# Source code for spatialmath.base.quaternions

    #!/usr/bin/env python3
    # -*- coding: utf-8 -*-
    """
    Created on Fri Apr 10 14:12:56 2020

    @author: Peter Corke
    """

    # This file is part of the SpatialMath toolbox for Python
    # https://github.com/petercorke/spatialmath-python
    #
    # MIT License
    #
    # Copyright (c) 1993-2020 Peter Corke
    #
    # Permission is hereby granted, free of charge, to any person obtaining a copy
    # of this software and associated documentation files (the "Software"), to deal
    # in the Software without restriction, including without limitation the rights
    # to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
    # copies of the Software, and to permit persons to whom the Software is
    # furnished to do so, subject to the following conditions:
    #
    # The above copyright notice and this permission notice shall be included in all
    # copies or substantial portions of the Software.
    #
    # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
    # IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
    # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
    # AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
    # LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
    # OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
    # SOFTWARE.

    # Contributors:
    #
    #     1. Luis Fernando Lara Tobar and Peter Corke, 2008
    #     2. Josh Carrigg Hodson, Aditya Dua, Chee Ho Chan, 2017 (robopy)
    #     3. Peter Corke, 2020

    import sys
    import math
    import numpy as np
    from spatialmath import base as tr
    from spatialmath.base import argcheck

    _eps = np.finfo(np.float64).eps

    def eye():
        """
        Create an identity quaternion

        :return: an identity quaternion
        :rtype: numpy.ndarray, shape=(4,)

        Creates an identity quaternion, with the scalar part equal to one, and
        a zero vector value.

        """
        return np.r_[1, 0, 0, 0]

    def pure(v):
        """
        Create a pure quaternion

        :arg v: vector from a 3-vector
        :type v: array_like
        :return: pure quaternion
        :rtype: numpy.ndarray, shape=(4,)

        Creates a pure quaternion, with a zero scalar value and the vector part
        equal to the passed vector value.

        """
        v = argcheck.getvector(v, 3)
        return np.r_[0, v]

    def qnorm(q):
        r"""
        Norm of a quaternion

        :arg q: input quaternion as a 4-vector
        :type v: : array_like
        :return: norm of the quaternion
        :rtype: float

        Returns the norm, length or magnitude of the input quaternion which is
        :math:`\sqrt{s^2 + v_x^2 + v_y^2 + v_z^2}`

        :seealso: unit

        """
        q = argcheck.getvector(q, 4)
        return np.linalg.norm(q)

    def unit(q, tol=10):
        """
        Create a unit quaternion

        :arg v: quaterion as a 4-vector
        :type v: array_like
        :return: a pure quaternion
        :rtype: numpy.ndarray, shape=(4,)

        Creates a unit quaternion, with unit norm, by scaling the input quaternion.

        .. seealso:: norm
        """
        q = argcheck.getvector(q, 4)
        nm = np.linalg.norm(q)
        assert abs(nm) > tol * _eps, 'cannot normalize (near) zero length quaternion'
        return q / nm

    def isunit(q, tol=10):
        """
        Test if quaternion has unit length

        :param v: quaternion as a 4-vector
        :type v: array_like
        :param tol: tolerance in units of eps
        :type tol: float
        :return: whether quaternion has unit length
        :rtype: bool

        :seealso: unit
        """
        return tr.iszerovec(q)

    def isequal(q1, q2, tol=100, unitq=False):
        """
        Test if quaternions are equal

        :param q1: quaternion as a 4-vector
        :type q1: array_like
        :param q2: quaternion as a 4-vector
        :type q2: array_like
        :param unitq: quaternions are unit quaternions
        :type unitq: bool
        :param tol: tolerance in units of eps
        :type tol: float
        :return: whether quaternion has unit length
        :rtype: bool

        Tests if two quaternions are equal.

