Source code for sympy.geometry.line

"""Line-like geometrical entities.

Contains
========
LinearEntity
Line
Ray
Segment
LinearEntity2D
Line2D
Ray2D
Segment2D
LinearEntity3D
Line3D
Ray3D
Segment3D

"""
from __future__ import division, print_function

from sympy.core import S, sympify
from sympy.core.relational import Eq
from sympy.core.symbol import _symbol
from sympy.functions.elementary.trigonometric import (_pi_coeff as pi_coeff, acos, tan, atan2)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.logic.boolalg import And
from sympy.simplify.simplify import simplify
from sympy.geometry.exceptions import GeometryError
from sympy.core.decorators import deprecated
from sympy.sets import Intersection
from sympy.matrices import Matrix

from .entity import GeometryEntity, GeometrySet
from .point import Point, Point3D
from sympy.utilities.misc import Undecidable


[docs]class LinearEntity(GeometrySet): """A base class for all linear entities (Line, Ray and Segment) in n-dimensional Euclidean space. Attributes ========== ambient_dimension direction length p1 p2 points Notes ===== This is an abstract class and is not meant to be instantiated. See Also ======== sympy.geometry.entity.GeometryEntity """ def __new__(cls, p1, p2=None, **kwargs): p1, p2 = Point._normalize_dimension(p1, p2) if p1 == p2: # sometimes we return a single point if we are not given two unique # points. This is done in the specific subclass raise ValueError( "%s.__new__ requires two unique Points." % cls.__name__) if len(p1) != len(p2): raise ValueError( "%s.__new__ requires two Points of equal dimension." % cls.__name__) return GeometryEntity.__new__(cls, p1, p2, **kwargs) def __contains__(self, other): """Return a definitive answer or else raise an error if it cannot be determined that other is on the boundaries of self.""" result = self.contains(other) if result is not None: return result else: raise Undecidable( "can't decide whether '%s' contains '%s'" % (self, other)) def _span_test(self, other): """Test whether the point `other` lies in the positive span of `self`. A point x is 'in front' of a point y if x.dot(y) >= 0. Return -1 if `other` is behind `self.p1`, 0 if `other` is `self.p1` and and 1 if `other` is in front of `self.p1`.""" if self.p1 == other: return 0 rel_pos = other - self.p1 d = self.direction if d.dot(rel_pos) > 0: return 1 return -1 @property def ambient_dimension(self): """A property method that returns the dimension of LinearEntity object. Parameters ========== p1 : LinearEntity Returns ======= dimension : integer Examples ======== >>> from sympy import Point, Line >>> p1, p2 = Point(0, 0), Point(1, 1) >>> l1 = Line(p1, p2) >>> l1.ambient_dimension 2 >>> from sympy import Point, Line >>> p1, p2 = Point(0, 0, 0), Point(1, 1, 1) >>> l1 = Line(p1, p2) >>> l1.ambient_dimension 3 """ return len(self.p1)
[docs] def angle_between(l1, l2): """The smallest of the two angles formed at the intersection of two linear entities. Parameters ========== l1 : LinearEntity l2 : LinearEntity Returns ======= angle : angle in radians Notes ===== From the dot product of vectors v1 and v2 it is known that: ``dot(v1, v2) = |v1|*|v2|*cos(A)`` where A is the angle formed between the two vectors. We can get the directional vectors of the two lines and readily find the angle between the two using the above formula. See Also ======== is_perpendicular, Ray2D.closing_angle Examples ======== >>> from sympy import Point, Line >>> p1, p2, p3 = Point(0, 0), Point(0, 4), Point(2, 0) >>> l1, l2 = Line(p1, p2), Line(p1, p3) >>> l1.angle_between(l2) pi/2 >>> from sympy import Point3D, Line3D >>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(-1, 2, 0) >>> l1, l2 = Line3D(p1, p2), Line3D(p2, p3) >>> l1.angle_between(l2) acos(-sqrt(2)/3) """ if not isinstance(l1, LinearEntity) and not isinstance(l2, LinearEntity): raise TypeError('Must pass only LinearEntity objects') v1, v2 = l1.direction, l2.direction return acos(v1.dot(v2)/(abs(v1)*abs(v2)))
[docs] def arbitrary_point(self, parameter='t'): """A parameterized point on the Line. Parameters ========== parameter : str, optional The name of the parameter which will be used for the parametric point. The default value is 't'. When this parameter is 0, the first point used to define the line will be returned, and when it is 1 the second point will be returned. Returns ======= point : Point Raises ====== ValueError When ``parameter`` already appears in the Line's definition. See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Point, Line >>> p1, p2 = Point(1, 0), Point(5, 3) >>> l1 = Line(p1, p2) >>> l1.arbitrary_point() Point2D(4*t + 1, 3*t) >>> from sympy import Point3D, Line3D >>> p1, p2 = Point3D(1, 0, 0), Point3D(5, 3, 1) >>> l1 = Line3D(p1, p2) >>> l1.arbitrary_point() Point3D(4*t + 1, 3*t, t) """ t = _symbol(parameter, real=True) if t.name in (f.name for f in self.free_symbols): raise ValueError(filldedent(''' Symbol %s already appears in object and cannot be used as a parameter. ''' % t.name)) # multiply on the right so the variable gets # combined with the coordinates of the point return self.p1 + (self.p2 - self.p1)*t
[docs] @staticmethod def are_concurrent(*lines): """Is a sequence of linear entities concurrent? Two or more linear entities are concurrent if they all intersect at a single point. Parameters ========== lines : a sequence of linear entities. Returns ======= True : if the set of linear entities intersect in one point False : otherwise. See Also ======== sympy.geometry.util.intersection Examples ======== >>> from sympy import Point, Line, Line3D >>> p1, p2 = Point(0, 0), Point(3, 5) >>> p3, p4 = Point(-2, -2), Point(0, 2) >>> l1, l2, l3 = Line(p1, p2), Line(p1, p3), Line(p1, p4) >>> Line.are_concurrent(l1, l2, l3) True >>> l4 = Line(p2, p3) >>> Line.are_concurrent(l2, l3, l4) False >>> from sympy import Point3D, Line3D >>> p1, p2 = Point3D(0, 0, 0), Point3D(3, 5, 2) >>> p3, p4 = Point3D(-2, -2, -2), Point3D(0, 2, 1) >>> l1, l2, l3 = Line3D(p1, p2), Line3D(p1, p3), Line3D(p1, p4) >>> Line3D.are_concurrent(l1, l2, l3) True >>> l4 = Line3D(p2, p3) >>> Line3D.are_concurrent(l2, l3, l4) False """ common_points = Intersection(*lines) if common_points.is_FiniteSet and len(common_points) == 1: return True return False
[docs] def contains(self, other): """Subclasses should implement this method and should return True if other is on the boundaries of self; False if not on the boundaries of self; None if a determination cannot be made.""" raise NotImplementedError()
