ddg.math.euclidean module

Utility functions for Euclidean space.

It includes an implementation of Catmull–Rom splines.

ddg.math.euclidean.intersect_diags(v1, v2, v3, v4)

For a quadrilateral(v1,v2,v3,v4) that lies in a plane, get the intersection point of the diagonals.

The four points are given as an argument in positive cyclic order.

v4-----v3
 |\   /|
 |  x  |
 |/   \|
v1-----v2

Parameters:

v1, v2, v3, v4numpy.ndarray of shape (n,)

Four points of quadrilateral vertices in positive cyclic order.

Returns:

numpy.ndarray of shape (n,)

Notes

Works for arbitrarily many dimensions as long as it is contained in a plane.

Examples

>>> from ddg.math.euclidean import intersect_diags
>>> intersect_diags((1, 0, 0), (10, 0, 0), (0, 1, 0), (0, 10, 0))
array([ 1.04455446, -0.04455446,  0.        ])

ddg.math.euclidean.intersect_edges(v1, v2, v3, v4)

For a quadrilateral (v1,v2,v3,v4) that lies in a plane, get the intersection of the two opposite edges (v1,v2) and (v3,v4).

v4-----v3--
 |     |    \
 |     |     x
 |     |    /
v1-----v2--

Parameters:

v1, v2, v3, v4numpy.ndarray of shape (n,)

Four points of quadrilateral vertices in positive cyclic order.

Returns:

numpy.ndarray of shape (n,)

Notes

Works for arbitrarily many dimensions as long as it is contained in a plane.

Examples

>>> from ddg.math.euclidean import intersect_edges
>>> intersect_edges((1, 0, 0), (10, 0, 0), (0, 1, 0), (0, 10, 0))
array([0., 0., 0.])

ddg.math.euclidean.focal_points(v1, v2, v3, v4)

For a quadrilateral (v1, v2, v3, v4), get the focal points of the quadrilateral.

These are the intersecting points of all edges.

   p2
   / \
  /   \
v4-----v3--
 |     |    \
 |     |     p1
 |     |    /
v1-----v2--

Parameters:

v1, v2, v3, v4numpy.ndarray of shape (n,)

Coordinates of the given quadrilateral.

Returns:

p1, p2numpy.ndarray of shape (n,)

Focal points of the given quadrilateral.

Examples

>>> from ddg.math.euclidean import focal_points
>>> focal_points((1, 0, 0), (10, 0, 0), (0, 1, 0), (0, 10, 0))
(array([0., 0., 0.]), array([0.90909091, 0.90909091, 0.        ]))

ddg.math.euclidean.diagonal_triangle(v1, v2, v3, v4)

For a quadrilateral (v1, v2, v3, v4), get the diagonal triangle of the quadrilateral.

These points determine the triangle formed by the intersection of all diagonals.

   p3
   / \
  /   \
v4-----v3--
 |\   /|    \
 |  p1 |     p2
 |/   \|    /
v1-----v2--

Parameters:

v1, v2, v3, v4numpy.ndarray of shape (n,)

Coordinates of the given quadrilateral.

Returns:

p1, p2, p3numpy.ndarray of shape (n,)

Diagonal triangle of the given quadrilateral.

Examples

>>> import numpy as np
>>> from ddg.math.euclidean import diagonal_triangle
>>> np.array(diagonal_triangle((1, 0, 0), (10, 0, 0), (0, 1, 0), (0, 10, 0)))
array([[ 1.04455446, -0.04455446,  0.        ],
       [ 0.        ,  0.        ,  0.        ],
       [ 0.90909091,  0.90909091,  0.        ]])

ddg.math.euclidean.christoffel_dual_quad(v1, v2, v3, v4)

Compute the Christoffel dual quadrilateral of a given quadrilateral.

Corresponding edges and non-corresponding diagonals are parallel. This determines the dual quadrilateral up to scaling.

Parameters:

v1, v2, v3, v4numpy.ndarray of shape (n,)

Coordinates of the given quadrilateral.

Returns:

V1, V2, V3, V4numpy.ndarray of shape (n,)

Coordinates of the dual quadrilateral.

