# Copyright (c) Facebook, Inc. and its affiliates. All rights reserved.


import torch


HAT_INV_SKEW_SYMMETRIC_TOL = 1e-5


def so3_relative_angle(R1, R2, cos_angle: bool = False):
    """
    Calculates the relative angle (in radians) between pairs of
    rotation matrices `R1` and `R2` with `angle = acos(0.5 * Trace(R1 R2^T)-1)`

    .. note::
        This corresponds to a geodesic distance on the 3D manifold of rotation
        matrices.

    Args:
        R1: Batch of rotation matrices of shape `(minibatch, 3, 3)`.
        R2: Batch of rotation matrices of shape `(minibatch, 3, 3)`.
        cos_angle: If==True return cosine of the relative angle rather than
                   the angle itself. This can avoid the unstable
                   calculation of `acos`.

    Returns:
        Corresponding rotation angles of shape `(minibatch,)`.
        If `cos_angle==True`, returns the cosine of the angles.

    Raises:
        ValueError if `R1` or `R2` is of incorrect shape.
        ValueError if `R1` or `R2` has an unexpected trace.
    """
    R12 = torch.bmm(R1, R2.permute(0, 2, 1))
    return so3_rotation_angle(R12, cos_angle=cos_angle)


def so3_rotation_angle(R, eps: float = 1e-4, cos_angle: bool = False):
    """
    Calculates angles (in radians) of a batch of rotation matrices `R` with
    `angle = acos(0.5 * (Trace(R)-1))`. The trace of the
    input matrices is checked to be in the valid range `[-1-eps,3+eps]`.
    The `eps` argument is a small constant that allows for small errors
    caused by limited machine precision.

    Args:
        R: Batch of rotation matrices of shape `(minibatch, 3, 3)`.
        eps: Tolerance for the valid trace check.
        cos_angle: If==True return cosine of the rotation angles rather than
                   the angle itself. This can avoid the unstable
                   calculation of `acos`.

    Returns:
        Corresponding rotation angles of shape `(minibatch,)`.
        If `cos_angle==True`, returns the cosine of the angles.

    Raises:
        ValueError if `R` is of incorrect shape.
        ValueError if `R` has an unexpected trace.
    """

    N, dim1, dim2 = R.shape
    if dim1 != 3 or dim2 != 3:
        raise ValueError("Input has to be a batch of 3x3 Tensors.")

    rot_trace = R[:, 0, 0] + R[:, 1, 1] + R[:, 2, 2]

    if ((rot_trace < -1.0 - eps) + (rot_trace > 3.0 + eps)).any():
        raise ValueError("A matrix has trace outside valid range [-1-eps,3+eps].")

    # clamp to valid range
    rot_trace = torch.clamp(rot_trace, -1.0, 3.0)

    # phi ... rotation angle
    phi = 0.5 * (rot_trace - 1.0)

    if cos_angle:
        return phi
    else:
        return phi.acos()


def so3_exponential_map(log_rot, eps: float = 0.0001):
    """
    Convert a batch of logarithmic representations of rotation matrices `log_rot`
    to a batch of 3x3 rotation matrices using Rodrigues formula [1].

    In the logarithmic representation, each rotation matrix is represented as
    a 3-dimensional vector (`log_rot`) who's l2-norm and direction correspond
    to the magnitude of the rotation angle and the axis of rotation respectively.

    The conversion has a singularity around `log(R) = 0`
    which is handled by clamping controlled with the `eps` argument.

    Args:
        log_rot: Batch of vectors of shape `(minibatch , 3)`.
        eps: A float constant handling the conversion singularity.

    Returns:
        Batch of rotation matrices of shape `(minibatch , 3 , 3)`.

    Raises:
        ValueError if `log_rot` is of incorrect shape.

