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

References

https://stackoverflow.com/a/10290011

`circumcenter(v1, v2, v3)`

Circumcenter of the triangle formed by three points

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

`closest_point_on_plane(P, orig, N)`

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
`orig`	`Vec`	a point belonging to the plane	required
`N`	`Vec`	normal vector of the plane	required

Returns:

Name	Type	Description
`Vec`	`Vec`	P projected onto the plane

`closest_point_on_segment(P, A, B)`

Computes the closest point from p on segment [A;B], and the feature of the segment on which it lies.

Parameters:

Name	Type	Description	Default
`P`	`Vec`	Query point	required
`A`	`Vec`	First segment extremity	required
`B`	`Vec`	Second segment extremity	required

Returns:

Name	Type	Description
`Vec`		the point Q on [A;B] that minimizes its distance with P.
`int`		the feature of the segment on which Q lies: 0 if Q is in the interior of the segment, 1 if Q is the extremity A, 2 if Q is the extremity B.

`closest_point_on_triangle(P, A, B, C)`

Computes the closest point from point p to triangle ABC, and the feature of the triangle on which it lies. Uses the Voronoi region tests from Ericson, Real-Time Collision Detection, section 5.1.5.

Parameters:

Name	Type	Description	Default
`P`	`Vec`	Query point	required
`A,B,C`	`Vec`	vertices of the triangle	required

Returns:

Name	Type	Description
`Vec`		the point Q in triangle ABC that minimizes its distance with P.
`int`		the feature of the triangle on which Q lies: 0 if Q is in the interior of the face, 1, 2 or 3 if Q is the vertex A, B or C respectively, 4, 5 or 6 if Q is in the interior of the edge AB, BC or CA respectively.

`cotan(A, B, C)`

cotangent of the angle \(\hat{ABC}\)

Parameters:

Name	Type	Description	Default
`A`	`Vec`	point A	required
`B`	`Vec`	point B	required
`C`	`Vec`	point C	required

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(A, B, C)`

Computes a 3x3 determinant using the rule of Sarrus from three column vectors.

Parameters:

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

Returns:

Name	Type	Description
`float`	`float`	the determinant of [A\|B\|C]

`distance(A, B)`

Distance between A and B.

Parameters:

Name	Type	Description	Default
`A`	`Vec`	first point	required
`B`	`Vec`	second point	required

Returns:

Name	Type	Description
`float`	`float`	distance

`distance2(A, B)`

Squared euclidean distance between points A and B.

Parameters:

Name	Type	Description	Default
`A`	`Vec`	first vector	required
`B`	`Vec`	second point	required

Returns:

Name	Type	Description
`float`	`float`	distance squared.

`distance_to_plane(P, orig, N)`

Computes the distance of point P to the plane of normal N passing through point orig.

Parameters:

Name	Type	Description	Default
`P`	`Vec`	query position to project	required
`orig`	`Vec`	a point belonging to the plane	required
`N`	`Vec`	normal vector of the plane. Does not need to be normalized.	required

Raises:

Type	Description
`ValueError`	if the norm of N is smaller than 1e-10.

Returns:

Name	Type	Description
`float`	`float`	the euclidean distance from P to the plane.

`distance_to_segment(P, A, B)`

Computes the distance of point P to the segment [A;B]. Alias for distance(P, closest_point_on_segment(P,A,B)[0])

Parameters:

Name	Type	Description	Default
`P`	`Vec`	Query point	required
`A`	`Vec`	First segment extremity	required
`B`	`Vec`	Second segment extremity	required

Returns:

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

`distance_to_triangle(P, A, B, C)`

Computes the distance of point P to the triangle ABC. Alias for distance(P, closest_point_on_triangle(P,A,B,C)[0])

Parameters:

Name	Type	Description	Default
`P`	`Vec`	query point	required
`A,B,C`	`Vec`	vertices of the triangle.	required

Returns:

Name	Type	Description
`float`	`float`	float: the euclidean distance from P to triangle ABC.

`face_basis(A, B, C)`

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`	`Vec`	three points forming a triangle.	required

Returns:

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

`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

`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