Examples

>>> import numpy as np
>>> from ddg.math.euclidean import christoffel_dual_quad
>>> np.array(christoffel_dual_quad((1, 0, 0), (10, 0, 0), (0, 1, 0), (0, 10, 0)))
array([[  0.        ,   0.        ,   0.        ],
       [ 15.94937978,   0.        ,   0.        ],
       [ 15.19348501,   0.07558948,   0.        ],
       [  1.5715474 , -15.71547395,  -0.        ]])

ddg.math.euclidean.christoffel_dual_vertex_star(O, A, B, C, D, E, F, G, H)

Computes (the nine) vertex coordinates of the Christoffel dual locally around a vertex

D-----C-----B               Ds-----Cs-----Bs
|     |     |               |      |      |
|     |     |               |      |      |
E-----O-----A       ->      Es-----Os-----As
|     |     |               |      |      |
|     |     |               |      |      |
F-----G-----H               Fs-----Gs-----Hs

Parameters:

O: numpy.ndarray

A: numpy.ndarray

B: numpy.ndarray

C: numpy.ndarray

D: numpy.ndarray

E: numpy.ndarray

F: numpy.ndarray

G: numpy.ndarray

H: numpy.ndarray

Returns:

tuple of length 9

(Os, As, Bs, Cs, Ds, Es, Fs, Gs, Hs)

Notes

This function assumes local \(\mathbb{Z}^2\) combinatorics.

ddg.math.euclidean.face_normal_via_cross_product_of_diagonals(A, B, C, D)

Computes a normal of a face. The face is given as

D ----- C
|       |
|       |
|       |
A ----- B,

the normal is given by np.cross(d1, d2), the cross product of the two diagonals.

The intersection point of the diagonals is the first entry of the output, their cross product the second entry.

Parameters:

A: numpy.ndarray

B: numpy.ndarray

C: numpy.ndarray

D: numpy.ndarray

Returns:

tuple of length 2

(x, n) where x is the intersection point of the

diagonals and n the normal vector.

Notes

This function assumes that the face is a quadrilateral (see picture, not necessarily planar).

ddg.math.euclidean.vertex_normal_via_diagonal_intersection_points(O, A, B, C, D, E, F, G, H)

Computes a vertex normal of the vertex \(f(n1, n2)\) .

If M1, M2, M3 and M4 denote the intersection points of diagonals of adjacent faces to \(f(n1, n2)\) ,

D-----C-----B
|\   /|\   /|
|  M2 |  M1 |
|/   \|/   \|
E-----O-----A,     where O = :math:`f(n1, n2)` ,
|\   /|\   /|
| M3  |  M4 |
|/   \|/   \|
F-----G-----H

then the vertex normal is given by np.cross(M1 - M3, M4 - M2).

The point \(f(n1, n2)\) the first entry of the output, the normal is the second entry of the output.

Parameters:

O: numpy.ndarray

A: numpy.ndarray

B: numpy.ndarray

C: numpy.ndarray

D: numpy.ndarray

E: numpy.ndarray

F: numpy.ndarray

G: numpy.ndarray

H: numpy.ndarray

Returns:

tuple of length 2

(x, n) where x is the point \(f(n1, n2)\) and n the normal vector.

Notes

This function assumes local \(\mathbb{Z}^2\) combinatorics of the net around the vertex \(f(n1, n2)\) (see picture). If f is a discrete Koenigs net, M1, M2, M3 and M4 are co-planar. In this case the vertex normal is the face normal of the corresponding dual face of the Doliwa dual.

ddg.math.euclidean.circumcenter(v0, v1, v2)

Computes the center of the circle through the three given points in two-dimensional or three-dimensional space.

Parameters:

v0, v1, v2array_like of shape (2,) or (3,)

Points in R^2 or R^3

Returns:

numpy.ndarray of shape (2,) or (3,)

Center of the circle through the points v0, v1, v2

Examples

>>> from ddg.math.euclidean import circumcenter
>>> circumcenter((1, 0, 0), (0, 1, 0), (0, 0, 1))
array([0.33333333, 0.33333333, 0.33333333])
>>> circumcenter((1, 0), (0, 1), (0, 0))
array([0.5, 0.5])

ddg.math.euclidean.circle_through_three_points(v0, v1, v2, atol=None, rtol=None)

Computes the center, radius and normal of the circle through the three given points in three-dimensional space.

Parameters:

v0, v1, v2array_like of shape (2,) or (3,)

Points in R^2 or R^3.

atol, rtolfloat (default=None)

If None, the global defaults will be used.

