# Source code for skspatial.objects.line

"""Module for the Line class."""
from __future__ import annotations

from typing import Optional

import numpy as np
from matplotlib.axes import Axes
from mpl_toolkits.mplot3d import Axes3D

from skspatial.objects._base_line_plane import _BaseLinePlane
from skspatial.objects.point import Point
from skspatial.objects.points import Points
from skspatial.objects.vector import Vector
from skspatial.plotting import _connect_points_2d
from skspatial.plotting import _connect_points_3d
from skspatial.transformation import transform_coordinates
from skspatial.typing import array_like

[docs]class Line(_BaseLinePlane): """ A line in space. The line is defined by a point and a direction vector. Parameters ---------- point : array_like Point on the line. direction : array_like Direction vector of the line. kwargs : dict, optional Additional keywords passed to :meth:Vector.is_zero. This method is used to ensure that the direction vector is not the zero vector. Attributes ---------- point : Point Point on the line. direction : Vector Unit direction vector. vector : Vector Same as the direction. dimension : int Dimension of the line. Raises ------ ValueError If the point and vector have different dimensions. If the vector is all zeros. Examples -------- >>> from skspatial.objects import Line >>> line = Line(point=[0, 0], direction=[3, 0]) >>> line Line(point=Point([0, 0]), direction=Vector([3, 0])) >>> line.direction Vector([3, 0]) The direction can also be accessed with the vector attribute. >>> line.vector Vector([3, 0]) The line dimension is the dimension of the point and vector. >>> line.dimension 2 >>> Line([0, 0], [1, 0, 0]) Traceback (most recent call last): ... ValueError: The point and vector must have the same dimension. >>> Line([1, 1], [0, 0]) Traceback (most recent call last): ... ValueError: The vector must not be the zero vector. """ def __init__(self, point: array_like, direction: array_like): super().__init__(point, direction) self.direction = self.vector
[docs] @classmethod def from_points(cls, point_a: array_like, point_b: array_like) -> Line: """ Instantiate a line from two points. Parameters ---------- point_a, point_b : array_like Two points defining the line. Returns ------- Line Line containing the two input points. Examples -------- >>> from skspatial.objects import Line >>> Line.from_points([0, 0], [1, 0]) Line(point=Point([0, 0]), direction=Vector([1, 0])) The order of the points affects the line point and direction vector. >>> Line.from_points([1, 0], [0, 0]) Line(point=Point([1, 0]), direction=Vector([-1, 0])) """ vector_ab = Vector.from_points(point_a, point_b) return cls(point_a, vector_ab)
[docs] @classmethod def from_slope(cls, slope: float, y_intercept: float) -> Line: r""" Instantiate a 2D line from a slope and Y-intercept. A 2D line can be represented by the equation .. math:: y = mx + b where :math:m is the slope and :math:p is the Y-intercept. Parameters ---------- slope : {int, float} Slope of the 2D line. y_intercept : {int, float} Y coordinate of the point where the line intersects the Y axis. Returns ------- Line A 2D Line object. Examples -------- >>> from skspatial.objects import Line >>> Line.from_slope(2, 0) Line(point=Point([0, 0]), direction=Vector([1, 2])) >>> Line.from_slope(-3, 5) Line(point=Point([0, 5]), direction=Vector([ 1, -3])) >>> line_a = Line.from_slope(1, 0) >>> line_b = Line.from_slope(0, 5) >>> line_a.intersect_line(line_b) Point([5., 5.]) """ point = [0, y_intercept] direction = [1, slope] return cls(point, direction)