@property def direction(self): """The direction vector of the LinearEntity. Returns ======= p : a Point; the ray from the origin to this point is the direction of `self` Examples ======== >>> from sympy.geometry import Line >>> a, b = (1, 1), (1, 3) >>> Line(a, b).direction Point2D(0, 2) >>> Line(b, a).direction Point2D(0, -2) This can be reported so the distance from the origin is 1: >>> Line(b, a).direction.unit Point2D(0, -1) See Also ======== sympy.geometry.point.Point.unit """ return self.p2 - self.p1
[docs] def intersection(self, other): """The intersection with another geometrical entity. Parameters ========== o : Point or LinearEntity Returns ======= intersection : list of geometrical entities See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Point, Line, Segment >>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(7, 7) >>> l1 = Line(p1, p2) >>> l1.intersection(p3) [Point2D(7, 7)] >>> p4, p5 = Point(5, 0), Point(0, 3) >>> l2 = Line(p4, p5) >>> l1.intersection(l2) [Point2D(15/8, 15/8)] >>> p6, p7 = Point(0, 5), Point(2, 6) >>> s1 = Segment(p6, p7) >>> l1.intersection(s1) [] >>> from sympy import Point3D, Line3D, Segment3D >>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(7, 7, 7) >>> l1 = Line3D(p1, p2) >>> l1.intersection(p3) [Point3D(7, 7, 7)] >>> l1 = Line3D(Point3D(4,19,12), Point3D(5,25,17)) >>> l2 = Line3D(Point3D(-3, -15, -19), direction_ratio=[2,8,8]) >>> l1.intersection(l2) [Point3D(1, 1, -3)] >>> p6, p7 = Point3D(0, 5, 2), Point3D(2, 6, 3) >>> s1 = Segment3D(p6, p7) >>> l1.intersection(s1) [] """ def intersect_parallel_rays(ray1, ray2): if ray1.direction.dot(ray2.direction) > 0: # rays point in the same direction # so return the one that is "in front" return [ray2] if ray1._span_test(ray2.p1) >= 0 else [ray1] else: # rays point in opposite directions st = ray1._span_test(ray2.p1) if st < 0: return [] elif st == 0: return [ray2.p1] return [Segment(ray1.p1, ray2.p1)] def intersect_parallel_ray_and_segment(ray, seg): st1, st2 = ray._span_test(seg.p1), ray._span_test(seg.p2) if st1 < 0 and st2 < 0: return [] elif st1 >= 0 and st2 >= 0: return [seg] elif st1 >= 0 and st2 < 0: return [Segment(ray.p1, seg.p1)] elif st1 <= 0 and st2 > 0: return [Segment(ray.p1, seg.p2)] def intersect_parallel_segments(seg1, seg2): if seg1.contains(seg2): return [seg2] if seg2.contains(seg1): return [seg1] # direct the segments so they're oriented the same way if seg1.direction.dot(seg2.direction) < 0: seg2 = Segment(seg2.p1, seg2.p2) # order the segments so seg1 is "behind" seg2 if seg1._span_test(seg2.p1) < 0: seg1, seg2 = seg2, seg1 if seg2._span_test(seg1.p2) < 0: return [] return [Segment(seg2.p1, seg1.p2)] if not isinstance(other, GeometryEntity): other = Point(other, dim=self.ambient_dimension) if other.is_Point: if self.contains(other): return [other] else: return [] elif isinstance(other, LinearEntity): # break into cases based on whether # the lines are parallel, non-parallel intersecting, or skew pts = Point._normalize_dimension(self.p1, self.p2, other.p1, other.p2) rank = Point.affine_rank(*pts) if rank == 1: # we're collinear if isinstance(self, Line): return [other] if isinstance(other, Line): return [self] if isinstance(self, Ray) and isinstance(other, Ray): return intersect_parallel_rays(self, other) if isinstance(self, Ray) and isinstance(other, Segment): return intersect_parallel_ray_and_segment(self, other) if isinstance(self, Segment) and isinstance(other, Ray): return intersect_parallel_ray_and_segment(other, self) if isinstance(self, Segment) and isinstance(other, Segment): return intersect_parallel_segments(self, other) elif rank == 2: # we're in the same plane l1 = Line(*pts[:2]) l2 = Line(*pts[2:]) # check to see if we're parallel. If we are, we can't # be intersecting, since the collinear case was already # handled if l1.direction.is_scalar_multiple(l2.direction): return [] # find the intersection as if everything were lines # by solving the equation t*d + p1 == s*d' + p1' m = Matrix([l1.direction, -l2.direction]).transpose() v = Matrix([l2.p1 - l1.p1]).transpose() # we cannot use m.solve(v) because that only works for square matrices m_rref, pivots = m.col_insert(2, v).rref(simplify=True) # rank == 2 ensures we have 2 pivots, but let's check anyway if len(pivots) != 2: raise GeometryError("Failed when solving Mx=b when M={} and b={}".format(m, v)) coeff = m_rref[0, 2] line_intersection = l1.direction*coeff + self.p1 # if we're both lines, we can skip a containment check if isinstance(self, Line) and isinstance(other, Line): return [line_intersection] if self.contains(line_intersection) and other.contains(line_intersection): return [line_intersection] return [] else: # we're skew return [] return other.intersection(self)
[docs] def is_parallel(l1, l2): """Are two linear entities parallel? Parameters ========== l1 : LinearEntity l2 : LinearEntity Returns ======= True : if l1 and l2 are parallel, False : otherwise. See Also ======== coefficients Examples ======== >>> from sympy import Point, Line >>> p1, p2 = Point(0, 0), Point(1, 1) >>> p3, p4 = Point(3, 4), Point(6, 7) >>> l1, l2 = Line(p1, p2), Line(p3, p4) >>> Line.is_parallel(l1, l2) True >>> p5 = Point(6, 6) >>> l3 = Line(p3, p5) >>> Line.is_parallel(l1, l3) False >>> from sympy import Point3D, Line3D >>> p1, p2 = Point3D(0, 0, 0), Point3D(3, 4, 5) >>> p3, p4 = Point3D(2, 1, 1), Point3D(8, 9, 11) >>> l1, l2 = Line3D(p1, p2), Line3D(p3, p4) >>> Line3D.is_parallel(l1, l2) True >>> p5 = Point3D(6, 6, 6) >>> l3 = Line3D(p3, p5) >>> Line3D.is_parallel(l1, l3) False """ if not isinstance(l1, LinearEntity) and not isinstance(l2, LinearEntity): raise TypeError('Must pass only LinearEntity objects') return l1.direction.is_scalar_multiple(l2.direction)
[docs] def is_perpendicular(l1, l2): """Are two linear entities perpendicular? Parameters ========== l1 : LinearEntity l2 : LinearEntity Returns ======= True : if l1 and l2 are perpendicular, False : otherwise. See Also ======== coefficients Examples ======== >>> from sympy import Point, Line >>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(-1, 1) >>> l1, l2 = Line(p1, p2), Line(p1, p3) >>> l1.is_perpendicular(l2) True >>> p4 = Point(5, 3) >>> l3 = Line(p1, p4) >>> l1.is_perpendicular(l3) False >>> from sympy import Point3D, Line3D >>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(-1, 2, 0) >>> l1, l2 = Line3D(p1, p2), Line3D(p2, p3) >>> l1.is_perpendicular(l2) False >>> p4 = Point3D(5, 3, 7) >>> l3 = Line3D(p1, p4) >>> l1.is_perpendicular(l3) False """ if not isinstance(l1, LinearEntity) and not isinstance(l2, LinearEntity): raise TypeError('Must pass only LinearEntity objects') return S.Zero.equals(l1.direction.dot(l2.direction))
[docs] def is_similar(self, other): """ Return True if self and other are contained in the same line. Examples ======== >>> from sympy import Point, Line >>> p1, p2, p3 = Point(0, 1), Point(3, 4), Point(2, 3) >>> l1 = Line(p1, p2) >>> l2 = Line(p1, p3) >>> l1.is_similar(l2) True """ l = Line(self.p1, self.p2) return l.contains(other)