Returns:

numpy.ndarray of shape (2,) (3,), float, numpy.ndarray of shape (3,) or float

Center, radius, normal of the circle through the three given points, respectively. If points are 2D then center will have shape (2,) and normal will be float.

Raises:

ValueError

If the three points given lie on a common line

Notes

Since the cross product of difference vectors of the points is computed, their dimension must be three.

If 2d points are given instead, the third component is assumed to be zero and the normal is returned as a value (type float), such that the normal vector is given by [0, 0, normal]

Examples

>>> from ddg.math.euclidean import circle_through_three_points
>>> center, radius, normal = circle_through_three_points(
...     (1, 0, 0), (0, 1, 0), (0, 0, 1)
... )
>>> center
array([0.33333333, 0.33333333, 0.33333333])
>>> radius
0.816496580927726
>>> normal
array([0.57735027, 0.57735027, 0.57735027])
>>> circle_through_three_points((1, 0), (0, 1), (2, 3))
(array([1.5, 1.5]), 1.5811388300841902, -1.0)

ddg.math.euclidean.sphere_through_four_points(p0, p1, p2, p3)

Computes the center and radius of sphere through the four given points in three-dimensional space.

Parameters:

p0, p1, p2, p3array_like of shape (3,)

Points in R^3.

Returns:

numpy.ndarray of shape (3,), float

Center, radius of the sphere respectively.

Examples

>>> from ddg.math.euclidean import sphere_through_four_points
>>> sphere_through_four_points((1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 0, 0))
([0.5, 0.5, 0.5], 0.8660254037844386)

ddg.math.euclidean.rotation_from_to(source, target, homogeneous=True, atol=None, rtol=None)

Returns a 3x3 rotation matrix taking the source vector to the target vector, or a 4x4 matrix of the type

      A | 0
T =   -----
      0 | 1

Parameters:

sourcearray_like of shape (3,)

Source vector.

targetarray_like of shape (3,)

Target vector.

homogeneous: bool, (default=True)

Determines whether 4x4 matrix or 3x3 matrix is returned.

atol, rtolfloat (default=None)

If None, the global defaults will be used.

Returns:

numpy.ndarray of shape either (3, 3) or (4, 4)

Rotation matrix taking source vector to target vector.

Raises:

ValueError

If at least one of the vectors is the zero vector (0, 0, 0).

Notes

If source and target are parallel the identety matirx is returned. If source and target are anti-parallel a correspoding 180 degree rotation is returned.

Examples

>>> from ddg.math.euclidean import rotation_from_to
>>> rotation_from_to((1, 0, 0), (0, 0, -1), homogeneous=False)
array([[ 0.,  0.,  1.],
       [ 0.,  1.,  0.],
       [-1.,  0.,  0.]])
>>> rotation_from_to((10, 0, 0), (0, 0, -1), homogeneous=False)
array([[ 0.,  0.,  1.],
       [ 0.,  1.,  0.],
       [-1.,  0.,  0.]])
>>> rotation_from_to((1, 0, 0), (0, 0, -1), homogeneous=True)
array([[ 0.,  0.,  1.,  0.],
       [ 0.,  1.,  0.,  0.],
       [-1.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  1.]])

ddg.math.euclidean.scale_rotation_from_to(source, target, homogeneous=True, atol=None, rtol=None)

Returns a 3x3 scale rotation matrix taking the source vector to the target vector, or a 4x4 matrix of the type

      A | 0
T =   -----
      0 | 1

Parameters:

sourcearray_like of shape (3,)

Source vector.

targetarray_like of shape (3,)

Target vector.

homogeneous: bool, (default=True)

Determines whether 4x4 matrix or 3x3 matrix is returned.

atol, rtolfloat (default=None)

If None, the global defaults will be used.

Returns:

numpy.ndarray of shape either (3, 3) or (4, 4)

Scale rotation matrix taking source vector to target vector.

Raises:

ValueError

If at least one of the vectors is the zero vector (0, 0, 0).

Examples

>>> from ddg.math.euclidean import scale_rotation_from_to
>>> scale_rotation_from_to((1, 0, 0), (0, 0, -1), homogeneous=False)
array([[ 0.,  0.,  1.],
       [ 0.,  1.,  0.],
       [-1.,  0.,  0.]])
>>> scale_rotation_from_to((10, 0, 0), (0, 0, -1), homogeneous=False)
array([[ 0. ,  0. ,  0.1],
       [ 0. ,  0.1,  0. ],
       [-0.1,  0. ,  0. ]])
>>> scale_rotation_from_to((1, 0, 0), (0, 0, -1), homogeneous=True)
array([[ 0.,  0.,  1.,  0.],
       [ 0.,  1.,  0.,  0.],
       [-1.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  1.]])

ddg.math.euclidean.rotation_angle_axis(rotation_vector, homogeneous=True)

This function parameterizes rotation using axis–angle representation.