        For unit-quaternions ``unitq=True`` the double mapping is taken into account,
        that is ``q`` and ``-q`` represent the same orientation and ``isequal(q, -q, unitq=True)`` will
        return ``True``.
        """
        q1 = argcheck.getvector(q1, 4)
        q2 = argcheck.getvector(q2, 4)

        if unit:
            return (np.sum(np.abs(q1 - q2)) < tol * _eps) or (np.sum(np.abs(q1 + q2)) < tol * _eps)
        else:
            return (np.sum(np.abs(q1 - q2)) < tol * _eps)

    def q2v(q):
        """
        Convert unit-quaternion to 3-vector

        :arg q: unit-quaternion as a 4-vector
        :type v: array_like
        :return: a unique 3-vector
        :rtype: numpy.ndarray, shape=(3,)

        Returns a unique 3-vector representing the input unit-quaternion. The sign
        of the scalar part is made positive, if necessary by multiplying the
        entire quaternion by -1, then the vector part is taken.

        .. warning:: There is no check that the passed value is a unit-quaternion.

        .. seealso:: v2q

        """
        q = argcheck.getvector(q, 4)
        if q[0] >= 0:
            return q[1:4]
        else:
            return -q[1:4]

    def v2q(v):
        r"""
        Convert 3-vector to unit-quaternion

        :arg v: vector part of unit quaternion, a 3-vector
        :type v: array_like
        :return: a unit quaternion
        :rtype: numpy.ndarray, shape=(4,)

        Returns a unit-quaternion reconsituted from just its vector part.  Assumes
        that the scalar part was positive, so :math:`s = \sqrt{1-||v||}`.

        .. seealso:: q2v
        """
        v = argcheck.getvector(v, 3)
        s = math.sqrt(1 - np.sum(v**2))
        return np.r_[s, v]

    def qqmul(q1, q2):
        """
        Quaternion multiplication

        :arg q0: left-hand quaternion as a 4-vector
        :type q0: : array_like
        :arg q1: right-hand quaternion as a 4-vector
        :type q1: array_like
        :return: quaternion product
        :rtype: numpy.ndarray, shape=(4,)

        This is the quaternion or Hamilton product.  If both operands are unit-quaternions then
        the product will be a unit-quaternion.

        :seealso: qvmul, inner, vvmul

        """
        q1 = argcheck.getvector(q1, 4)
        q2 = argcheck.getvector(q2, 4)
        s1 = q1[0]
        v1 = q1[1:4]
        s2 = q2[0]
        v2 = q2[1:4]

        return np.r_[s1 * s2 - np.dot(v1, v2), s1 * v2 + s2 * v1 + np.cross(v1, v2)]

    def inner(q1, q2):
        """
        Quaternion innert product

        :arg q0: quaternion as a 4-vector
        :type q0: : array_like
        :arg q1: uaternion as a 4-vector
        :type q1: array_like
        :return: inner product
        :rtype: numpy.ndarray, shape=(4,)

        This is the inner or dot product of two quaternions, it is the sum of the element-wise
        product.

        :seealso: qvmul

        """
        q1 = argcheck.getvector(q1, 4)
        q2 = argcheck.getvector(q2, 4)

        return np.dot(q1, q2)

    def qvmul(q, v):
        """
        Vector rotation

        :arg q: unit-quaternion as a 4-vector
        :type q: array_like
        :arg v: 3-vector to be rotated
        :type v: list, tuple, numpy.ndarray
        :return: rotated 3-vector
        :rtype: numpy.ndarray, shape=(3,)

        The vector `v` is rotated about the origin by the SO(3) equivalent of the unit
        quaternion.

        .. warning:: There is no check that the passed value is a unit-quaternions.

        :seealso: qvmul
        """
        q = argcheck.getvector(q, 4)
        v = argcheck.getvector(v, 3)
        qv = qqmul(q, qqmul(pure(v), conj(q)))
        return qv[1:4]

    def vvmul(qa, qb):
        """
        Quaternion multiplication

        :arg qa: left-hand quaternion as a 3-vector
        :type qa: : array_like
        :arg qb: right-hand quaternion as a 3-vector
        :type qb: array_like
        :return: quaternion product
        :rtype: numpy.ndarray, shape=(3,)

        This is the quaternion or Hamilton product of unit-quaternions defined only
        by their vector components.  The product will be a unit-quaternion, defined only
        by its vector component.

