DAMASK_EICMD/python/damask/_Lambert.py

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# Code below available according to the following conditions on
# https://github.com/MarDiehl/3Drotations
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# Copyright (c) 2017-2019, Martin Diehl/Max-Planck-Institut für Eisenforschung GmbH
# Copyright (c) 2013-2014, Marc De Graef/Carnegie Mellon University
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import numpy as np

sc   = np.pi**(1./6.)/6.**(1./6.)
beta = np.pi**(5./6.)/6.**(1./6.)/2.
R1   = (3.*np.pi/4.)**(1./3.)

def cube_to_ball(cube):
    """
    Map a point in a uniform refinable cubical grid to a point on a uniform refinable grid on a ball.

    Parameters
    ----------
    cube : numpy.ndarray
        coordinates of a point in a uniform refinable cubical grid.

    References
    ----------
    D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014
    https://doi.org/10.1088/0965-0393/22/7/075013

    """
    cube_ = np.clip(cube,None,np.pi**(2./3.) * 0.5) if np.isclose(np.abs(np.max(cube)),np.pi**(2./3.) * 0.5) else cube
    if np.abs(np.max(cube_))>np.pi**(2./3.) * 0.5:
        raise ValueError

    # transform to the sphere grid via the curved square, and intercept the zero point
    if np.allclose(cube_,0.0,rtol=0.0,atol=1.0e-16):
        ball = np.zeros(3)
    else:
        # get pyramide and scale by grid parameter ratio
        p = _get_order(cube_)
        XYZ = cube_[p[0]] * sc

        # intercept all the points along the z-axis
        if np.allclose(XYZ[0:2],0.0,rtol=0.0,atol=1.0e-16):
            ball = np.array([0.0, 0.0, np.sqrt(6.0/np.pi) * XYZ[2]])
        else:
            order = [1,0] if np.abs(XYZ[1]) <= np.abs(XYZ[0]) else [0,1]
            q = np.pi/12.0 * XYZ[order[0]]/XYZ[order[1]]
            c = np.cos(q)
            s = np.sin(q)
            q = R1*2.0**0.25/beta * XYZ[order[1]] / np.sqrt(np.sqrt(2.0)-c)
            T = np.array([ (np.sqrt(2.0)*c - 1.0), np.sqrt(2.0) * s]) * q

            # transform to sphere grid (inverse Lambert)
            # note that there is no need to worry about dividing by zero, since XYZ[2] can not become zero
            c = np.sum(T**2)
            s = c *         np.pi/24.0 /XYZ[2]**2
            c = c * np.sqrt(np.pi/24.0)/XYZ[2]
            q = np.sqrt( 1.0 - s )
            ball = np.array([ T[order[1]] * q, T[order[0]] * q, np.sqrt(6.0/np.pi) * XYZ[2] - c ])

        # reverse the coordinates back to the regular order according to the original pyramid number
        ball = ball[p[1]]

    return ball


def ball_to_cube(ball):
    """
    Map a point on a uniform refinable grid on a ball to a point in a uniform refinable cubical grid.

    Parameters
    ----------
    ball : numpy.ndarray
       coordinates of a point on a uniform refinable grid on a ball.

    References
    ----------
    D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014
    https://doi.org/10.1088/0965-0393/22/7/075013

    """
    ball_ = ball/np.linalg.norm(ball) if np.isclose(np.linalg.norm(ball),R1) else ball
    rs = np.linalg.norm(ball_)
    if rs > R1:
        raise ValueError

    if np.allclose(ball_,0.0,rtol=0.0,atol=1.0e-16):
        cube = np.zeros(3)
    else:
        p = _get_order(ball_)
        xyz3 = ball_[p[0]]

        # inverse M_3
        xyz2 = xyz3[0:2] * np.sqrt( 2.0*rs/(rs+np.abs(xyz3[2])) )

        # inverse M_2
        qxy = np.sum(xyz2**2)

        if np.isclose(qxy,0.0,rtol=0.0,atol=1.0e-16):
            Tinv = np.zeros(2)
        else:
            q2 = qxy + np.max(np.abs(xyz2))**2
            sq2 = np.sqrt(q2)
            q = (beta/np.sqrt(2.0)/R1) * np.sqrt(q2*qxy/(q2-np.max(np.abs(xyz2))*sq2))
            tt = np.clip((np.min(np.abs(xyz2))**2+np.max(np.abs(xyz2))*sq2)/np.sqrt(2.0)/qxy,-1.0,1.0)
            Tinv = np.array([1.0,np.arccos(tt)/np.pi*12.0]) if np.abs(xyz2[1]) <= np.abs(xyz2[0]) else \
                   np.array([np.arccos(tt)/np.pi*12.0,1.0])
            Tinv = q * np.where(xyz2<0.0,-Tinv,Tinv)

        # inverse M_1
        cube = np.array([ Tinv[0], Tinv[1],  (-1.0 if xyz3[2] < 0.0 else 1.0) * rs / np.sqrt(6.0/np.pi) ]) /sc
        # reverse the coordinates back to the regular order according to the original pyramid number
        cube = cube[p[1]]

    return cube


def _get_order(xyz):
    """
    Get order of the coordinates.

    Depending on the pyramid in which the point is located, the order need to be adjusted.

    Parameters
    ----------
    xyz : numpy.ndarray
       coordinates of a point on a uniform refinable grid on a ball or
       in a uniform refinable cubical grid.

    References
    ----------
    D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014
    https://doi.org/10.1088/0965-0393/22/7/075013

    """
    if   (abs(xyz[0])<= xyz[2]) and (abs(xyz[1])<= xyz[2]) or \
         (abs(xyz[0])<=-xyz[2]) and (abs(xyz[1])<=-xyz[2]):
        return [[0,1,2],[0,1,2]]
    elif (abs(xyz[2])<= xyz[0]) and (abs(xyz[1])<= xyz[0]) or \
         (abs(xyz[2])<=-xyz[0]) and (abs(xyz[1])<=-xyz[0]):
        return [[1,2,0],[2,0,1]]
    elif (abs(xyz[0])<= xyz[1]) and (abs(xyz[2])<= xyz[1]) or \
         (abs(xyz[0])<=-xyz[1]) and (abs(xyz[2])<=-xyz[1]):
        return [[2,0,1],[1,2,0]]