It returns either 4x4 matrix of the type

      A | 0
M =   -----
      0 | 1

where A represents the rotation matrix or A as a 3x3 rotation matrix.

Parameters:

rotation_vectorarray_like of shape (3,)

A 3-dimensional vector which is co-directional to the axis of rotation and whose norm gives the angle of rotation.

homogeneous: bool, (default=True)

Determines whether 4x4 matrix or 3x3 matrix is returned.

Returns:

numpy.ndarray of shape (3, 3) or (4, 4)

Examples

>>> from ddg.math.euclidean import rotation_angle_axis
>>> rotation_angle_axis((1, 0, 0), homogeneous=False)
array([[ 1.        ,  0.        ,  0.        ],
       [ 0.        ,  0.54030231, -0.84147098],
       [ 0.        ,  0.84147098,  0.54030231]])
>>> rotation_angle_axis((0, -1, 0), homogeneous=False)
array([[ 0.54030231, -0.        , -0.84147098],
       [ 0.        ,  1.        , -0.        ],
       [ 0.84147098,  0.        ,  0.54030231]])
>>> rotation_angle_axis((1, 0, 0), homogeneous=True)
array([[ 1.        ,  0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.54030231, -0.84147098,  0.        ],
       [ 0.        ,  0.84147098,  0.54030231,  0.        ],
       [ 0.        ,  0.        ,  0.        ,  1.        ]])

ddg.math.euclidean.scaleXYZ(xyz=(1, 1, 1), homogeneous=True)

Returns either 4x4 matrix of type

      A | 0
M =   -----
      0 | 1

where A denotes diagonal scaling matrix, or A as 3x3 diagonal scaling matrix.

Parameters:

xyzarray_like of shape (3,), (default=(1, 1, 1))

Scaling factors in x, y and z directions.

homogeneous: bool, (default=True)

Determines whether 4x4 matrix or 3x3 matrix is returned.

Returns:

numpy.ndarray of shape (3, 3) or (4, 4)

Scaling matrix in either affine or homogeneous coordinates.

Examples

>>> from ddg.math.euclidean import scaleXYZ
>>> scaleXYZ((0.5, 2, 15), homogeneous=False)
array([[ 0.5,  0. ,  0. ],
       [ 0. ,  2. ,  0. ],
       [ 0. ,  0. , 15. ]])
>>> scaleXYZ((0.5, 2, 15), homogeneous=True)
array([[ 0.5,  0. ,  0. ,  0. ],
       [ 0. ,  2. ,  0. ,  0. ],
       [ 0. ,  0. , 15. ,  0. ],
       [ 0. ,  0. ,  0. ,  1. ]])

ddg.math.euclidean.translation_to(b=(0, 0, 0))

Returns 4x4 matrix of type

      I | b
M =   -----
      0 | 1

where I denotes 3x3 identity matrix and b denotes 3x1 translation vector.

Parameters:

barray_like of shape (3,), (default=(0, 0, 0))

Translation vector.

Returns:

numpy.ndarray of shape (4, 4)

Examples

>>> from ddg.math.euclidean import translation_to
>>> translation_to((0.5, 10, 15))
array([[ 1. ,  0. ,  0. ,  0.5],
       [ 0. ,  1. ,  0. , 10. ],
       [ 0. ,  0. ,  1. , 15. ],
       [ 0. ,  0. ,  0. ,  1. ]])
>>> translation_to()
array([[1., 0., 0., 0.],
       [0., 1., 0., 0.],
       [0., 0., 1., 0.],
       [0., 0., 0., 1.]])

ddg.math.euclidean.reflection_in_a_hyperplane(normal, level)

Calculate the reflection matrix (in homogeneous coordinates) of a hyperplane.

Parameters:

normalarray_like of shape (n,)

Normal of the plane

levelfloat

Level of the plane

Returns:

reflection_matrixnp.ndarray of shape (n + 1, n + 1)

Raises:

ValueError

If the given normal does not have shape (n,).

