Module vectors

Collection of functions for vector operations.

Functions

float complex_inner_product(v1=None, v2_conj=None)
Calculate the inner product <v1|v2> for the two complex vectors v1 and v2.

random_unit_vector(vector)
Generate a random rotation axis.

numpy float64 array unit_vector_from_2point(point1, point2)
Generate the unit vector connecting point 1 to point 2.

float vector_angle_acos(vector1, vector2)
Calculate the angle between two N-dimensional vectors using the acos formula.

float vector_angle_atan2(vector1, vector2)
Calculate the angle between two N-dimensional vectors using the atan2 formula.

float vector_angle_complex_conjugate(v1=None, v2=None, v1_conj=None, v2_conj=None)
Calculate the inter-vector angle between two complex vectors using the arccos formula.

float vector_angle_normal(vector1, vector2, normal)
Calculate the directional angle between two N-dimensional vectors.

Variables

__package__ = 'lib.geometry'

Imports: acos, atan2, cos, pi, sin, array, cross, dot, float64, sqrt, norm, uniform

Function Details

complex_inner_product(v1=None, v2_conj=None)

Calculate the inner product <v1|v2> for the two complex vectors v1 and v2.

This is calculated as:

              ___
                      <v1|v2> =   >   v1i . v2i* ,
              /__
               i

where * is the complex conjugate.

Parameters:

• v1 (numpy rank-1 complex array) - The first vector.
• v2_conj (numpy rank-1 complex array) - The conjugate of the second vector. This is already in conjugate form to allow for non-standard definitions of the conjugate (for example Sm* = (-1)^m S-m).

Returns: float

The value of the inner product <v1|v2>.

random_unit_vector(vector)

Generate a random rotation axis.

Uniform point sampling on a unit sphere is used to generate a random axis orientation.

Parameters:

• vector (numpy 3D, rank-1 array) - The 3D rotation axis.

unit_vector_from_2point(point1, point2)

Generate the unit vector connecting point 1 to point 2.

Parameters:

• point1 (list of float or numpy array) - The first point.
• point2 (list of float or numpy array) - The second point.

Returns: numpy float64 array

The unit vector.

vector_angle_acos(vector1, vector2)

Calculate the angle between two N-dimensional vectors using the acos formula.

The formula is:

   angle = acos(dot(a / norm(a), b / norm(b))).

Parameters:

• vector1 (numpy rank-1 array) - The first vector.
• vector2 (numpy rank-1 array) - The second vector.

Returns: float

The angle between 0 and pi.

vector_angle_atan2(vector1, vector2)

Calculate the angle between two N-dimensional vectors using the atan2 formula.

The formula is:

   angle = atan2(norm(cross(a, b)), dot(a, b)).

This is more numerically stable for angles close to 0 or pi than the acos() formula.

Parameters:

• vector1 (numpy rank-1 array) - The first vector.
• vector2 (numpy rank-1 array) - The second vector.

Returns: float

The angle between 0 and pi.

vector_angle_complex_conjugate(v1=None, v2=None, v1_conj=None, v2_conj=None)

Calculate the inter-vector angle between two complex vectors using the arccos formula.

The formula is:

   theta = arccos(Re(<v1|v2>) / (|v1|.|v2|)) ,

where:

              ___
              \ 
   <v1|v2> =   >   v1i . v2i* ,
              /__
               i

and:

   |v1| = Re(<v1|v1>) .

Parameters:

• v1 (numpy rank-1 complex array) - The first vector.
• v2 (numpy rank-1 complex array) - The second vector.
• v1_conj (numpy rank-1 complex array) - The conjugate of the first vector. This is already in conjugate form to allow for non-standard definitions of the conjugate (for example Sm* = (-1)^m S-m).
• v2_conj (numpy rank-1 complex array) - The conjugate of the second vector. This is already in conjugate form to allow for non-standard definitions of the conjugate (for example Sm* = (-1)^m S-m).

Returns: float

The angle between 0 and pi.

vector_angle_normal(vector1, vector2, normal)

Calculate the directional angle between two N-dimensional vectors.

Parameters:

• vector1 (numpy rank-1 array) - The first vector.
• vector2 (numpy rank-1 array) - The second vector.
• normal (numpy rank-1 array) - The vector defining the plane, to determine the sign.

Returns: float

The angle between -pi and pi.