Procedures

Modifies axis and angle to ensure a right handed rotation with angle $\in [0, \pi)$.

Extracts axis and angle from a rotation matrix.

Generates a random orientation in axis-angle representation. The axis is a random vector drawn from a uniform distribution on the surface of a unit sphere. The current implementation in based on the algorithm from Allen & Tildesley p. 349.

Rotates vectors about axis by angle.

Returns the rotation matrix corresponding to an axis-angle representation.

Given two frames A and B such that the orientation of frame B with respect to frame A is given by an axis-angle representation, shifts second-order tensors from A to B or B to A.

Given two frames A and B such that the orientation of frame B with respect to frame A is given by an axis-angle representation, shifts third-order tensors from A to B or B to A.

Given two frames A and B such that the orientation of frame B with respect to frame A is given by an axis-angle representation, shifts vectors from A to B or B to A.

Returns the shifter matrix corresponding to an axis-angle representation.

Converts an axis-angle representation to a direction cosine matrix.

Coverts an axis-angle representation to Euler angles.

Converts an axis-angle representation to a unit quaternion.

Adds a square matrix and its transpose in place: $A_{ij} = A_{ij } + A_{ji}$

Rotates vectors such that a set of mutually orthogonal unit vectors aligns with another set of mutually orthogonal unit vectors.

Checks if two arrays are elementwise close within tolerance

Checks if two rank-1 floating point arrays of type double are close within tolerance.

Checks if two rank-2 floating point arrays of type double are close within tolerance.

Checks if two rank-3 floating point arrays of type double are close within tolerance.

Calculates the cross product between two 3-element vectors

Calculates the cross product matrix of a 3-element vector. The cross product matrix A of a is defined as a x b = A . b, where b is another 3-element vector.

Returns the direction cosine matrix of axes(i.e. frame) B with respect to axes(i.e. frame) A.

Rotates vectors by a direction cosine matrix.

Returns the rotation matrix corresponding to a direction cosine matrix.

Given two frames A and B such that the orientation of frame B with respect to frame A is given by a direction cosine matrix, shifts second-order tesnors from A to B or B to A.

Given two frames A and B such that the orientation of frame B with respect to frame A is given by a direction cosine matrix, shifts third-order tesnors from A to B or B to A.

Given two frames A and B such that the orientation of frame B with respect to frame A is given by a direction cosine matrix, shifts vectors from A to B or B to A.

Returns the shifter matrix corresponding to a direction cosine matrix.

Converts a direction cosine matrix to an axis-angle representation.

Converts a direction cosine matrix to Euler angles.

Converts a direction cosine matrix to a unit quaternion.

Returns the determinant of an (N x N) matrix, where N = 2, 3, or 4.

Calculates the eigenvalues of a symmetric 3x3 matrix A using Cardano's analytical algorithm. Only the diagonal and upper triangular parts of A are accessed. The access is read-only.

Calculates the eigenvalues of a 3 x 3 real symmetric matrix. The eigenvalues calculated are in decreasing order. Only the diagonal and lower triangular part of the matrix is accessed.

Rotates vectors with a set of Euler angles.

Returns the rotation matrix for a set of Euler angles.

Given two frames A and B such that the orientation of frame B with respect to frame A is given by three Euler angles, shifts second-order tensors from A to B or B to A.

Given two frames A and B such that the orientation of frame B with respect to frame A is given by three Euler angles, shifts third-order tensors from A to B or B to A.

Given two frames A and B such that the orientation of frame B with respect to frame A is given by three Euler angles, shifts vectors from A to B or B to A.

Returns the shifter matrix for a set of Euler angles.

Convert Euler angles to an axis-angle representation.

Convert Euler angles to a direction cosine matrix.

Convert one set of Euler angles to another.

Convert Euler angles to a unit quaternion.

Returns the diagonal elements of a square matrix.

Calculates the quadratic form x^T A x, where A is an n x n matrix and x is a vector of length n

Creates an identity matrix of size n x n.

Inverts a 3x3 matrix.

Checks if two floating point numbers of type double are close within tolerance.

Generates evenly spaced numbers over a specified interval. Both end points are included. If start < finish, the returned step size (if step is present) will be negative.

Generates numbers spaced evenly on a log scale. In linear space, the sequence starts at base ** start (base to the power of start) and ends with base ** stop

Returns true if a (3,3) matrix is a direction cosine matrix.

Returns .true. if a (3,3) matrix is a rotation matrix.

Multiplies a matrix with its transpose: $\mathbf{\mathrm{B}} = \mathbf{\mathrm{A}} \cdot \mathbf{\mathrm{A}}^T$

Orthogonalizes a set of vectors in-place using Gram-Schmidt orthonormalization

Calculates the outer product of two vectors, $c_{ij} = a_i b_j$.

Returns the angle between two unit quaternions

Conjugates a quaternion in-place.

Calculates the time derivative of a unit quaternion from angular velocity.

Returns the matrix mapping angular velocity to time derivative of a unit quaternion.

Calculates the angular velocity from a unit quaternion and its time derivative.

Returns the matrix mapping the derivative of a unit quaternion to angular velocity.

Returns the identity unit quaternion.

Interpolates between two unit quaternions.

Inverts a quaternion in-place.

Returns .true. if quaternion is normalized, i.e. if it is a unit quaternion

Normalizes a quaternion in-place.

Calculates the product of two quaternions

Returns a random unit quaternion.

Rotates vectors by a unit quaternion.

Returns the rotation matrix corresponding to a unit quaternion.

Given two frames A and B such that the orientation of frame B with respect to frame A is given by a unit quaternion, shifts second-order tensors from A to B or B to A.

Given two frames A and B such that the orientation of frame B with respect to frame A is given by a unit quaternion, shifts third-order tensors from A to B or B to A.

Given two frames A and B such that the orientation of frame B with respect to frame A is given by a unit quaternion, shifts vectors from A to B or B to A.

Returns the shifter matrix corresponding to a unit quaternion.

Converts a unit quaternion to an axis-angle representation.

Converts a unit quaternion to a direction cosine matrix.

Converts a unit quaternion to an Euler angle sequence.

Returns the scalar triple product a.(b x c)

Calculates the difference of a square matrix and its transpose in place: $A_{ij} = A_{ij } - A_{ji}$

Swaps two arrays

Returns the trace of a square matrix

Normalizes a vector in-place. If the magnitude of the vector is nearly zero, no normalization takes place and the vector is returned as is with a warning message.

Returns the vector triple product d = a x (b x c)