Examples

>>> import numpy as np
>>> from ddg.math.euclidean import reflection_in_a_hyperplane as re
>>> M = re((1, 0), 0)
>>> M
array([[-1.,  0.,  0.],
       [ 0.,  1.,  0.],
       [ 0.,  0.,  1.]])
>>> p = (-5, 10, 1)
>>> reflected_p = M.dot(p)
>>> reflected_p
array([ 5., 10.,  1.])

>>> M = re((1, 1, 1), 5)
>>> M
array([[ 0.33333333, -0.66666667, -0.66666667,  5.77350269],
       [-0.66666667,  0.33333333, -0.66666667,  5.77350269],
       [-0.66666667, -0.66666667,  0.33333333,  5.77350269],
       [ 0.        ,  0.        ,  0.        ,  1.        ]])
>>> p = (0, 0, 0, 1)
>>> reflected_p = M.dot(p)
>>> reflected_p
array([5.77350269, 5.77350269, 5.77350269, 1.        ])

ddg.math.euclidean.catmull_rom_spline(P0, P1, P2, P3, alpha=0.5, nPoints=100)

Computes the points of the Catmull-Rom spline segment, using four control points.

Parameters:

P0array_like of shape (2,) or (3,)

First control point, which is used to determine curvature of spline segment between P1 and P2.

P1array_like of shape (2,) or (3,)

Second control point, which forms one end of the spline segment.

P2array_like of shape (2,) or (3,)

Third control point, which forms the other end of the spline segment.

P3array_like of shape (2,) or (3,)

Fourth control point, which is used to determine curvature of spline segment between P1 and P2.

alphafloat (default=0.5)

Spline Knot parameter, ranges from 0 to 1.

nPointsint (default=100)

Number of points making up the resulting spline segment.

Returns:

numpy.ndarray of shape (nPoints, 2) or (nPoints, 3)

The Catmull-Rom spline segment between points P1 and P2.

Notes

Depending on the value of alpha, we get uniform (alpha=0), centripetal (alpha=0.5) and chordal (alpha=1) parameterization of Catmull–Rom spline.

P1 and P2 are the start- and end point respectively. P0 and P3 influence the way the spline transitions from P1 to P2. More specifically:

• P0 influences the curvature of the spline segment between P1 and P2, specifically the curvature at the beginning of the segment near P1.

• P3 influences the curvature of the spline segment between P1 and P2, specifically the curvature at the end of the segment near P2.

In addition, you can influence the Spline Knot parameter alpha, which influences the shape of the resulting curve. It’s a number that can be set between 0 and 1. Specifically, here is what happens for certain values

• alpha=0, in this the curve behaves like a uniform Catmull-Rom spline. It may not preserve the proportions or distances between control points. It’s useful when you want to create a smoothly connected curve but don’t need to maintain specific proportions.

• alpha=0.5, now the curve behaves like a centripetal Catmull-Rom spline. It takes into account the distances between control points, creating a smoother curve while preserving some proportions. It’s a good choice for most cases when you want a well-behaved, smooth curve.

• alpha=1, the curve will behave like a chordal Catmull-Rom spline. It heavily emphasizes the positions of control points and can create sharper angles near control points. It’s useful when you want the curve to closely follow the positions of the control points, even if it means less smoothness.

Examples

>>> from ddg.math.euclidean import catmull_rom_spline
>>> catmull_rom_spline((10, -1), (1, 0), (2, 1), (0, 0), alpha=0.5, nPoints=4)
array([[1.        , 0.        ],
       [1.22712485, 0.36686251],
       [1.74106915, 0.77179467],
       [2.        , 1.        ]])
>>> catmull_rom_spline((10, -1), (1, 0), (2, 1), (0, 0), alpha=1, nPoints=4)
array([[1.        , 0.        ],
       [1.35019312, 0.3632956 ],
       [1.77259491, 0.75192064],
       [2.        , 1.        ]])
>>> catmull_rom_spline((10, -1), (1, 0), (2, 1), (0, 0), alpha=0, nPoints=4)
array([[1.        , 0.        ],
       [0.7037037 , 0.40740741],
       [1.51851852, 0.81481481],
       [2.        , 1.        ]])
>>> catmull_rom_spline(
...     (10, -1, 0), (1, 0, 0), (2, 1, -1), (0, 0, 0), alpha=0.5, nPoints=4
... )
array([[ 1.        ,  0.        ,  0.        ],
       [ 1.20156546,  0.37025892, -0.35054431],
       [ 1.73689456,  0.77856239, -0.76870509],
       [ 2.        ,  1.        , -1.        ]])

ddg.math.euclidean.catmull_rom_curve(P, alpha=0.5, nPoints=4)

Calculate Catmull Rom for a chain of points and return the combined curve.

