# Source code for pytorch3d.transforms.math

```# Copyright (c) Meta Platforms, Inc. and affiliates.
#
# LICENSE file in the root directory of this source tree.

import math
from typing import Tuple

import torch

DEFAULT_ACOS_BOUND: float = 1.0 - 1e-4

[docs]def acos_linear_extrapolation(
x: torch.Tensor,
bounds: Tuple[float, float] = (-DEFAULT_ACOS_BOUND, DEFAULT_ACOS_BOUND),
) -> torch.Tensor:
"""
Implements `arccos(x)` which is linearly extrapolated outside `x`'s original
domain of `(-1, 1)`. This allows for stable backpropagation in case `x`
is not guaranteed to be strictly within `(-1, 1)`.

More specifically:
```
bounds=(lower_bound, upper_bound)
if lower_bound <= x <= upper_bound:
acos_linear_extrapolation(x) = acos(x)
elif x <= lower_bound: # 1st order Taylor approximation
acos_linear_extrapolation(x)
= acos(lower_bound) + dacos/dx(lower_bound) * (x - lower_bound)
else:  # x >= upper_bound
acos_linear_extrapolation(x)
= acos(upper_bound) + dacos/dx(upper_bound) * (x - upper_bound)
```

Args:
x: Input `Tensor`.
bounds: A float 2-tuple defining the region for the
linear extrapolation of `acos`.
The first/second element of `bound`
describes the lower/upper bound that defines the lower/upper
extrapolation region, i.e. the region where
`x <= bound[0]`/`bound[1] <= x`.
Note that all elements of `bound` have to be within (-1, 1).
Returns:
acos_linear_extrapolation: `Tensor` containing the extrapolated `arccos(x)`.
"""

lower_bound, upper_bound = bounds

if lower_bound > upper_bound:
raise ValueError("lower bound has to be smaller or equal to upper bound.")

if lower_bound <= -1.0 or upper_bound >= 1.0:
raise ValueError("Both lower bound and upper bound have to be within (-1, 1).")

# init an empty tensor and define the domain sets
acos_extrap = torch.empty_like(x)
x_upper = x >= upper_bound
x_lower = x <= lower_bound
x_mid = (~x_upper) & (~x_lower)

# acos calculation for upper_bound < x < lower_bound
acos_extrap[x_mid] = torch.acos(x[x_mid])
# the linear extrapolation for x >= upper_bound
acos_extrap[x_upper] = _acos_linear_approximation(x[x_upper], upper_bound)
# the linear extrapolation for x <= lower_bound
acos_extrap[x_lower] = _acos_linear_approximation(x[x_lower], lower_bound)

return acos_extrap

def _acos_linear_approximation(x: torch.Tensor, x0: float) -> torch.Tensor:
"""
Calculates the 1st order Taylor expansion of `arccos(x)` around `x0`.
"""
return (x - x0) * _dacos_dx(x0) + math.acos(x0)

def _dacos_dx(x: float) -> float:
"""
Calculates the derivative of `arccos(x)` w.r.t. `x`.
"""
return (-1.0) / math.sqrt(1.0 - x * x)
```