        :seealso: qvmul, inner
        """
        t6 = math.sqrt(1.0 - np.sum(qa**2))
        t11 = math.sqrt(1.0 - np.sum(qb**2))
        return np.r_[qa[1] * qb[2] - qb[1] * qa[2] + qb[0] * t6 + qa[0] * t11, -qa[0] * qb[2] + qb[0] * qa[2] + qb[1] * t6 + qa[1] * t11, qa[0] * qb[1] - qb[0] * qa[1] + qb[2] * t6 + qa[2] * t11]

    def pow(q, power):
        """
        Raise quaternion to a power

        :arg q: quaternion as a 4-vector
        :type v: array_like
        :arg power: exponent
        :type power: int
        :return: input quaternion raised to the specified power
        :rtype: numpy.ndarray, shape=(4,)

        Raises a quaternion to the specified power using repeated multiplication.

        Notes:

        - power must be an integer
        - power can be negative, in which case the conjugate is taken

        """
        q = argcheck.getvector(q, 4)
        assert isinstance(power, int), "Power must be an integer"
        qr = eye()
        for i in range(0, abs(power)):
            qr = qqmul(qr, q)

        if power < 0:
            qr = conj(qr)

        return qr

    def conj(q):
        """
        Quaternion conjugate

        :arg q: quaternion as a 4-vector
        :type v: array_like
        :return: conjugate of input quaternion
        :rtype: numpy.ndarray, shape=(4,)

        Conjugate of quaternion, the vector part is negated.

        """
        q = argcheck.getvector(q, 4)
        return np.r_[q[0], -q[1:4]]

    def q2r(q):
        """
        Convert unit-quaternion to SO(3) rotation matrix

        :arg q: unit-quaternion as a 4-vector
        :type v: array_like
        :return: corresponding SO(3) rotation matrix
        :rtype: numpy.ndarray, shape=(3,3)

        Returns an SO(3) rotation matrix corresponding to this unit-quaternion.

        .. warning:: There is no check that the passed value is a unit-quaternion.

        :seealso: r2q

        """
        q = argcheck.getvector(q, 4)
        s = q[0]
        x = q[1]
        y = q[2]
        z = q[3]
        return np.array([[1 - 2 * (y ** 2 + z ** 2), 2 * (x * y - s * z), 2 * (x * z + s * y)],
                         [2 * (x * y + s * z), 1 - 2 * (x ** 2 + z ** 2), 2 * (y * z - s * x)],
                         [2 * (x * z - s * y), 2 * (y * z + s * x), 1 - 2 * (x ** 2 + y ** 2)]])

    def r2q(R, check=True):
        """
        Convert SO(3) rotation matrix to unit-quaternion

        :arg R: rotation matrix
        :type R: numpy.ndarray, shape=(3,3)
        :return: unit-quaternion
        :rtype: numpy.ndarray, shape=(3,)

        Returns a unit-quaternion corresponding to the input SO(3) rotation matrix.

        .. warning:: There is no check that the passed matrix is a valid rotation matrix.

        :seealso: q2r

        """
        assert R.shape == (3, 3) and tr.isR(R), "Argument must be 3x3 rotation matrix"
        qs = math.sqrt(np.trace(R) + 1) / 2.0
        kx = R[2, 1] - R[1, 2]  # Oz - Ay
        ky = R[0, 2] - R[2, 0]  # Ax - Nz
        kz = R[1, 0] - R[0, 1]  # Ny - Ox

        if (R[0, 0] >= R[1, 1]) and (R[0, 0] >= R[2, 2]):
            kx1 = R[0, 0] - R[1, 1] - R[2, 2] + 1  # Nx - Oy - Az + 1
            ky1 = R[1, 0] + R[0, 1]  # Ny + Ox
            kz1 = R[2, 0] + R[0, 2]  # Nz + Ax
            add = (kx >= 0)
        elif R[1, 1] >= R[2, 2]:
            kx1 = R[1, 0] + R[0, 1]  # Ny + Ox
            ky1 = R[1, 1] - R[0, 0] - R[2, 2] + 1  # Oy - Nx - Az + 1
            kz1 = R[2, 1] + R[1, 2]  # Oz + Ay
            add = (ky >= 0)
        else:
            kx1 = R[2, 0] + R[0, 2]  # Nz + Ax
            ky1 = R[2, 1] + R[1, 2]  # Oz + Ay
            kz1 = R[2, 2] - R[0, 0] - R[1, 1] + 1  # Az - Nx - Oy + 1
            add = (kz >= 0)