Parameters:

Parray_like of array_likes each of shape (2,) or (3,)

Chain of points from which the quadruples for the catmull_rom_spline function are taken.

alphafloat (default=0.5)

Spline Knot parameter, ranges from 0 to 1.

nPointsint (default=4)

The number of points to include in each curve segment.

Returns:

numpy.ndarray of shape (m, 2) or (m, 3)

Catmull-Rom curve made up of n-1 spline segments, each containing nPoints, i.e m = 2 - n + nPoints*(n-1) where n denotes the length of given list of points.

Notes

The resulting curve passes through all the given points.

The centripetal parameterization (alpha=0.5) of Catmull-Rom curve is the only parameterization that guarantees that the curves do not form cusps or self-intersections within its segments.

Examples

>>> from ddg.math.euclidean import catmull_rom_curve
>>> catmull_rom_curve(((10, -1), (1, 0), (2, 1), (0, 0)), alpha=0.5, nPoints=4)
array([[10.        , -1.        ],
       [ 6.3878198 , -0.74792157],
       [ 2.7756396 , -0.49584315],
       [ 1.        ,  0.        ],
       [ 1.22712485,  0.36686251],
       [ 1.74106915,  0.77179467],
       [ 2.        ,  1.        ],
       [ 1.60214111,  0.8529531 ],
       [ 0.80107055,  0.42647655],
       [ 0.        ,  0.        ]])

ddg.math.euclidean.extend_to_onb(vectors, index=0)

Extends a given list of vectors to an orthonormal basis for n-dimensional space.

Parameters:

vectorslist of numpy.ndarray of shape (n,) or a numpy.ndarray of shape (k, n)

List of linearly independent vectors to complete into an orthonormal basis.

indexint (default=0)

If only one vector is given, it’s possible to specify its index in the resulting orthonormal basis. If more than one vector is given, the first k columns of the result will have the same span as the given family of vectors.

Returns:

numpy.ndarray of shape (n, n)

Columns of resulting matrix form an orthonormal basis.

Raises:

ValueError

If vectors are given as numpy.ndarray of dimension greater than 2.

ValueError

If the given family of vectors is linearly dependent.

Notes

The given vector(s) will be normalized.

This function uses Gram-Schmidt, so it might be a bit unstable.

If the vectors are given as a 2-dimensional numpy.ndarray, the rows of the matrix will be considered as input.

Examples

>>> from ddg.math.euclidean import extend_to_onb
>>> extend_to_onb(((1, 0, 1)))
array([[ 0.70710678,  0.70710678,  0.        ],
       [ 0.        ,  0.        ,  1.        ],
       [ 0.70710678, -0.70710678,  0.        ]])
>>> extend_to_onb(((1, 0, 1)), index=2)
array([[ 0.        ,  0.70710678,  0.70710678],
       [ 1.        ,  0.        ,  0.        ],
       [ 0.        , -0.70710678,  0.70710678]])
>>> extend_to_onb(((15, 0, 0), (0, 15, 0), (0, 0, 15)))
array([[1., 0., 0.],
       [0., 1., 0.],
       [0., 0., 1.]])
>>> extend_to_onb(((1, 0, 1), (10, 15, 2)))
array([[ 0.70710678,  0.24951314,  0.66162164],
       [ 0.        ,  0.93567429, -0.35286487],
       [ 0.70710678, -0.24951314, -0.66162164]])
>>> extend_to_onb(((1, 0, 0, 0, 0)), index=2)
array([[0., 0., 1., 0., 0.],
       [0., 1., 0., 0., 0.],
       [1., 0., 0., 0., 0.],
       [0., 0., 0., 1., 0.],
       [0., 0., 0., 0., 1.]])

ddg.math.euclidean.distance(p0, p1)

Computes the Euclidean distance between two given point in n-dimensional space.

Parameters:

p0, p1array_like of shape (n,)

Points to calculate distance of.

Returns:

float

Euclidean distance, i.e l2 norm of difference between the given two point.