[docs] def is_coplanar(self, other: Line, **kwargs: float) -> bool: """ Check if the line is coplanar with another. Parameters ---------- other : Line Other line. kwargs : dict, optional Additional keywords passed to :func:numpy.linalg.matrix_rank Returns ------- bool True if the line is coplanar; false otherwise. Raises ------ TypeError If the input is not a line. References ---------- http://mathworld.wolfram.com/Coplanar.html Examples -------- >>> from skspatial.objects import Line >>> line_a = Line(point=[0, 0, 0], direction=[1, 0, 0]) >>> line_b = Line([-5, 3, 0], [7, 1, 0]) >>> line_c = Line([0, 0, 0], [0, 0, 1]) >>> line_a.is_coplanar(line_b) True >>> line_a.is_coplanar(line_c) True >>> line_b.is_coplanar(line_c) False The input must be another line. >>> from skspatial.objects import Plane >>> line_a.is_coplanar(Plane(line_a.point, line_a.vector)) Traceback (most recent call last): ... TypeError: The input must also be a line. """ if not isinstance(other, type(self)): raise TypeError("The input must also be a line.") point_1 = self.point point_2 = self.to_point() point_3 = other.point point_4 = other.to_point() points = Points([point_1, point_2, point_3, point_4]) return points.are_coplanar(**kwargs)
[docs] def to_point(self, t: float = 1) -> Point: r""" Return a point along the line using a parameter t. Parameters ---------- t : {int, float} Parameter that defines the new point along the line. Returns ------- Point New point along the line. Notes ----- The new point :math:p is computed as: .. math:: p = \mathtt{line.point} + t \cdot \mathtt{line.direction} Examples -------- >>> from skspatial.objects import Line >>> line = Line(point=[0, 0], direction=[2, 0]) >>> line.to_point() Point([2, 0]) >>> line.to_point(t=2) Point([4, 0]) """ vector_along_line = t * self.direction return self.point + vector_along_line
[docs] def project_point(self, point: array_like) -> Point: """ Project a point onto the line. Parameters ---------- point : array_like Input point. Returns ------- Point Projection of the point onto the line. Examples -------- >>> from skspatial.objects import Line >>> Line(point=[0, 0], direction=[8, 0]).project_point([5, 5]) Point([5., 0.]) >>> Line(point=[0, 0, 0], direction=[1, 1, 0]).project_point([5, 5, 3]) Point([5., 5., 0.]) """ # Vector from the point on the line to the point in space. vector_to_point = Vector.from_points(self.point, point) # Project the vector onto the line. vector_projected = self.direction.project_vector(vector_to_point) # Add the projected vector to the point on the line. return self.point + vector_projected
[docs] def project_vector(self, vector: array_like) -> Vector: """ Project a vector onto the line. Parameters ---------- vector : array_like Input vector. Returns ------- Vector Projection of the vector onto the line. Examples -------- >>> from skspatial.objects import Line >>> line = Line([-1, 5, 3], [3, 4, 5]) >>> line.project_vector([1, 1, 1]) Vector([0.72, 0.96, 1.2 ]) """ return self.direction.project_vector(vector)
[docs] def side_point(self, point: array_like) -> int: """ Find the side of the line where a point lies. The line and point must be 2D. Parameters ---------- point : array_like Input point. Returns ------- int -1 if the point is left of the line. 0 if the point is on the line. 1 if the point is right of the line. Examples -------- >>> from skspatial.objects import Line >>> line = Line([0, 0], [1, 1]) The point is on the line. >>> line.side_point([2, 2]) 0 The point is to the right of the line. >>> line.side_point([5, 3]) 1 The point is to the left of the line. >>> line.side_point([5, 10]) -1 """ vector_to_point = Vector.from_points(self.point, point) return self.direction.side_vector(vector_to_point)
[docs] def distance_point(self, point: array_like) -> np.float64: """ Return the distance from a point to the line. This is the distance from the point to its projection on the line. Parameters ---------- point : array_like Input point. Returns ------- np.float64 Distance from the point to the line. Examples -------- >>> from skspatial.objects import Line >>> line = Line([0, 0], [1, 0]) >>> line.distance_point([0, 0]) 0.0 >>> line.distance_point([5, 0]) 0.0 >>> line.distance_point([5, -5]) 5.0 >>> line = Line([5, 2, -3], [3, 8, 2]) >>> line.distance_point([5, -5, 3]).round(3) 7.737 """ point_projected = self.project_point(point) return point_projected.distance_point(point)