@property def length(self): """ The length of the line. Examples ======== >>> from sympy import Point, Line >>> p1, p2 = Point(0, 0), Point(3, 5) >>> l1 = Line(p1, p2) >>> l1.length oo """ return S.Infinity @property def p1(self): """The first defining point of a linear entity. See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Point, Line >>> p1, p2 = Point(0, 0), Point(5, 3) >>> l = Line(p1, p2) >>> l.p1 Point2D(0, 0) """ return self.args[0] @property def p2(self): """The second defining point of a linear entity. See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Point, Line >>> p1, p2 = Point(0, 0), Point(5, 3) >>> l = Line(p1, p2) >>> l.p2 Point2D(5, 3) """ return self.args[1]
[docs] def parallel_line(self, p): """Create a new Line parallel to this linear entity which passes through the point `p`. Parameters ========== p : Point Returns ======= line : Line See Also ======== is_parallel Examples ======== >>> from sympy import Point, Line >>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2) >>> l1 = Line(p1, p2) >>> l2 = l1.parallel_line(p3) >>> p3 in l2 True >>> l1.is_parallel(l2) True >>> from sympy import Point3D, Line3D >>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(2, 3, 4), Point3D(-2, 2, 0) >>> l1 = Line3D(p1, p2) >>> l2 = l1.parallel_line(p3) >>> p3 in l2 True >>> l1.is_parallel(l2) True """ p = Point(p, dim=self.ambient_dimension) return Line(p, p + self.direction)
[docs] def perpendicular_line(self, p): """Create a new Line perpendicular to this linear entity which passes through the point `p`. Parameters ========== p : Point Returns ======= line : Line See Also ======== is_perpendicular, perpendicular_segment Examples ======== >>> from sympy import Point, Line >>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2) >>> l1 = Line(p1, p2) >>> l2 = l1.perpendicular_line(p3) >>> p3 in l2 True >>> l1.is_perpendicular(l2) True >>> from sympy import Point3D, Line3D >>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(2, 3, 4), Point3D(-2, 2, 0) >>> l1 = Line3D(p1, p2) >>> l2 = l1.perpendicular_line(p3) >>> p3 in l2 True >>> l1.is_perpendicular(l2) True """ p = Point(p, dim=self.ambient_dimension) if p in self: p = p + self.direction.orthogonal_direction return Line(p, self.projection(p))
[docs] def perpendicular_segment(self, p): """Create a perpendicular line segment from `p` to this line. The enpoints of the segment are ``p`` and the closest point in the line containing self. (If self is not a line, the point might not be in self.) Parameters ========== p : Point Returns ======= segment : Segment Notes ===== Returns `p` itself if `p` is on this linear entity. See Also ======== perpendicular_line Examples ======== >>> from sympy import Point, Line >>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 2) >>> l1 = Line(p1, p2) >>> s1 = l1.perpendicular_segment(p3) >>> l1.is_perpendicular(s1) True >>> p3 in s1 True >>> l1.perpendicular_segment(Point(4, 0)) Segment2D(Point2D(2, 2), Point2D(4, 0)) >>> from sympy import Point3D, Line3D >>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, 2, 0) >>> l1 = Line3D(p1, p2) >>> s1 = l1.perpendicular_segment(p3) >>> l1.is_perpendicular(s1) True >>> p3 in s1 True >>> l1.perpendicular_segment(Point3D(4, 0, 0)) Segment3D(Point3D(4/3, 4/3, 4/3), Point3D(4, 0, 0)) """ p = Point(p, dim=self.ambient_dimension) if p in self: return p l = self.perpendicular_line(p) # The intersection should be unique, so unpack the singleton p2, = Intersection(Line(self.p1, self.p2), l) return Segment(p, p2)
@property def points(self): """The two points used to define this linear entity. Returns ======= points : tuple of Points See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Point, Line >>> p1, p2 = Point(0, 0), Point(5, 11) >>> l1 = Line(p1, p2) >>> l1.points (Point2D(0, 0), Point2D(5, 11)) """ return (self.p1, self.p2)
[docs] def projection(self, other): """Project a point, line, ray, or segment onto this linear entity. Parameters ========== other : Point or LinearEntity (Line, Ray, Segment) Returns ======= projection : Point or LinearEntity (Line, Ray, Segment) The return type matches the type of the parameter ``other``. Raises ====== GeometryError When method is unable to perform projection. Notes ===== A projection involves taking the two points that define the linear entity and projecting those points onto a Line and then reforming the linear entity using these projections. A point P is projected onto a line L by finding the point on L that is closest to P. This point is the intersection of L and the line perpendicular to L that passes through P. See Also ======== sympy.geometry.point.Point, perpendicular_line Examples ======== >>> from sympy import Point, Line, Segment, Rational >>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(Rational(1, 2), 0) >>> l1 = Line(p1, p2) >>> l1.projection(p3) Point2D(1/4, 1/4) >>> p4, p5 = Point(10, 0), Point(12, 1) >>> s1 = Segment(p4, p5) >>> l1.projection(s1) Segment2D(Point2D(5, 5), Point2D(13/2, 13/2)) >>> p1, p2, p3 = Point(0, 0, 1), Point(1, 1, 2), Point(2, 0, 1) >>> l1 = Line(p1, p2) >>> l1.projection(p3) Point3D(2/3, 2/3, 5/3) >>> p4, p5 = Point(10, 0, 1), Point(12, 1, 3) >>> s1 = Segment(p4, p5) >>> l1.projection(s1) Segment3D(Point3D(10/3, 10/3, 13/3), Point3D(5, 5, 6)) """ if not isinstance(other, GeometryEntity): other = Point(other, dim=self.ambient_dimension) def proj_point(p): return Point.project(p - self.p1, self.direction) + self.p1 if isinstance(other, Point): return proj_point(other) elif isinstance(other, LinearEntity): p1, p2 = proj_point(other.p1), proj_point(other.p2) # test to see if we're degenerate if p1 == p2: return p1 projected = other.__class__(p1, p2) projected = Intersection(self, projected) # if we happen to have intersected in only a point, return that if projected.is_FiniteSet and len(projected) == 1: # projected is a set of size 1, so unpack it in `a` a, = projected return a return projected raise GeometryError( "Do not know how to project %s onto %s" % (other, self))
[docs] def random_point(self): """A random point on a LinearEntity. Returns ======= point : Point See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Point, Line >>> p1, p2 = Point(0, 0), Point(5, 3) >>> l1 = Line(p1, p2) >>> p3 = l1.random_point() >>> # random point - don't know its coords in advance >>> p3 # doctest: +ELLIPSIS Point2D(...) >>> # point should belong to the line >>> p3 in l1 True """ from random import randint from sympy.functions import floor # The lower and upper lower, upper = -2**32 - 1, 2**32 if isinstance(self, Ray): lower = 0 if isinstance(self, Segment): lower = 0 upper = floor(self.length) t = randint(lower, upper) return self.direction*t/abs(self.direction) + self.p1
