Geometry Utilities

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

circumcenter(v1, v2, v3)

Circumcenter of the triangle formed by three points in the plane

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

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]

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

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

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

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

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

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