# Source code for kornia.geometry.line

```# kornia.geometry.line module inspired by Eigen::geometry::ParametrizedLine
# https://gitlab.com/libeigen/eigen/-/blob/master/Eigen/src/Geometry/ParametrizedLine.h
from typing import Iterator, Optional, Tuple, Union

import torch

from kornia.core import Module, Parameter, Tensor, normalize, where
from kornia.core.check import KORNIA_CHECK, KORNIA_CHECK_IS_TENSOR, KORNIA_CHECK_SHAPE
from kornia.geometry.linalg import batched_dot_product, squared_norm
from kornia.geometry.plane import Hyperplane
from kornia.utils.helpers import _torch_svd_cast

__all__ = ["ParametrizedLine", "fit_line"]

[docs]class ParametrizedLine(Module):
"""Class that describes a parametrize line.

A parametrized line is defined by an origin point :math:`o` and a unit
direction vector :math:`d` such that the line corresponds to the set

.. math::

l(t) = o + t * d
"""

[docs]    def __init__(self, origin: Tensor, direction: Tensor) -> None:
"""Initializes a parametrized line of direction and origin.

Args:
origin: any point on the line of any dimension.
direction: the normalized vector direction of any dimension.

Example:
>>> o = torch.tensor([0.0, 0.0])
>>> d = torch.tensor([1.0, 1.0])
>>> l = ParametrizedLine(o, d)
"""
super().__init__()
self._origin = Parameter(origin)
self._direction = Parameter(direction)

def __str__(self) -> str:
return f"Origin: {self.origin}\nDirection: {self.direction}"

def __repr__(self) -> str:
return str(self)

def __getitem__(self, idx: int) -> Tensor:
return self.origin if idx == 0 else self.direction

def __iter__(self) -> Iterator[Tensor]:
yield from (self.origin, self.direction)

@property
def origin(self) -> Tensor:
"""Return the line origin point."""
return self._origin

@property
def direction(self) -> Tensor:
"""Return the line direction vector."""
return self._direction

[docs]    def dim(self) -> int:
"""Return the dimension in which the line holds."""
return self.direction.shape[-1]

[docs]    @classmethod
def through(cls, p0: Tensor, p1: Tensor) -> "ParametrizedLine":
"""Constructs a parametrized line going from a point :math:`p0` to :math:`p1`.

Args:
p0: tensor with first point :math:`(B, D)` where `D` is the point dimension.
p1: tensor with second point :math:`(B, D)` where `D` is the point dimension.

Example:
>>> p0 = torch.tensor([0.0, 0.0])
>>> p1 = torch.tensor([1.0, 1.0])
>>> l = ParametrizedLine.through(p0, p1)
"""
return ParametrizedLine(p0, normalize((p1 - p0), p=2, dim=-1))

[docs]    def point_at(self, t: Union[float, Tensor]) -> Tensor:
"""The point at :math:`t` along this line.

Args:
t: step along the line.

Return:
tensor with the point.

Example:
>>> p0 = torch.tensor([0.0, 0.0])
>>> p1 = torch.tensor([1.0, 1.0])
>>> l = ParametrizedLine.through(p0, p1)
>>> p2 = l.point_at(0.1)
"""
return self.origin + self.direction * t

[docs]    def projection(self, point: Tensor) -> Tensor:
"""Return the projection of a point onto the line.

Args:
point: the point to be projected.
"""
return self.origin + (self.direction @ (point - self.origin)) * self.direction

# TODO: improve order and speed
[docs]    def squared_distance(self, point: Tensor) -> Tensor:
"""Return the squared distance of a point to its projection onte the line.

Args:
point: the point to calculate the distance onto the line.
"""
diff: Tensor = point - self.origin
return squared_norm(diff - (self.direction @ diff) * self.direction)

# TODO: improve order and speed
[docs]    def distance(self, point: Tensor) -> Tensor:
"""Return the distance of a point to its projections onto the line.

Args:
point: the point to calculate the distance into the line.
"""
return self.squared_distance(point).sqrt()

# TODO(edgar) implement the following:
# - intersection
# - intersection_parameter
# - intersection_point

[docs]    def intersect(self, plane: Hyperplane, eps: float = 1e-6) -> Tuple[Tensor, Tensor]:
"""Return the intersection point between the line and a given plane.

Args:
plane: the plane to compute the intersection point.

Return:
- the lambda value used to compute the look at point.
- the intersected point.
"""
dot_prod = batched_dot_product(plane.normal.data, self.direction.data)

# TODO: add check for dot product
res_lambda = where(
-(plane.offset + batched_dot_product(plane.normal.data, self.origin.data)) / dot_prod,
torch.empty_like(dot_prod),
)

res_point = self.point_at(res_lambda)
return res_lambda, res_point

[docs]def fit_line(points: Tensor, weights: Optional[Tensor] = None) -> ParametrizedLine:
"""Fit a line from a set of points.

Args:
points: tensor containing a batch of sets of n-dimensional points. The  expected
shape of the tensor is :math:`(B, N, D)`.
weights: weights to use to solve the equations system. The  expected
shape of the tensor is :math:`(B, N)`.

Return:
A tensor containing the direction of the fited line of shape :math:`(B, D)`.

Example:
>>> points = torch.rand(2, 10, 3)
>>> weights = torch.ones(2, 10)
>>> line = fit_line(points, weights)
>>> line.direction.shape
torch.Size([2, 3])
"""
KORNIA_CHECK_IS_TENSOR(points, "points must be a tensor")
KORNIA_CHECK_SHAPE(points, ["B", "N", "D"])

mean = points.mean(-2, True)
A = points - mean

if weights is not None:
KORNIA_CHECK_IS_TENSOR(weights, "weights must be a tensor")
KORNIA_CHECK_SHAPE(weights, ["B", "N"])
KORNIA_CHECK(points.shape[0] == weights.shape[0])
A = A.transpose(-2, -1) @ torch.diag_embed(weights) @ A
else:
A = A.transpose(-2, -1) @ A

# NOTE: not optimal for 2d points, but for now works for other dimensions
_, _, V = _torch_svd_cast(A)
V = V.transpose(-2, -1)

# the first left eigenvector is the direction on the fited line
direction = V[..., 0, :]  # BxD
origin = mean[..., 0, :]  # BxD

return ParametrizedLine(origin, direction)
```