[docs] def distance_line(self, other: Line) -> np.float64: """ Return the shortest distance from the line to another. Parameters ---------- other : Line Other line. Returns ------- np.float64 Distance between the lines. References ---------- http://mathworld.wolfram.com/Line-LineDistance.html Examples -------- There are three cases: 1. The lines intersect (i.e., they are coplanar and not parallel). >>> from skspatial.objects import Line >>> line_a = Line([1, 2], [4, 3]) >>> line_b = Line([-4, 1], [7, 23]) >>> line_a.distance_line(line_b) 0.0 2. The lines are parallel. >>> line_a = Line([0, 0], [1, 0]) >>> line_b = Line([0, 5], [-1, 0]) >>> line_a.distance_line(line_b) 5.0 3. The lines are skew. >>> line_a = Line([0, 0, 0], [1, 0, 1]) >>> line_b = Line([1, 0, 0], [1, 1, 1]) >>> line_a.distance_line(line_b).round(3) 0.707 """ if self.direction.is_parallel(other.direction): # The lines are parallel. # The distance between the lines is the distance from line point B to line A. distance = self.distance_point(other.point) elif self.is_coplanar(other): # The lines must intersect, since they are coplanar and not parallel. distance = np.float64(0) else: # The lines are skew. vector_ab = Vector.from_points(self.point, other.point) vector_perpendicular = self.direction.cross(other.direction) distance = abs(vector_ab.dot(vector_perpendicular)) / vector_perpendicular.norm() return distance
[docs] def intersect_line(self, other: Line, **kwargs) -> Point: """ Intersect the line with another. The lines must be coplanar and not parallel. Parameters ---------- other : Line Other line. kwargs : dict, optional Additional keywords passed to :meth:Vector.is_parallel. Returns ------- Point The point at the intersection. Raises ------ ValueError If the lines don't have the same dimension. If the line dimension is greater than three. If the lines are parallel. If the lines are not coplanar. References ---------- http://mathworld.wolfram.com/Line-LineIntersection.html Examples -------- >>> from skspatial.objects import Line >>> line_a = Line([0, 0], [1, 0]) >>> line_b = Line([5, 5], [0, 1]) >>> line_a.intersect_line(line_b) Point([5., 0.]) >>> line_a = Line([0, 0, 0], [1, 1, 1]) >>> line_b = Line([5, 5, 0], [0, 0, -8]) >>> line_a.intersect_line(line_b) Point([5., 5., 5.]) >>> line_a = Line([0, 0, 0], [1, 0, 0]) >>> line_b = Line([0, 0], [1, 1]) >>> line_a.intersect_line(line_b) Traceback (most recent call last): ... ValueError: The lines must have the same dimension. >>> line_a = Line(4 * [0], [1, 0, 0, 0]) >>> line_b = Line(4 * [0], [0, 0, 0, 1]) >>> line_a.intersect_line(line_b) Traceback (most recent call last): ... ValueError: The line dimension cannot be greater than 3. >>> line_a = Line([0, 0], [0, 1]) >>> line_b = Line([0, 1], [0, 1]) >>> line_a = Line([0, 0], [1, 0]) >>> line_b = Line([0, 1], [2, 0]) >>> line_a.intersect_line(line_b) Traceback (most recent call last): ... ValueError: The lines must not be parallel. >>> line_a = Line([1, 2, 3], [-4, 1, 1]) >>> line_b = Line([4, 5, 6], [3, 1, 5]) >>> line_a.intersect_line(line_b) Traceback (most recent call last): ... ValueError: The lines must be coplanar. """ if self.dimension != other.dimension: raise ValueError("The lines must have the same dimension.") if self.dimension > 3 or other.dimension > 3: raise ValueError("The line dimension cannot be greater than 3.") if self.direction.is_parallel(other.direction, **kwargs): raise ValueError("The lines must not be parallel.") if not self.is_coplanar(other): raise ValueError("The lines must be coplanar.") # Vector from line A to line B. vector_ab = Vector.from_points(self.point, other.point) # Vector perpendicular to both lines. vector_perpendicular = self.direction.cross(other.direction) num = vector_ab.cross(other.direction).dot(vector_perpendicular) denom = vector_perpendicular.norm() ** 2 # Vector along line A to the intersection point. vector_a_scaled = num / denom * self.direction return self.point + vector_a_scaled