[docs]class Line(LinearEntity): """An infinite line in space. A line is declared with two distinct points. A 2D line may be declared with a point and slope and a 3D line may be defined with a point and a direction ratio. Parameters ========== p1 : Point p2 : Point slope : sympy expression direction_ratio : list Notes ===== `Line` will automatically subclass to `Line2D` or `Line3D` based on the dimension of `p1`. The `slope` argument is only relevant for `Line2D` and the `direction_ratio` argument is only relevant for `Line3D`. See Also ======== sympy.geometry.point.Point sympy.geometry.line.Line2D sympy.geometry.line.Line3D Examples ======== >>> from sympy import Point >>> from sympy.geometry import Line, Segment >>> L = Line(Point(2,3), Point(3,5)) >>> L Line2D(Point2D(2, 3), Point2D(3, 5)) >>> L.points (Point2D(2, 3), Point2D(3, 5)) >>> L.equation() -2*x + y + 1 >>> L.coefficients (-2, 1, 1) Instantiate with keyword ``slope``: >>> Line(Point(0, 0), slope=0) Line2D(Point2D(0, 0), Point2D(1, 0)) Instantiate with another linear object >>> s = Segment((0, 0), (0, 1)) >>> Line(s).equation() x """ def __new__(cls, p1, p2=None, **kwargs): if isinstance(p1, LinearEntity): if p2: raise ValueError('If p1 is a LinearEntity, p2 must be None.') dim = len(p1.p1) else: p1 = Point(p1) dim = len(p1) if p2 is not None or isinstance(p2, Point) and p2.ambient_dimension != dim: p2 = Point(p2) if dim == 2: return Line2D(p1, p2, **kwargs) elif dim == 3: return Line3D(p1, p2, **kwargs) return LinearEntity.__new__(cls, p1, p2, **kwargs)
[docs] def contains(self, other): """ Return True if `other` is on this Line, or False otherwise. Examples ======== >>> from sympy import Line,Point >>> p1, p2 = Point(0, 1), Point(3, 4) >>> l = Line(p1, p2) >>> l.contains(p1) True >>> l.contains((0, 1)) True >>> l.contains((0, 0)) False >>> a = (0, 0, 0) >>> b = (1, 1, 1) >>> c = (2, 2, 2) >>> l1 = Line(a, b) >>> l2 = Line(b, a) >>> l1 == l2 False >>> l1 in l2 True """ if not isinstance(other, GeometryEntity): other = Point(other, dim=self.ambient_dimension) if isinstance(other, Point): return Point.is_collinear(other, self.p1, self.p2) if isinstance(other, LinearEntity): return Point.is_collinear(self.p1, self.p2, other.p1, other.p2) return False
[docs] def distance(self, other): """ Finds the shortest distance between a line and a point. Raises ====== NotImplementedError is raised if `other` is not a Point Examples ======== >>> from sympy import Point, Line >>> p1, p2 = Point(0, 0), Point(1, 1) >>> s = Line(p1, p2) >>> s.distance(Point(-1, 1)) sqrt(2) >>> s.distance((-1, 2)) 3*sqrt(2)/2 >>> p1, p2 = Point(0, 0, 0), Point(1, 1, 1) >>> s = Line(p1, p2) >>> s.distance(Point(-1, 1, 1)) 2*sqrt(6)/3 >>> s.distance((-1, 1, 1)) 2*sqrt(6)/3 """ if not isinstance(other, GeometryEntity): other = Point(other, dim=self.ambient_dimension) if self.contains(other): return S.Zero return self.perpendicular_segment(other).length
@deprecated(useinstead="equals", issue=12860, deprecated_since_version="1.0") def equal(self, other): return self.equals(other)
[docs] def equals(self, other): """Returns True if self and other are the same mathematical entities""" if not isinstance(other, Line): return False return Point.is_collinear(self.p1, other.p1, self.p2, other.p2)
[docs] def plot_interval(self, parameter='t'): """The plot interval for the default geometric plot of line. Gives values that will produce a line that is +/- 5 units long (where a unit is the distance between the two points that define the line). Parameters ========== parameter : str, optional Default value is 't'. Returns ======= plot_interval : list (plot interval) [parameter, lower_bound, upper_bound] Examples ======== >>> from sympy import Point, Line >>> p1, p2 = Point(0, 0), Point(5, 3) >>> l1 = Line(p1, p2) >>> l1.plot_interval() [t, -5, 5] """ t = _symbol(parameter, real=True) return [t, -5, 5]
[docs]class Ray(LinearEntity): """A Ray is a semi-line in the space with a source point and a direction. Parameters ========== p1 : Point The source of the Ray p2 : Point or radian value This point determines the direction in which the Ray propagates. If given as an angle it is interpreted in radians with the positive direction being ccw. Attributes ========== source See Also ======== sympy.geometry.line.Ray2D sympy.geometry.line.Ray3D sympy.geometry.point.Point sympy.geometry.line.Line Notes ===== `Ray` will automatically subclass to `Ray2D` or `Ray3D` based on the dimension of `p1`. Examples ======== >>> from sympy import Point, pi >>> from sympy.geometry import Ray >>> r = Ray(Point(2, 3), Point(3, 5)) >>> r Ray2D(Point2D(2, 3), Point2D(3, 5)) >>> r.points (Point2D(2, 3), Point2D(3, 5)) >>> r.source Point2D(2, 3) >>> r.xdirection oo >>> r.ydirection oo >>> r.slope 2 >>> Ray(Point(0, 0), angle=pi/4).slope 1 """ def __new__(cls, p1, p2=None, **kwargs): p1 = Point(p1) if p2 is not None: p1, p2 = Point._normalize_dimension(p1, Point(p2)) dim = len(p1) if dim == 2: return Ray2D(p1, p2, **kwargs) elif dim == 3: return Ray3D(p1, p2, **kwargs) return LinearEntity.__new__(cls, p1, *pts, **kwargs) def _svg(self, scale_factor=1., fill_color="#66cc99"): """Returns SVG path element for the LinearEntity. Parameters ========== scale_factor : float Multiplication factor for the SVG stroke-width. Default is 1. fill_color : str, optional Hex string for fill color. Default is "#66cc99". """ from sympy.core.evalf import N verts = (N(self.p1), N(self.p2)) coords = ["{0},{1}".format(p.x, p.y) for p in verts] path = "M {0} L {1}".format(coords[0], " L ".join(coords[1:])) return ( '<path fill-rule="evenodd" fill="{2}" stroke="#555555" ' 'stroke-width="{0}" opacity="0.6" d="{1}" ' 'marker-start="url(#markerCircle)" marker-end="url(#markerArrow)"/>' ).format(2. * scale_factor, path, fill_color)
[docs] def contains(self, other): """ Is other GeometryEntity contained in this Ray? Examples ======== >>> from sympy import Ray,Point,Segment >>> p1, p2 = Point(0, 0), Point(4, 4) >>> r = Ray(p1, p2) >>> r.contains(p1) True >>> r.contains((1, 1)) True >>> r.contains((1, 3)) False >>> s = Segment((1, 1), (2, 2)) >>> r.contains(s) True >>> s = Segment((1, 2), (2, 5)) >>> r.contains(s) False >>> r1 = Ray((2, 2), (3, 3)) >>> r.contains(r1) True >>> r1 = Ray((2, 2), (3, 5)) >>> r.contains(r1) False """ if not isinstance(other, GeometryEntity): other = Point(other, dim=self.ambient_dimension) if isinstance(other, Point): if Point.is_collinear(self.p1, self.p2, other): # if we're in the direction of the ray, our # direction vector dot the ray's direction vector # should be non-negative return bool((self.p2 - self.p1).dot(other - self.p1) >= S.Zero) return False elif isinstance(other, Ray): if Point.is_collinear(self.p1, self.p2, other.p1, other.p2): return bool((self.p2 - self.p1).dot(other.p2 - other.p1) > S.Zero) return False elif isinstance(other, Segment): return other.p1 in self and other.p2 in self # No other known entity can be contained in a Ray return False
