of the grid, where k is the number of faces.
coordinates : np.ndarray
Default coordinates which start at (0,0) (or (0,0,0)) and expand
to the specified values of shape.
The shape of the array is (n, 2) if the input shape has length 2,
otherwise it is (n, 3).
Raises
------
ValueError
If the given shape does not have .ndim 2 or 3.
Examples
--------
>>> from ddg.math.grids import quad_grid
>>> faces, coords = quad_grid((2, 3))
>>> faces
array([[0, 1, 3, 2],
[2, 3, 5, 4]])
>>> coords
array([[0, 0],
[1, 0],
[0, 1],
[1, 1],
[0, 2],
[1, 2]])
In the 3D case we get
>>> from ddg.math.grids import quad_grid
>>> faces, coords = quad_grid((2, 2, 2))
>>> faces
array([[0, 1, 3, 2],
[4, 5, 7, 6],
[0, 1, 5, 4],
[2, 3, 7, 6],
[0, 2, 6, 4],
[1, 3, 7, 5]])
>>> coords
array([[0, 0, 0],
[1, 0, 0],
[0, 1, 0],
[1, 1, 0],
[0, 0, 1],
[1, 0, 1],
[0, 1, 1],
[1, 1, 1]])
"""
# 2D
if len(shape) == 2:
shape = (shape[1], shape[0])
n, m = shape
grid_indices = np.arange(m * n).reshape(shape)
faces = _quad_faces2D(grid_indices)
# Create xx and yy which define a complete unit index grid
# with the shape (n, m).
xx, yy = np.mgrid[0:n, 0:m]
# Use the indices to create unit coordinates for the vertices.
coordinates = np.vstack([yy.ravel(), xx.ravel()]).T
# 3D
elif len(shape) == 3:
shape = (shape[2], shape[1], shape[0])
n, m, l = shape
grid_indices = np.arange(m * n * l).reshape(shape)
faces = _quad_faces3D(grid_indices)
# Same as the 2D case just with another dimension.
xx, yy, zz = np.mgrid[0:n, 0:m, 0:l]
coordinates = np.vstack([zz.ravel(), yy.ravel(), xx.ravel()]).T
else:
raise ValueError("Quad grid must have .ndim 2 or 3")
return faces, coordinates
[docs]def triangulate_quads(quads):
"""
Subdivide quads into lower left and upper right triangles.
Parameters
----------
quads : np.ndarray of shape (n, 4)
The quad faces of the grid, where n is the number of faces.
Returns
-------
np.ndarray of shape (n*2, 3)
The triangle faces of the given quad faces, where n*2 is the number
of triangle faces.
Examples
--------
>>> import numpy as np
>>> from ddg.math.grids import triangulate_quads
>>> quad_faces = np.array([[0, 1, 4, 3], [1, 2, 5, 4]])
>>> triangulate_quads(quad_faces)
array([[0, 1, 4],
[1, 2, 5],
[0, 4, 3],
[1, 5, 4]])
"""
lower_left_triangles = quads[:, (0, 1, 2)]
upper_right_triangles = quads[:, (0, 2, 3)]
return np.vstack((lower_left_triangles, upper_right_triangles))
[docs]def triangle_grid(shape):
r"""
Create a triangle grid of the input shape.
The input shape defines the amount of vertices in each direction.
Both 2D and 3D are supported. Returns the faces and default coordinates.
Currently, the 3D version only supports `(m, n, 1)` for the shape.
A shape of (2, 3) will return the following combinatorics ::
o-----o
/ \ /
/ \ /
o-----o
/ \ /
/ \ /
o-----o
Parameters
----------
shape : tuple of length 2 or 3
Shape of the grid. The first entry defines the number of vertices
in the first coordinate axis, the second the number of vertices
in the second coordinate axid and the third (if given) the
number of vertices in the third coordinate axis.
Returns
-------
faces : np.ndarray of shape (k, 3)
The faces of the triangle grid, where k is the number of faces.
coordinates : np.ndarray
The default coordinates which start at (0,0) (or (0,0,0)) and expand
to the specified values of shape.
The shape of the array is (n, 2) if the input shape has length 2,
otherwise it is (n, 3).
The coordinates get spanned by the basis vectors
.. math::
b_1 = (2, 0), \quad b_2 = (1, 2)
See examples for more detail.
Raises
------
ValueError
If the given shape has .ndim 3 and shape[2] != 1.