[docs] @classmethod def best_fit(cls, points: array_like, tol: Optional[float] = None, **kwargs) -> Line: """ Return the line of best fit for a set of points. Parameters ---------- points : array_like Input points. tol : float | None, optional Keyword passed to :meth:Points.are_collinear (default None). kwargs : dict, optional Additional keywords passed to :func:numpy.linalg.svd Returns ------- Line The line of best fit. Raises ------ ValueError If the points are concurrent. Examples -------- >>> from skspatial.objects import Line >>> points = [[0, 0], [1, 2], [2, 1], [2, 3], [3, 2]] >>> line = Line.best_fit(points) The point on the line is the centroid of the points. >>> line.point Point([1.6, 1.6]) The line direction is a unit vector. >>> line.direction.round(3) Vector([0.707, 0.707]) """ points_spatial = Points(points) if points_spatial.are_concurrent(tol=tol): raise ValueError("The points must not be concurrent.") points_centered, centroid = points_spatial.mean_center(return_centroid=True) _, _, vh = np.linalg.svd(points_centered, **kwargs) direction = vh[0, :] return cls(centroid, direction)
[docs] def transform_points(self, points: array_like) -> np.ndarray: """ Transform points to a one-dimensional coordinate system defined by the line. The point on the line acts as the origin of the coordinate system. This is analogous is projecting the points onto the line, then computing the signed distance from the line point to the projections. Parameters ---------- points : (N, D) array_like Array of N points with dimension D. Returns ------- ndarray (N,) array of N coordinates. Examples -------- >>> from skspatial.objects import Line >>> points = [[-1, 1], [0, 1], [1, 1], [2, 1]] >>> Line([0, 0], [1, 0]).transform_points(points) array([-1., 0., 1., 2.]) The point on the line acts as the origin of the coordinates. >>> Line([1, 0], [1, 0]).transform_points(points) array([-2., -1., 0., 1.]) The sign of the coordinates depends on the direction of the line. >>> Line([0, 0], [-1, 0]).transform_points(points) array([ 1., 0., -1., -2.]) The magnitude of the direction vector is irrelevant. >>> Line([0, 0], [5, 0]).transform_points(points) array([-1., 0., 1., 2.]) """ # Basis vector of the subspace (the line). vectors_basis = [self.direction.unit()] column = transform_coordinates(points, self.point.to_array(), vectors_basis) return column.flatten()
[docs] def plot_2d(self, ax_2d: Axes, t_1: float = 0, t_2: float = 1, **kwargs) -> None: """ Plot a 2D line. The line is plotted by connecting two 2D points. Parameters ---------- ax_2d : Axes Instance of :class:~matplotlib.axes.Axes. t_1, t_2 : {int, float} Parameters to determine points 1 and 2 along the line. These are passed to :meth:Line.to_point. Defaults are 0 and 1. kwargs : dict, optional Additional keywords passed to :meth:~matplotlib.axes.Axes.plot. Examples -------- .. plot:: :include-source: >>> import matplotlib.pyplot as plt >>> from skspatial.objects import Line >>> _, ax = plt.subplots() >>> line = Line([1, 2], [3, 4]) >>> line.plot_2d(ax, t_1=-2, t_2=3, c='k') >>> line.point.plot_2d(ax, c='r', s=100, zorder=3) >>> grid = ax.grid() """ point_1 = self.to_point(t_1) point_2 = self.to_point(t_2) _connect_points_2d(ax_2d, point_1, point_2, **kwargs)
[docs] def plot_3d(self, ax_3d: Axes3D, t_1: float = 0, t_2: float = 1, **kwargs) -> None: """ Plot a 3D line. The line is plotted by connecting two 3D points. Parameters ---------- ax_3d : Axes3D Instance of :class:~mpl_toolkits.mplot3d.axes3d.Axes3D. t_1, t_2 : {int, float} Parameters to determine points 1 and 2 along the line. These are passed to :meth:Line.to_point. Defaults are 0 and 1. kwargs : dict, optional Additional keywords passed to :meth:~mpl_toolkits.mplot3d.axes3d.Axes3D.plot. Examples -------- .. plot:: :include-source: >>> import matplotlib.pyplot as plt >>> from mpl_toolkits.mplot3d import Axes3D >>> from skspatial.objects import Line >>> fig = plt.figure() >>> ax = fig.add_subplot(111, projection='3d') >>> line = Line([1, 2, 3], [0, 1, 1]) >>> line.plot_3d(ax, c='k') >>> line.point.plot_3d(ax, s=100) """ point_1 = self.to_point(t_1) point_2 = self.to_point(t_2) _connect_points_3d(ax_3d, point_1, point_2, **kwargs)