[docs] def distance(self, other): """ Finds the shortest distance between the ray and a point. Raises ====== NotImplementedError is raised if `other` is not a Point Examples ======== >>> from sympy import Point, Ray >>> p1, p2 = Point(0, 0), Point(1, 1) >>> s = Ray(p1, p2) >>> s.distance(Point(-1, -1)) sqrt(2) >>> s.distance((-1, 2)) 3*sqrt(2)/2 >>> p1, p2 = Point(0, 0, 0), Point(1, 1, 2) >>> s = Ray(p1, p2) >>> s Ray3D(Point3D(0, 0, 0), Point3D(1, 1, 2)) >>> s.distance(Point(-1, -1, 2)) 4*sqrt(3)/3 >>> s.distance((-1, -1, 2)) 4*sqrt(3)/3 """ if not isinstance(other, GeometryEntity): other = Point(other, dim=self.ambient_dimension) if self.contains(other): return S.Zero proj = Line(self.p1, self.p2).projection(other) if self.contains(proj): return abs(other - proj) else: return abs(other - self.source)
[docs] def equals(self, other): """Returns True if self and other are the same mathematical entities""" if not isinstance(other, Ray): return False return self.source == other.source and other.p2 in self
[docs] def plot_interval(self, parameter='t'): """The plot interval for the default geometric plot of the Ray. Gives values that will produce a ray that is 10 units long (where a unit is the distance between the two points that define the ray). Parameters ========== parameter : str, optional Default value is 't'. Returns ======= plot_interval : list [parameter, lower_bound, upper_bound] Examples ======== >>> from sympy import Ray, pi >>> r = Ray((0, 0), angle=pi/4) >>> r.plot_interval() [t, 0, 10] """ t = _symbol(parameter, real=True) return [t, 0, 10]
@property def source(self): """The point from which the ray emanates. See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Point, Ray >>> p1, p2 = Point(0, 0), Point(4, 1) >>> r1 = Ray(p1, p2) >>> r1.source Point2D(0, 0) >>> p1, p2 = Point(0, 0, 0), Point(4, 1, 5) >>> r1 = Ray(p2, p1) >>> r1.source Point3D(4, 1, 5) """ return self.p1
[docs]class Segment(LinearEntity): """An undirected line segment in space. Parameters ========== p1 : Point p2 : Point Attributes ========== length : number or sympy expression midpoint : Point See Also ======== sympy.geometry.line.Segment2D sympy.geometry.line.Segment3D sympy.geometry.point.Point sympy.geometry.line.Line Notes ===== If 2D or 3D points are used to define `Segment`, it will be automatically subclassed to `Segment2D` or `Segment3D`. Examples ======== >>> from sympy import Point >>> from sympy.geometry import Segment >>> Segment((1, 0), (1, 1)) # tuples are interpreted as pts Segment2D(Point2D(1, 0), Point2D(1, 1)) >>> s = Segment(Point(4, 3), Point(1, 1)) >>> s Segment2D(Point2D(1, 1), Point2D(4, 3)) >>> s.points (Point2D(1, 1), Point2D(4, 3)) >>> s.slope 2/3 >>> s.length sqrt(13) >>> s.midpoint Point2D(5/2, 2) >>> Segment((1, 0, 0), (1, 1, 1)) # tuples are interpreted as pts Segment3D(Point3D(1, 0, 0), Point3D(1, 1, 1)) >>> s = Segment(Point(4, 3, 9), Point(1, 1, 7)) >>> s Segment3D(Point3D(1, 1, 7), Point3D(4, 3, 9)) >>> s.points (Point3D(1, 1, 7), Point3D(4, 3, 9)) >>> s.length sqrt(17) >>> s.midpoint Point3D(5/2, 2, 8) """ def __new__(cls, p1, p2, **kwargs): p1, p2 = Point._normalize_dimension(Point(p1), Point(p2)) dim = len(p1) if dim == 2: return Segment2D(p1, p2, **kwargs) elif dim == 3: return Segment3D(p1, p2, **kwargs) return LinearEntity.__new__(cls, p1, p2, **kwargs)
[docs] def contains(self, other): """ Is the other GeometryEntity contained within this Segment? Examples ======== >>> from sympy import Point, Segment >>> p1, p2 = Point(0, 1), Point(3, 4) >>> s = Segment(p1, p2) >>> s2 = Segment(p2, p1) >>> s.contains(s2) True >>> from sympy import Point3D, Segment3D >>> p1, p2 = Point3D(0, 1, 1), Point3D(3, 4, 5) >>> s = Segment3D(p1, p2) >>> s2 = Segment3D(p2, p1) >>> s.contains(s2) True >>> s.contains((p1 + p2) / 2) True """ if not isinstance(other, GeometryEntity): other = Point(other, dim=self.ambient_dimension) if isinstance(other, Point): if Point.is_collinear(other, self.p1, self.p2): d1, d2 = other - self.p1, other - self.p2 d = self.p2 - self.p1 # without the call to simplify, sympy cannot tell that an expression # like (a+b)*(a/2+b/2) is always non-negative. If it cannot be # determined, raise an Undecidable error try: # the triangle inequality says that |d1|+|d2| >= |d| and is strict # only if other lies in the line segment return bool(simplify(Eq(abs(d1) + abs(d2) - abs(d), 0))) except TypeError: raise Undecidable("Cannot determine if {} is in {}".format(other, self)) if isinstance(other, Segment): return other.p1 in self and other.p2 in self return False
[docs] def distance(self, other): """ Finds the shortest distance between a line segment and a point. Raises ====== NotImplementedError is raised if `other` is not a Point Examples ======== >>> from sympy import Point, Segment >>> p1, p2 = Point(0, 1), Point(3, 4) >>> s = Segment(p1, p2) >>> s.distance(Point(10, 15)) sqrt(170) >>> s.distance((0, 12)) sqrt(73) >>> from sympy import Point3D, Segment3D >>> p1, p2 = Point3D(0, 0, 3), Point3D(1, 1, 4) >>> s = Segment3D(p1, p2) >>> s.distance(Point3D(10, 15, 12)) sqrt(341) >>> s.distance((10, 15, 12)) sqrt(341) """ if not isinstance(other, GeometryEntity): other = Point(other, dim=self.ambient_dimension) if isinstance(other, Point): vp1 = other - self.p1 vp2 = other - self.p2 dot_prod_sign_1 = self.direction.dot(vp1) >= 0 dot_prod_sign_2 = self.direction.dot(vp2) <= 0 if dot_prod_sign_1 and dot_prod_sign_2: return Line(self.p1, self.p2).distance(other) if dot_prod_sign_1 and not dot_prod_sign_2: return abs(vp2) if not dot_prod_sign_1 and dot_prod_sign_2: return abs(vp1) raise NotImplementedError()
@property def length(self): """The length of the line segment. See Also ======== sympy.geometry.point.Point.distance Examples ======== >>> from sympy import Point, Segment >>> p1, p2 = Point(0, 0), Point(4, 3) >>> s1 = Segment(p1, p2) >>> s1.length 5 >>> from sympy import Point3D, Segment3D >>> p1, p2 = Point3D(0, 0, 0), Point3D(4, 3, 3) >>> s1 = Segment3D(p1, p2) >>> s1.length sqrt(34) """ return Point.distance(self.p1, self.p2) @property def midpoint(self): """The midpoint of the line segment. See Also ======== sympy.geometry.point.Point.midpoint Examples ======== >>> from sympy import Point, Segment >>> p1, p2 = Point(0, 0), Point(4, 3) >>> s1 = Segment(p1, p2) >>> s1.midpoint Point2D(2, 3/2) >>> from sympy import Point3D, Segment3D >>> p1, p2 = Point3D(0, 0, 0), Point3D(4, 3, 3) >>> s1 = Segment3D(p1, p2) >>> s1.midpoint Point3D(2, 3/2, 3/2) """ return Point.midpoint(self.p1, self.p2)