Examples

>>> from ddg.math.euclidean import distance
>>> distance((1, 0, 0), (0, 1, 0))
1.4142135623730951
>>> distance((15, 15, 15), (0, 0, 0))
25.98076211353316
>>> distance((1, 0, 0, 0, 0), (15, 5, 10, 2, -100))
101.61200716450787

ddg.math.euclidean.distance_lines(a, v, b, w)

Calculate the shortest distance between the lines a + sv and b + tw. This only works in 3D-space. The lines are given as

l_1 = a + sv l_2 = b + tw

Parameters:

a, bnumpy.ndarray of shape (3,)

Initial vector of line.

v, warray_like of shape (3,)

Direction vector of line.

Returns:

float

Shortest distance of line.

Examples

>>> import numpy as np
>>> from ddg.math.euclidean import distance_lines
>>> a1 = np.array([0, 0, 0])
>>> v1 = np.array([1, 0, 0])
>>> b1 = np.array([0, 1, 0])
>>> w1 = np.array([0, 0, 1])
>>> distance_lines(a1, v1, b1, w1)
1.0
>>> a2 = np.array([-1, -1, -1])
>>> v2 = np.array([1, 1, 1])
>>> b2 = np.array([0, 0, 0])
>>> w2 = np.array([0, 0, 1])
>>> distance_lines(a2, v2, b2, w2)
0.0

ddg.math.euclidean.skew_symmetric_matrix(v)

Returns the skew-symmetric matrix

 0  | -v3  |  v2
----+------+----
 v3 |  0   | -v1
----+------+----
-v2 |  v1  |  0

Parameters:

vnumpy.ndarray of shape (3,)

Returns:

numpy.ndarray of shape (3, 3)

Examples

>>> import numpy as np
>>> from ddg.math.euclidean import skew_symmetric_matrix
>>> skew_symmetric_matrix(np.array((1, 2, 3)))
array([[ 0, -3,  2],
       [ 3,  0, -1],
       [-2,  1,  0]])

ddg.math.euclidean.normalize(v)

Normalizes a given vector in n-dimensional space.

Parameters:

varray_like of shape (n,)

The vector to normalize.

Returns:

numpy.ndarray of shape (n,)

The normalized vector if it is non-zero, the zero vector otherwise.

Examples

>>> from ddg.math.euclidean import normalize
>>> normalize((15, 0, 0))
array([1., 0., 0.])
>>> normalize((1, 0, 1, 0))
array([0.70710678, 0.        , 0.70710678, 0.        ])
>>> normalize((0, 0, 0, 0, 15, 0))
array([0., 0., 0., 0., 1., 0.])

ddg.math.euclidean.embed(v, level=0, index=-1)

Embeds a coordinate vector by inserting the given level.

Parameters:

varray_like of shape (n,)

The vector to be embedded.

levelfloat (default=0)

The value to insert.

indexint (default=-1)

Index of inserted value in the output.

Returns:

numpy.ndarray of shape (n+1,)

Examples

>>> from ddg.math.euclidean import embed
>>> embed((1, 2, 3, 4), level=0, index=-1)
array([1, 2, 3, 4, 0])
>>> embed((1, 2, 3, 4), level=15, index=2)
array([ 1,  2, 15,  3,  4])

ddg.math.euclidean.angle_bisector_orientation_preserving(n1, d1, n2, d2)

Calculate the orientation preserving angle bisector of two hyperplanes.

The hyperplanes are represented by a n-dimensional normal vector and a level.

Parameters:

n1array_like of shape (n,)

Normal of first hyperplane.

d1float

Level of first hyperplane (distace to origin).

n2array_like of shape (n,)

Normal of second hyperplane.

d2float

Level of second hyperplane (distace to origin).

Returns:

n, dnumpy.ndarray of shape (n,), float

Normal and level of orientation preserving angle bisector.

ddg.math.euclidean.angle_bisector_orientation_reversing(n1, d1, n2, d2)

Calculate the orientation reversing angle bisector of two hyperplanes.

The hyperplanes are represented by a n-dimensional normal vector and a level.

Parameters:

n1array_like of shape (n,)

Normal of first hyperplane.

d1float

Level of first hyperplane (distace to origin).

n2array_like of shape (n,)

Normal of second hyperplane.

d2float

Level of second hyperplane (distace to origin).

Returns:

n, dnumpy.ndarray of shape (n,), float

Normal and level of orientation reversing angle bisector.