        if add:
            kx = kx + kx1
            ky = ky + ky1
            kz = kz + kz1
        else:
            kx = kx - kx1
            ky = ky - ky1
            kz = kz - kz1

        kv = np.r_[kx, ky, kz]
        nm = np.linalg.norm(kv)
        if abs(nm) < 100 * _eps:
            return eye()
        else:
            return np.r_[qs, (math.sqrt(1.0 - qs ** 2) / nm) * kv]

    def slerp(q0, q1, s, shortest=False):
        """
        Quaternion conjugate

        :arg q0: initial unit quaternion as a 4-vector
        :type q0: array_like
        :arg q1: final unit quaternion as a 4-vector
        :type q1: array_like
        :arg s: interpolation coefficient in the range [0,1]
        :type s: float
        :arg shortest: choose shortest distance [default False]
        :type shortest: bool
        :return: interpolated unit-quaternion
        :rtype: numpy.ndarray, shape=(4,)

        An interpolated quaternion between ``q0`` when ``s`` = 0 to ``q1`` when ``s`` = 1.

        Interpolation is performed on a great circle on a 4D hypersphere. This is
        a rotation about a single fixed axis in space which yields the straightest
        and shortest path between two points.

        For large rotations the path may be the *long way around* the circle,
        the option ``'shortest'`` ensures always the shortest path.

        .. warning:: There is no check that the passed values are unit-quaternions.

        """
        assert 0 <= s <= 1, 's must be in the interval [0,1]'
        q0 = argcheck.getvector(q0, 4)
        q1 = argcheck.getvector(q1, 4)

        if s == 0:
            return q0
        elif s == 1:
            return q1

        dot = np.dot(q0, q1)

        # If the dot product is negative, the quaternions
        # have opposite handed-ness and slerp won't take
        # the shorter path. Fix by reversing one quaternion.
        if shortest:
            if dot < 0:
                q0 = - q0
                dot = -dot

        dot = np.clip(dot, -1, 1)  # Clip within domain of acos()
        theta = math.acos(dot)  # theta is the angle between rotation vectors
        if abs(theta) > 10*_eps:
            s0 = math.sin((1 - s) * theta)
            s1 = math.sin(s * theta)
            return ((q0 * s0) + (q1 * s1)) / math.sin(theta)
        else:
            # quaternions are identical
            return q0

    def rand():
        """
        Random unit-quaternion

        :return: random unit-quaternion
        :rtype: numpy.ndarray, shape=(4,)

        Computes a uniformly distributed random unit-quaternion which can be
        considered equivalent to a random SO(3) rotation.
        """
        u = np.random.uniform(low=0, high=1, size=3)  # get 3 random numbers in [0,1]
        return np.r_[
            math.sqrt(1 - u[0]) * math.sin(2 * math.pi * u[1]),
            math.sqrt(1 - u[0]) * math.cos(2 * math.pi * u[1]),
            math.sqrt(u[0]) * math.sin(2 * math.pi * u[2]),
            math.sqrt(u[0]) * math.cos(2 * math.pi * u[2])]

    def matrix(q):
        """
        Convert to 4x4 matrix equivalent

        :arg q: quaternion as a 4-vector
        :type v: array_like
        :return: equivalent matrix
        :rtype: numpy.ndarray, shape=(4,4)

        Hamilton multiplication between two quaternions can be considered as a
        matrix-vector product, the left-hand quaternion is represented by an
        equivalent 4x4 matrix and the right-hand quaternion as 4x1 column vector.