[docs] def perpendicular_bisector(self, p=None): """The perpendicular bisector of this segment. If no point is specified or the point specified is not on the bisector then the bisector is returned as a Line. Otherwise a Segment is returned that joins the point specified and the intersection of the bisector and the segment. Parameters ========== p : Point Returns ======= bisector : Line or Segment See Also ======== LinearEntity.perpendicular_segment Examples ======== >>> from sympy import Point, Segment >>> p1, p2, p3 = Point(0, 0), Point(6, 6), Point(5, 1) >>> s1 = Segment(p1, p2) >>> s1.perpendicular_bisector() Line2D(Point2D(3, 3), Point2D(-3, 9)) >>> s1.perpendicular_bisector(p3) Segment2D(Point2D(3, 3), Point2D(5, 1)) """ l = self.perpendicular_line(self.midpoint) if p is not None: p2 = Point(p, dim=self.ambient_dimension) if p2 in l: return Segment(self.midpoint, p2) return l
[docs] def plot_interval(self, parameter='t'): """The plot interval for the default geometric plot of the Segment gives values that will produce the full segment in a plot. Parameters ========== parameter : str, optional Default value is 't'. Returns ======= plot_interval : list [parameter, lower_bound, upper_bound] Examples ======== >>> from sympy import Point, Segment >>> p1, p2 = Point(0, 0), Point(5, 3) >>> s1 = Segment(p1, p2) >>> s1.plot_interval() [t, 0, 1] """ t = _symbol(parameter, real=True) return [t, 0, 1]
[docs]class LinearEntity2D(LinearEntity): """A base class for all linear entities (line, ray and segment) in a 2-dimensional Euclidean space. Attributes ========== p1 p2 coefficients slope points Notes ===== This is an abstract class and is not meant to be instantiated. See Also ======== sympy.geometry.entity.GeometryEntity """ @property def bounds(self): """Return a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure. """ verts = self.points xs = [p.x for p in verts] ys = [p.y for p in verts] return (min(xs), min(ys), max(xs), max(ys))
[docs] def perpendicular_line(self, p): """Create a new Line perpendicular to this linear entity which passes through the point `p`. Parameters ========== p : Point Returns ======= line : Line See Also ======== is_perpendicular, perpendicular_segment Examples ======== >>> from sympy import Point, Line >>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2) >>> l1 = Line(p1, p2) >>> l2 = l1.perpendicular_line(p3) >>> p3 in l2 True >>> l1.is_perpendicular(l2) True """ p = Point(p, dim=self.ambient_dimension) # any two lines in R^2 intersect, so blindly making # a line through p in an orthogonal direction will work return Line(p, p + self.direction.orthogonal_direction)
@property def slope(self): """The slope of this linear entity, or infinity if vertical. Returns ======= slope : number or sympy expression See Also ======== coefficients Examples ======== >>> from sympy import Point, Line >>> p1, p2 = Point(0, 0), Point(3, 5) >>> l1 = Line(p1, p2) >>> l1.slope 5/3 >>> p3 = Point(0, 4) >>> l2 = Line(p1, p3) >>> l2.slope oo """ d1, d2 = (self.p1 - self.p2).args if d1 == 0: return S.Infinity return simplify(d2/d1)
[docs]class Line2D(LinearEntity2D, Line): """An infinite line in space 2D. A line is declared with two distinct points or a point and slope as defined using keyword `slope`. Parameters ========== p1 : Point pt : Point slope : sympy expression See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Point >>> from sympy.abc import L >>> from sympy.geometry import Line, Segment >>> L = Line(Point(2,3), Point(3,5)) >>> L Line2D(Point2D(2, 3), Point2D(3, 5)) >>> L.points (Point2D(2, 3), Point2D(3, 5)) >>> L.equation() -2*x + y + 1 >>> L.coefficients (-2, 1, 1) Instantiate with keyword ``slope``: >>> Line(Point(0, 0), slope=0) Line2D(Point2D(0, 0), Point2D(1, 0)) Instantiate with another linear object >>> s = Segment((0, 0), (0, 1)) >>> Line(s).equation() x """ def __new__(cls, p1, pt=None, slope=None, **kwargs): if isinstance(p1, LinearEntity): if pt is not None: raise ValueError('When p1 is a LinearEntity, pt should be None') p1, pt = Point._normalize_dimension(*p1.args, dim=2) else: p1 = Point(p1, dim=2) if pt is not None and slope is None: try: p2 = Point(pt, dim=2) except (NotImplementedError, TypeError, ValueError): raise ValueError(filldedent(''' The 2nd argument was not a valid Point. If it was a slope, enter it with keyword "slope". ''')) elif slope is not None and pt is None: slope = sympify(slope) if slope.is_finite is False: # when infinite slope, don't change x dx = 0 dy = 1 else: # go over 1 up slope dx = 1 dy = slope # XXX avoiding simplification by adding to coords directly p2 = Point(p1.x + dx, p1.y + dy, evaluate=False) else: raise ValueError('A 2nd Point or keyword "slope" must be used.') return LinearEntity2D.__new__(cls, p1, p2, **kwargs) def _svg(self, scale_factor=1., fill_color="#66cc99"): """Returns SVG path element for the LinearEntity. Parameters ========== scale_factor : float Multiplication factor for the SVG stroke-width. Default is 1. fill_color : str, optional Hex string for fill color. Default is "#66cc99". """ from sympy.core.evalf import N verts = (N(self.p1), N(self.p2)) coords = ["{0},{1}".format(p.x, p.y) for p in verts] path = "M {0} L {1}".format(coords[0], " L ".join(coords[1:])) return ( '<path fill-rule="evenodd" fill="{2}" stroke="#555555" ' 'stroke-width="{0}" opacity="0.6" d="{1}" ' 'marker-start="url(#markerReverseArrow)" marker-end="url(#markerArrow)"/>' ).format(2. * scale_factor, path, fill_color) @property def coefficients(self): """The coefficients (`a`, `b`, `c`) for `ax + by + c = 0`. See Also ======== sympy.geometry.line.Line.equation Examples ======== >>> from sympy import Point, Line >>> from sympy.abc import x, y >>> p1, p2 = Point(0, 0), Point(5, 3) >>> l = Line(p1, p2) >>> l.coefficients (-3, 5, 0) >>> p3 = Point(x, y) >>> l2 = Line(p1, p3) >>> l2.coefficients (-y, x, 0) """ p1, p2 = self.points if p1.x == p2.x: return (S.One, S.Zero, -p1.x) elif p1.y == p2.y: return (S.Zero, S.One, -p1.y) return tuple([simplify(i) for i in (self.p1.y - self.p2.y, self.p2.x - self.p1.x, self.p1.x*self.p2.y - self.p1.y*self.p2.x)])
[docs] def equation(self, x='x', y='y'): """The equation of the line: ax + by + c. Parameters ========== x : str, optional The name to use for the x-axis, default value is 'x'. y : str, optional The name to use for the y-axis, default value is 'y'. Returns ======= equation : sympy expression See Also ======== LinearEntity.coefficients Examples ======== >>> from sympy import Point, Line >>> p1, p2 = Point(1, 0), Point(5, 3) >>> l1 = Line(p1, p2) >>> l1.equation() -3*x + 4*y + 3 """ x = _symbol(x, real=True) y = _symbol(y, real=True) p1, p2 = self.points if p1.x == p2.x: return x - p1.x elif p1.y == p2.y: return y - p1.y a, b, c = self.coefficients return a*x + b*y + c