ValueError
If the given shape does not have .ndim 2 or 3.
Notes
-----
Currently supports planar 3D grids only. Meaning you can only put in a shape of
(m, n, 1) or use (m, n) for 2D grids.
Examples
--------
>>> from ddg.math.grids import triangle_grid
>>> faces, coords = triangle_grid((2, 3))
>>> faces
array([[0, 1, 2],
[2, 3, 4],
[1, 3, 2],
[3, 5, 4]])
>>> coords
array([[0, 0],
[2, 0],
[1, 2],
[3, 2],
[2, 4],
[4, 4]])
In the 3D case we get
>>> faces, coords = triangle_grid((2, 3, 1))
>>> faces
array([[0, 1, 2],
[2, 3, 4],
[1, 3, 2],
[3, 5, 4]])
>>> coords
array([[0, 0, 0],
[2, 0, 0],
[1, 2, 0],
[3, 2, 0],
[2, 4, 0],
[4, 4, 0]])
"""
if len(shape) == 3 and shape[2] != 1:
raise ValueError(
"Only supports planar grid in 3D space as of now. "
+ "Use shape of (m, n, 1) or (m, n)."
)
if len(shape) == 2:
m, n = shape[1], shape[0]
grid_indices = np.arange(m * n).reshape((m, n))
faces = _triangle_faces2D(grid_indices)
xx, yy = np.mgrid[0:m, 0:n]
A = np.vstack([yy.ravel(), xx.ravel()]).T
# B is a matrix of basis vectors for transforming A, see docstring
B = np.array(((2, 1), (0, 2)))
# Apply basis transformation on the unit coordinates A
coordinates = A @ B.T
elif len(shape) == 3:
m, n, k = shape[2], shape[1], shape[0]
grid_indices = np.arange(m * n * k).reshape(n, k)
faces = _triangle_faces2D(grid_indices)
xx, yy, zz = np.mgrid[0:m, 0:n, 0:k]
A = np.vstack([zz.ravel(), yy.ravel(), xx.ravel()]).T
# B is a matrix of basis vectors for transforming A, see docstring
B = np.array(((2, 1, 0), (0, 2, 0), (0, 0, 1)))
# Apply basis transformation on the unit coordinates A
coordinates = A @ B.T
else:
raise ValueError("Triangle grid must have .ndim 2 or 3")
return faces, coordinates
[docs]def triangulated_quad_grid(shape):
r"""
Create a triangled quad grid of the input shape.
The input shape defines the number of vertices in each direction.
Both 2D and 3D are supported. Returns the faces and default coordinates.
A shape of (2, 3) will return the following combinatorics ::
o-----o
| / |
| / |
o-----o
| / |
| / |
o-----o
Parameters
----------
shape : tuple of length 2 or 3
Shape of the grid. The first entry defines the number of vertices
in the first coordinate axis, the second the number of vertices
in the second coordinate axid and the third (if given) the
number of vertices in the third coordinate axis.
Returns
-------
faces : np.ndarray of shape (k, 3)
The faces of the triangle grid, where k is the number of faces.
coordinates : np.ndarray
The default coordinates which start at (0,0) (or (0,0,0)) and expand
to the specified values of shape.
The shape of the array is (n, 2) if the input shape has length 2,
otherwise it is (n, 3).
Examples
--------
>>> from ddg.math.grids import triangulated_quad_grid
>>> faces, coords = triangulated_quad_grid((2, 3))
>>> faces
array([[0, 1, 3],
[2, 3, 5],
[0, 3, 2],
[2, 5, 4]])
>>> coords
array([[0, 0],
[1, 0],
[0, 1],
[1, 1],
[0, 2],
[1, 2]])
In the 3D case we get
>>> faces, coords = triangulated_quad_grid((2, 2, 2))
>>> faces
array([[0, 1, 3],
[4, 5, 7],
[0, 1, 5],
[2, 3, 7],
[0, 2, 6],
[1, 3, 7],
[0, 3, 2],
[4, 7, 6],
[0, 5, 4],
[2, 7, 6],
[0, 6, 4],
[1, 7, 5]])
>>> coords
array([[0, 0, 0],
[1, 0, 0],
[0, 1, 0],
[1, 1, 0],
[0, 0, 1],
[1, 0, 1],
[0, 1, 1],
[1, 1, 1]])
"""
quads, coordinates = quad_grid(shape)
faces = triangulate_quads(quads)
return faces, coordinates