        :seealso: qqmul

        """
        q = argcheck.getvector(q, 4)
        s = q[0]
        x = q[1]
        y = q[2]
        z = q[3]
        return np.array([[s, -x, -y, -z],
                         [x, s, -z, y],
                         [y, z, s, -x],
                         [z, -y, x, s]])

    def dot(q, w):
        """
        Rate of change of unit-quaternion

        :arg q0: unit-quaternion as a 4-vector
        :type q0: array_like
        :arg w: angular velocity in world frame as a 3-vector
        :type w: array_like
        :return: rate of change of unit quaternion
        :rtype: numpy.ndarray, shape=(4,)

        ``dot(q, w)`` is the rate of change of the elements of the unit quaternion ``q``
        which represents the orientation of a body frame with angular velocity ``w`` in
        the world frame.

        .. warning:: There is no check that the passed values are unit-quaternions.

        """
        q = argcheck.getvector(q, 4)
        w = argcheck.getvector(w, 3)
        E = q[0] * (np.eye(3, 3)) - tr.skew(q[1:4])
        return 0.5 * np.r_[-np.dot(q[1:4], w), E@w]

    def dotb(q, w):
        """
        Rate of change of unit-quaternion

        :arg q0: unit-quaternion as a 4-vector
        :type q0: array_like
        :arg w: angular velocity in body frame as a 3-vector
        :type w: array_like
        :return: rate of change of unit quaternion
        :rtype: numpy.ndarray, shape=(4,)

        ``dot(q, w)`` is the rate of change of the elements of the unit quaternion ``q``
        which represents the orientation of a body frame with angular velocity ``w`` in
        the body frame.

        .. warning:: There is no check that the passed values are unit-quaternions.

        """
        q = argcheck.getvector(q, 4)
        w = argcheck.getvector(w, 3)
        E = q[0] * (np.eye(3, 3)) + tr.skew(q[1:4])
        return 0.5 * np.r_[-np.dot(q[1:4], w), E@w]

    def angle(q1, q2):
        """
        Angle between two unit-quaternions

        :arg q0: unit-quaternion as a 4-vector
        :type q0: array_like
        :arg q1: unit-quaternion as a 4-vector
        :type q1: array_like
        :return: angle between the rotations [radians]
        :rtype: float

        If each of the input quaternions is considered a rotated coordinate
        frame, then the angle is the smallest rotation required about a fixed
        axis, to rotate the first frame into the second.

        References:  Metrics for 3D rotations: comparison and analysis,
        Du Q. Huynh, % J.Math Imaging Vis. DOFI 10.1007/s10851-009-0161-2.

        .. warning:: There is no check that the passed values are unit-quaternions.

        """
        # TODO different methods

        q1 = argcheck.getvector(q1, 4)
        q2 = argcheck.getvector(q2, 4)
        return 2.0 * math.atan2(norm(q1 - q2), norm(q1 + q2))

    def qprint(q, delim=('<', '>'), fmt='%f', file=sys.stdout):
        """
        Format a quaternion

        :arg q: unit-quaternion as a 4-vector
        :type q: array_like
        :arg delim: 2-list of delimeters [default ('<', '>')]
        :type delim: list or tuple of strings
        :arg fmt: printf-style format soecifier [default '%f']
        :type fmt: str
        :arg file: destination for formatted string [default sys.stdout]
        :type file: file object
        :return: formatted string
        :rtype: str

        Format the quaternion in a human-readable form as::

            S  D1  VX VY VZ D2

        where S, VX, VY, VZ are the quaternion elements, and D1 and D2 are a pair
        of delimeters given by `delim`.

        By default the string is written to `sys.stdout`.

        If `file=None` then a string is returned.

        """
        q = argcheck.getvector(q, 4)
        template = "# %s #, #, # %s".replace('#', fmt)
        s = template % (q[0], delim[0], q[1], q[2], q[3], delim[1])
        if file:
            file.write(s + '\n')
        else:
            return s

    if __name__ == '__main__':  # pragma: no cover
        import pathlib
        import os.path

        exec(open(os.path.join(pathlib.Path(__file__).parent.absolute(), "test_quaternions.py")).read())