[docs]class Ray2D(LinearEntity2D, Ray): """ A Ray is a semi-line in the space with a source point and a direction. Parameters ========== p1 : Point The source of the Ray p2 : Point or radian value This point determines the direction in which the Ray propagates. If given as an angle it is interpreted in radians with the positive direction being ccw. Attributes ========== source xdirection ydirection See Also ======== sympy.geometry.point.Point, Line Examples ======== >>> from sympy import Point, pi >>> from sympy.geometry import Ray >>> r = Ray(Point(2, 3), Point(3, 5)) >>> r Ray2D(Point2D(2, 3), Point2D(3, 5)) >>> r.points (Point2D(2, 3), Point2D(3, 5)) >>> r.source Point2D(2, 3) >>> r.xdirection oo >>> r.ydirection oo >>> r.slope 2 >>> Ray(Point(0, 0), angle=pi/4).slope 1 """ def __new__(cls, p1, pt=None, angle=None, **kwargs): p1 = Point(p1, dim=2) if pt is not None and angle is None: try: p2 = Point(pt, dim=2) except (NotImplementedError, TypeError, ValueError): from sympy.utilities.misc import filldedent raise ValueError(filldedent(''' The 2nd argument was not a valid Point; if it was meant to be an angle it should be given with keyword "angle".''')) if p1 == p2: raise ValueError('A Ray requires two distinct points.') elif angle is not None and pt is None: # we need to know if the angle is an odd multiple of pi/2 c = pi_coeff(sympify(angle)) p2 = None if c is not None: if c.is_Rational: if c.q == 2: if c.p == 1: p2 = p1 + Point(0, 1) elif c.p == 3: p2 = p1 + Point(0, -1) elif c.q == 1: if c.p == 0: p2 = p1 + Point(1, 0) elif c.p == 1: p2 = p1 + Point(-1, 0) if p2 is None: c *= S.Pi else: c = angle % (2*S.Pi) if not p2: m = 2*c/S.Pi left = And(1 < m, m < 3) # is it in quadrant 2 or 3? x = Piecewise((-1, left), (Piecewise((0, Eq(m % 1, 0)), (1, True)), True)) y = Piecewise((-tan(c), left), (Piecewise((1, Eq(m, 1)), (-1, Eq(m, 3)), (tan(c), True)), True)) p2 = p1 + Point(x, y) else: raise ValueError('A 2nd point or keyword "angle" must be used.') return LinearEntity2D.__new__(cls, p1, p2, **kwargs) @property def xdirection(self): """The x direction of the ray. Positive infinity if the ray points in the positive x direction, negative infinity if the ray points in the negative x direction, or 0 if the ray is vertical. See Also ======== ydirection Examples ======== >>> from sympy import Point, Ray >>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, -1) >>> r1, r2 = Ray(p1, p2), Ray(p1, p3) >>> r1.xdirection oo >>> r2.xdirection 0 """ if self.p1.x < self.p2.x: return S.Infinity elif self.p1.x == self.p2.x: return S.Zero else: return S.NegativeInfinity @property def ydirection(self): """The y direction of the ray. Positive infinity if the ray points in the positive y direction, negative infinity if the ray points in the negative y direction, or 0 if the ray is horizontal. See Also ======== xdirection Examples ======== >>> from sympy import Point, Ray >>> p1, p2, p3 = Point(0, 0), Point(-1, -1), Point(-1, 0) >>> r1, r2 = Ray(p1, p2), Ray(p1, p3) >>> r1.ydirection -oo >>> r2.ydirection 0 """ if self.p1.y < self.p2.y: return S.Infinity elif self.p1.y == self.p2.y: return S.Zero else: return S.NegativeInfinity
[docs] def closing_angle(r1, r2): """Return the angle by which r2 must be rotated so it faces the same direction as r1. Parameters ========== r1 : Ray2D r2 : Ray2D Returns ======= angle : angle in radians (ccw angle is positive) See Also ======== LinearEntity.angle_between Examples ======== >>> from sympy import Ray, pi >>> r1 = Ray((0, 0), (1, 0)) >>> r2 = r1.rotate(-pi/2) >>> angle = r1.closing_angle(r2); angle pi/2 >>> r2.rotate(angle).direction.unit == r1.direction.unit True >>> r2.closing_angle(r1) -pi/2 """ if not all(isinstance(r, Ray2D) for r in (r1, r2)): # although the direction property is defined for # all linear entities, only the Ray is truly a # directed object raise TypeError('Both arguments must be Ray2D objects.') a1 = atan2(*list(reversed(r1.direction.args))) a2 = atan2(*list(reversed(r2.direction.args))) return a1 - a2
[docs]class Segment2D(LinearEntity2D, Segment): """An undirected line segment in 2D space. Parameters ========== p1 : Point p2 : Point Attributes ========== length : number or sympy expression midpoint : Point See Also ======== sympy.geometry.point.Point, Line Examples ======== >>> from sympy import Point >>> from sympy.geometry import Segment >>> Segment((1, 0), (1, 1)) # tuples are interpreted as pts Segment2D(Point2D(1, 0), Point2D(1, 1)) >>> s = Segment(Point(4, 3), Point(1, 1)) >>> s Segment2D(Point2D(1, 1), Point2D(4, 3)) >>> s.points (Point2D(1, 1), Point2D(4, 3)) >>> s.slope 2/3 >>> s.length sqrt(13) >>> s.midpoint Point2D(5/2, 2) """ def __new__(cls, p1, p2, **kwargs): # Reorder the two points under the following ordering: # if p1.x != p2.x then p1.x < p2.x # if p1.x == p2.x then p1.y < p2.y p1 = Point(p1, dim=2) p2 = Point(p2, dim=2) if p1 == p2: return p1 if (p1.x > p2.x) == True: p1, p2 = p2, p1 elif (p1.x == p2.x) == True and (p1.y > p2.y) == True: p1, p2 = p2, p1 return LinearEntity2D.__new__(cls, p1, p2, **kwargs) def _svg(self, scale_factor=1., fill_color="#66cc99"): """Returns SVG path element for the LinearEntity. Parameters ========== scale_factor : float Multiplication factor for the SVG stroke-width. Default is 1. fill_color : str, optional Hex string for fill color. Default is "#66cc99". """ from sympy.core.evalf import N verts = (N(self.p1), N(self.p2)) coords = ["{0},{1}".format(p.x, p.y) for p in verts] path = "M {0} L {1}".format(coords[0], " L ".join(coords[1:])) return ( '<path fill-rule="evenodd" fill="{2}" stroke="#555555" ' 'stroke-width="{0}" opacity="0.6" d="{1}" />' ).format(2. * scale_factor, path, fill_color)
[docs]class LinearEntity3D(LinearEntity): """An base class for all linear entities (line, ray and segment) in a 3-dimensional Euclidean space. Attributes ========== p1 p2 direction_ratio direction_cosine points Notes ===== This is a base class and is not meant to be instantiated. """ def __new__(cls, p1, p2, **kwargs): p1 = Point3D(p1, dim=3) p2 = Point3D(p2, dim=3) if p1 == p2: # if it makes sense to return a Point, handle in subclass raise ValueError( "%s.__new__ requires two unique Points." % cls.__name__) return GeometryEntity.__new__(cls, p1, p2, **kwargs) ambient_dimension = 3 @property def direction_ratio(self): """The direction ratio of a given line in 3D. See Also ======== sympy.geometry.line.Line.equation Examples ======== >>> from sympy import Point3D, Line3D >>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1) >>> l = Line3D(p1, p2) >>> l.direction_ratio [5, 3, 1] """ p1, p2 = self.points return p1.direction_ratio(p2) @property def direction_cosine(self): """The normalized direction ratio of a given line in 3D. See Also ======== sympy.geometry.line.Line.equation Examples ======== >>> from sympy import Point3D, Line3D >>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1) >>> l = Line3D(p1, p2) >>> l.direction_cosine [sqrt(35)/7, 3*sqrt(35)/35, sqrt(35)/35] >>> sum(i**2 for i in _) 1 """ p1, p2 = self.points return p1.direction_cosine(p2)
