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

from typing import Tuple

import torch

from ..transforms import acos_linear_extrapolation

HAT_INV_SKEW_SYMMETRIC_TOL = 1e-5


def so3_relative_angle(
    R1: torch.Tensor,
    R2: torch.Tensor,
    cos_angle: bool = False,
    cos_bound: float = 1e-4,
    eps: float = 1e-4,
) -> torch.Tensor:
    """
    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`.
        cos_bound: Clamps the cosine of the relative rotation angle to
            [-1 + cos_bound, 1 - cos_bound] to avoid non-finite outputs/gradients
            of the `acos` call. Note that the non-finite outputs/gradients
            are returned when the angle is requested (i.e. `cos_angle==False`)
            and the rotation angle is close to 0 or π.
        eps: Tolerance for the valid trace check of the relative rotation matrix
            in `so3_rotation_angle`.
    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, cos_bound=cos_bound, eps=eps)


def so3_rotation_angle(
    R: torch.Tensor,
    eps: float = 1e-4,
    cos_angle: bool = False,
    cos_bound: float = 1e-4,
) -> torch.Tensor:
    """
    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`.
        cos_bound: Clamps the cosine of the rotation angle to
            [-1 + cos_bound, 1 - cos_bound] to avoid non-finite outputs/gradients
            of the `acos` call. Note that the non-finite outputs/gradients
            are returned when the angle is requested (i.e. `cos_angle==False`)
            and the rotation angle is close to 0 or π.

    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].")

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

    if cos_angle:
        return phi_cos
    else:
        if cos_bound > 0.0:
            return acos_linear_extrapolation(phi_cos, 1.0 - cos_bound)
        else:
            return torch.acos(phi_cos)


def so3_exp_map(log_rot: torch.Tensor, eps: float = 0.0001) -> torch.Tensor:
    """
    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
    """
    return _so3_exp_map(log_rot, eps=eps)[0]


so3_exponential_map = so3_exp_map


def _so3_exp_map(
    log_rot: torch.Tensor, eps: float = 0.0001
) -> Tuple[torch.Tensor, torch.Tensor, torch.Tensor, torch.Tensor]:
    """
    A helper function that computes the so3 exponential map and,
    apart from the rotation matrix, also returns intermediate variables
    that can be re-used in other functions.
    """
    _, 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)
    skews_square = torch.bmm(skews, skews)

    R = (
        # pyre-fixme[16]: `float` has no attribute `__getitem__`.
        fac1[:, None, None] * skews
        + fac2[:, None, None] * skews_square
        + torch.eye(3, dtype=log_rot.dtype, device=log_rot.device)[None]
    )

    return R, rot_angles, skews, skews_square


def so3_log_map(
    R: torch.Tensor, eps: float = 0.0001, cos_bound: float = 1e-4
) -> torch.Tensor:
    """
    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` and `cos_bound` arguments.

    Args:
        R: batch of rotation matrices of shape `(minibatch, 3, 3)`.
        eps: A float constant handling the conversion singularity.
        cos_bound: Clamps the cosine of the rotation angle to
            [-1 + cos_bound, 1 - cos_bound] to avoid non-finite outputs/gradients
            of the `acos` call when computing `so3_rotation_angle`.
            Note that the non-finite outputs/gradients are returned when
            the rotation angle is close to 0 or π.

    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, cos_bound=cos_bound, eps=eps)

    phi_sin = torch.sin(phi)

    # We want to avoid a tiny denominator of phi_factor = phi / (2.0 * phi_sin).
    # Hence, for phi_sin.abs() <= 0.5 * eps, we approximate phi_factor with
    # 2nd order Taylor expansion: phi_factor = 0.5 + (1.0 / 12) * phi**2
    phi_factor = torch.empty_like(phi)
    ok_denom = phi_sin.abs() > (0.5 * eps)
    phi_factor[~ok_denom] = 0.5 + (phi[~ok_denom] ** 2) * (1.0 / 12)
    phi_factor[ok_denom] = phi[ok_denom] / (2.0 * phi_sin[ok_denom])

    log_rot_hat = phi_factor[:, None, None] * (R - R.permute(0, 2, 1))

    log_rot = hat_inv(log_rot_hat)

    return log_rot


def hat_inv(h: torch.Tensor) -> torch.Tensor:
    """
    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 = torch.abs(h + h.permute(0, 2, 1)).max()
    if float(ss_diff) > HAT_INV_SKEW_SYMMETRIC_TOL:
        raise ValueError("One of input matrices is 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: torch.Tensor) -> torch.Tensor:
    """
    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 = torch.zeros((N, 3, 3), dtype=v.dtype, device=v.device)

    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