    [1] https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula
    """

    _, dim = log_rot.shape
    if dim != 3:
        raise ValueError("Input tensor shape has to be Nx3.")

    nrms = (log_rot * log_rot).sum(1)
    # phis ... rotation angles
    rot_angles = torch.clamp(nrms, eps).sqrt()
    rot_angles_inv = 1.0 / rot_angles
    fac1 = rot_angles_inv * rot_angles.sin()
    fac2 = rot_angles_inv * rot_angles_inv * (1.0 - rot_angles.cos())
    skews = hat(log_rot)

    R = (
        fac1[:, None, None] * skews
        + fac2[:, None, None] * torch.bmm(skews, skews)
        + torch.eye(3, dtype=log_rot.dtype, device=log_rot.device)[None]
    )

    return R


def so3_log_map(R, eps: float = 0.0001):
    """
    Convert a batch of 3x3 rotation matrices `R`
    to a batch of 3-dimensional matrix logarithms of rotation matrices
    The conversion has a singularity around `(R=I)` which is handled
    by clamping controlled with the `eps` argument.

    Args:
        R: batch of rotation matrices of shape `(minibatch, 3, 3)`.
        eps: A float constant handling the conversion singularity.

    Returns:
        Batch of logarithms of input rotation matrices
        of shape `(minibatch, 3)`.

    Raises:
        ValueError if `R` is of incorrect shape.
        ValueError if `R` has an unexpected trace.
    """

    N, dim1, dim2 = R.shape
    if dim1 != 3 or dim2 != 3:
        raise ValueError("Input has to be a batch of 3x3 Tensors.")

    phi = so3_rotation_angle(R)

    phi_sin = phi.sin()

    phi_denom = (
        torch.clamp(phi_sin.abs(), eps) * phi_sin.sign()
        + (phi_sin == 0).type_as(phi) * eps
    )

    log_rot_hat = (phi / (2.0 * phi_denom))[:, None, None] * (R - R.permute(0, 2, 1))
    log_rot = hat_inv(log_rot_hat)

    return log_rot


def hat_inv(h):
    """
    Compute the inverse Hat operator [1] of a batch of 3x3 matrices.

    Args:
        h: Batch of skew-symmetric matrices of shape `(minibatch, 3, 3)`.

    Returns:
        Batch of 3d vectors of shape `(minibatch, 3, 3)`.

    Raises:
        ValueError if `h` is of incorrect shape.
        ValueError if `h` not skew-symmetric.

    [1] https://en.wikipedia.org/wiki/Hat_operator
    """

    N, dim1, dim2 = h.shape
    if dim1 != 3 or dim2 != 3:
        raise ValueError("Input has to be a batch of 3x3 Tensors.")

    ss_diff = (h + h.permute(0, 2, 1)).abs().max()
    if float(ss_diff) > HAT_INV_SKEW_SYMMETRIC_TOL:
        raise ValueError("One of input matrices not skew-symmetric.")

    x = h[:, 2, 1]
    y = h[:, 0, 2]
    z = h[:, 1, 0]

    v = torch.stack((x, y, z), dim=1)

    return v


def hat(v):
    """
    Compute the Hat operator [1] of a batch of 3D vectors.

    Args:
        v: Batch of vectors of shape `(minibatch , 3)`.

    Returns:
        Batch of skew-symmetric matrices of shape
        `(minibatch, 3 , 3)` where each matrix is of the form:
            `[    0  -v_z   v_y ]
             [  v_z     0  -v_x ]
             [ -v_y   v_x     0 ]`

    Raises:
        ValueError if `v` is of incorrect shape.

    [1] https://en.wikipedia.org/wiki/Hat_operator
    """

    N, dim = v.shape
    if dim != 3:
        raise ValueError("Input vectors have to be 3-dimensional.")

    h = v.new_zeros(N, 3, 3)

    x, y, z = v.unbind(1)

    h[:, 0, 1] = -z
    h[:, 0, 2] = y
    h[:, 1, 0] = z
    h[:, 1, 2] = -x
    h[:, 2, 0] = -y
    h[:, 2, 1] = x

    return h