[docs]class Line3D(LinearEntity3D, Line): """An infinite 3D line in space. A line is declared with two distinct points or a point and direction_ratio as defined using keyword `direction_ratio`. Parameters ========== p1 : Point3D pt : Point3D direction_ratio : list See Also ======== sympy.geometry.point.Point3D sympy.geometry.line.Line sympy.geometry.line.Line2D Examples ======== >>> from sympy import Point3D >>> from sympy.geometry import Line3D, Segment3D >>> L = Line3D(Point3D(2, 3, 4), Point3D(3, 5, 1)) >>> L Line3D(Point3D(2, 3, 4), Point3D(3, 5, 1)) >>> L.points (Point3D(2, 3, 4), Point3D(3, 5, 1)) """ def __new__(cls, p1, pt=None, direction_ratio=[], **kwargs): if isinstance(p1, LinearEntity3D): if pt is not None: raise ValueError('if p1 is a LinearEntity, pt must be None.') p1, pt = p1.args else: p1 = Point(p1, dim=3) if pt is not None and len(direction_ratio) == 0: pt = Point(pt, dim=3) elif len(direction_ratio) == 3 and pt is None: pt = Point3D(p1.x + direction_ratio[0], p1.y + direction_ratio[1], p1.z + direction_ratio[2]) else: raise ValueError('A 2nd Point or keyword "direction_ratio" must ' 'be used.') return LinearEntity3D.__new__(cls, p1, pt, **kwargs)
[docs] def equation(self, x='x', y='y', z='z', k='k'): """The equation of the line in 3D Parameters ========== x : str, optional The name to use for the x-axis, default value is 'x'. y : str, optional The name to use for the y-axis, default value is 'y'. z : str, optional The name to use for the x-axis, default value is 'z'. Returns ======= equation : tuple Examples ======== >>> from sympy import Point3D, Line3D >>> p1, p2 = Point3D(1, 0, 0), Point3D(5, 3, 0) >>> l1 = Line3D(p1, p2) >>> l1.equation() (x/4 - 1/4, y/3, zoo*z, k) """ x, y, z, k = [_symbol(i, real=True) for i in (x, y, z, k)] p1, p2 = self.points a = p1.direction_ratio(p2) return (((x - p1.x)/a[0]), ((y - p1.y)/a[1]), ((z - p1.z)/a[2]), k)
[docs]class Ray3D(LinearEntity3D, Ray): """ A Ray is a semi-line in the space with a source point and a direction. Parameters ========== p1 : Point3D The source of the Ray p2 : Point or a direction vector direction_ratio: Determines the direction in which the Ray propagates. Attributes ========== source xdirection ydirection zdirection See Also ======== sympy.geometry.point.Point3D, Line3D Examples ======== >>> from sympy import Point3D >>> from sympy.geometry import Ray3D >>> r = Ray3D(Point3D(2, 3, 4), Point3D(3, 5, 0)) >>> r Ray3D(Point3D(2, 3, 4), Point3D(3, 5, 0)) >>> r.points (Point3D(2, 3, 4), Point3D(3, 5, 0)) >>> r.source Point3D(2, 3, 4) >>> r.xdirection oo >>> r.ydirection oo >>> r.direction_ratio [1, 2, -4] """ def __new__(cls, p1, pt=None, direction_ratio=[], **kwargs): from sympy.utilities.misc import filldedent if isinstance(p1, LinearEntity3D): if pt is not None: raise ValueError('If p1 is a LinearEntity, pt must be None') p1, pt = p1.args else: p1 = Point(p1, dim=3) if pt is not None and len(direction_ratio) == 0: pt = Point(pt, dim=3) elif len(direction_ratio) == 3 and pt is None: pt = Point3D(p1.x + direction_ratio[0], p1.y + direction_ratio[1], p1.z + direction_ratio[2]) else: raise ValueError(filldedent(''' A 2nd Point or keyword "direction_ratio" must be used. ''')) return LinearEntity3D.__new__(cls, p1, pt, **kwargs) @property def xdirection(self): """The x direction of the ray. Positive infinity if the ray points in the positive x direction, negative infinity if the ray points in the negative x direction, or 0 if the ray is vertical. See Also ======== ydirection Examples ======== >>> from sympy import Point3D, Ray3D >>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, -1, 0) >>> r1, r2 = Ray3D(p1, p2), Ray3D(p1, p3) >>> r1.xdirection oo >>> r2.xdirection 0 """ if self.p1.x < self.p2.x: return S.Infinity elif self.p1.x == self.p2.x: return S.Zero else: return S.NegativeInfinity @property def ydirection(self): """The y direction of the ray. Positive infinity if the ray points in the positive y direction, negative infinity if the ray points in the negative y direction, or 0 if the ray is horizontal. See Also ======== xdirection Examples ======== >>> from sympy import Point3D, Ray3D >>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(-1, -1, -1), Point3D(-1, 0, 0) >>> r1, r2 = Ray3D(p1, p2), Ray3D(p1, p3) >>> r1.ydirection -oo >>> r2.ydirection 0 """ if self.p1.y < self.p2.y: return S.Infinity elif self.p1.y == self.p2.y: return S.Zero else: return S.NegativeInfinity @property def zdirection(self): """The z direction of the ray. Positive infinity if the ray points in the positive z direction, negative infinity if the ray points in the negative z direction, or 0 if the ray is horizontal. See Also ======== xdirection Examples ======== >>> from sympy import Point3D, Ray3D >>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(-1, -1, -1), Point3D(-1, 0, 0) >>> r1, r2 = Ray3D(p1, p2), Ray3D(p1, p3) >>> r1.ydirection -oo >>> r2.ydirection 0 >>> r2.zdirection 0 """ if self.p1.z < self.p2.z: return S.Infinity elif self.p1.z == self.p2.z: return S.Zero else: return S.NegativeInfinity
[docs]class Segment3D(LinearEntity3D, Segment): """A undirected line segment in a 3D space. Parameters ========== p1 : Point3D p2 : Point3D Attributes ========== length : number or sympy expression midpoint : Point3D See Also ======== sympy.geometry.point.Point3D, Line3D Examples ======== >>> from sympy import Point3D >>> from sympy.geometry import Segment3D >>> Segment3D((1, 0, 0), (1, 1, 1)) # tuples are interpreted as pts Segment3D(Point3D(1, 0, 0), Point3D(1, 1, 1)) >>> s = Segment3D(Point3D(4, 3, 9), Point3D(1, 1, 7)) >>> s Segment3D(Point3D(1, 1, 7), Point3D(4, 3, 9)) >>> s.points (Point3D(1, 1, 7), Point3D(4, 3, 9)) >>> s.length sqrt(17) >>> s.midpoint Point3D(5/2, 2, 8) """ def __new__(cls, p1, p2, **kwargs): # Reorder the two points under the following ordering: # if p1.x != p2.x then p1.x < p2.x # if p1.x == p2.x then p1.y < p2.y # The z-coordinate will not come into picture while ordering p1 = Point(p1, dim=3) p2 = Point(p2, dim=3) if p1 == p2: return p1 if (p1.x > p2.x) == True: p1, p2 = p2, p1 elif (p1.x == p2.x) == True and (p1.y > p2.y) == True: p1, p2 = p2, p1 return LinearEntity3D.__new__(cls, p1, p2, **kwargs)