Source code for sympy.geometry.line

"""Line-like geometrical entities.

Contains
========
LinearEntity
Line
Ray
Segment

"""
from sympy.core import S, C, sympify, Dummy
from sympy.functions.elementary.trigonometric import _pi_coeff as pi_coeff
from sympy.core.logic import fuzzy_and
from sympy.simplify import simplify
from sympy.solvers import solve
from sympy.geometry.exceptions import GeometryError
from .entity import GeometryEntity
from .point import Point
from .util import _symbol

# TODO: this should be placed elsewhere and reused in other modules
class Undecidable(ValueError):
    pass

[docs]class LinearEntity(GeometryEntity): """An abstract 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. Subclasses should implement the following methods: * __eq__ * contains See Also ======== sympy.geometry.entity.GeometryEntity """ def __new__(cls, p1, p2, **kwargs): p1 = Point(p1) p2 = Point(p2) if p1 == p2: # Rolygon returns lower priority classes...should LinearEntity, too? return p1 # raise ValueError("%s.__new__ requires two unique Points." % cls.__name__) return GeometryEntity.__new__(cls, p1, p2, **kwargs) @property
[docs] 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 Point(0, 0) """ return self.args[0]
@property
[docs] 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 Point(5, 3) """ return self.args[1]
@property
[docs] def coefficients(self): """The coefficients (`a`, `b`, `c`) for the linear equation `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 is_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 are concurrent, False : otherwise. Notes ===== Simply take the first two lines and find their intersection. If there is no intersection, then the first two lines were parallel and had no intersection so concurrency is impossible amongst the whole set. Otherwise, check to see if the intersection point of the first two lines is a member on the rest of the lines. If so, the lines are concurrent. See Also ======== sympy.geometry.util.intersection Examples ======== >>> from sympy import Point, Line >>> 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) >>> l1.is_concurrent(l2, l3) True >>> l4 = Line(p2, p3) >>> l4.is_concurrent(l2, l3) False """ # Concurrency requires intersection at a single point; One linear # entity cannot be concurrent. if len(lines) <= 1: return False try: # Get the intersection (if parallel) p = lines[0].intersection(lines[1]) if len(p) == 0: return False # Make sure the intersection is on every linear entity for line in lines[2:]: if p[0] not in line: return False return True except AttributeError: return False
[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 """ try: a1, b1, c1 = l1.coefficients a2, b2, c2 = l2.coefficients return bool(simplify(a1*b2 - b1*a2) == 0) except AttributeError: return False
[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 """ try: a1, b1, c1 = l1.coefficients a2, b2, c2 = l2.coefficients return bool(simplify(a1*a2 + b1*b2) == 0) except AttributeError: return False
[docs] def angle_between(l1, l2): """The angle formed between the 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 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 """ v1 = l1.p2 - l1.p1 v2 = l2.p2 - l2.p1 return C.acos(v1.dot(v2)/(abs(v1)*abs(v2)))
[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 """ d = self.p1 - self.p2 return Line(p, p + d)
[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 """ d1, d2 = (self.p1 - self.p2).args if d2 == 0: # If a horizontal line if p.y == self.p1.y: # if p is on this linear entity return Line(p, p + Point(0, 1)) else: p2 = Point(p.x, self.p1.y) return Line(p, p2) else: p2 = Point(p.x - d2, p.y + d1) return Line(p, p2)
[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)) Segment(Point(2, 2), Point(4, 0)) """ if p in self: return p pl = self.perpendicular_line(p) p2 = Line(self).intersection(pl)[0] return Segment(p, p2)
@property
[docs] 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
[docs] 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)
@property
[docs] 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 (Point(0, 0), Point(5, 11)) """ return (self.p1, self.p2)
[docs] def projection(self, o): """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 is done by creating a perpendicular line through P and L and finding its intersection with L. 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) Point(1/4, 1/4) >>> p4, p5 = Point(10, 0), Point(12, 1) >>> s1 = Segment(p4, p5) >>> l1.projection(s1) Segment(Point(5, 5), Point(13/2, 13/2)) """ tline = Line(self.p1, self.p2) def _project(p): """Project a point onto the line representing self.""" if p in tline: return p l1 = tline.perpendicular_line(p) return tline.intersection(l1)[0] projected = None if isinstance(o, Point): return _project(o) elif isinstance(o, LinearEntity): n_p1 = _project(o.p1) n_p2 = _project(o.p2) if n_p1 == n_p2: projected = n_p1 else: projected = o.__class__(n_p1, n_p2) # Didn't know how to project so raise an error if projected is None: n1 = self.__class__.__name__ n2 = o.__class__.__name__ raise GeometryError("Do not know how to project %s onto %s" % (n2, n1)) return self.intersection(projected)[0]
[docs] def intersection(self, o): """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) [Point(7, 7)] >>> p4, p5 = Point(5, 0), Point(0, 3) >>> l2 = Line(p4, p5) >>> l1.intersection(l2) [Point(15/8, 15/8)] >>> p6, p7 = Point(0, 5), Point(2, 6) >>> s1 = Segment(p6, p7) >>> l1.intersection(s1) [] """ if isinstance(o, Point): if o in self: return [o] else: return [] elif isinstance(o, LinearEntity): a1, b1, c1 = self.coefficients a2, b2, c2 = o.coefficients t = simplify(a1*b2 - a2*b1) if t.equals(0) is not False: # assume they are parallel if isinstance(self, Line): if o.p1 in self: return [o] return [] elif isinstance(o, Line): if self.p1 in o: return [self] return [] elif isinstance(self, Ray): if isinstance(o, Ray): # case 1, rays in the same direction if self.xdirection == o.xdirection: if self.source.x < o.source.x: return [o] return [self] # case 2, rays in the opposite directions else: if o.source in self: if self.source == o.source: return [self.source] return [Segment(o.source, self.source)] return [] elif isinstance(o, Segment): if o.p1 in self: if o.p2 in self: return [o] return [Segment(o.p1, self.source)] elif o.p2 in self: return [Segment(o.p2, self.source)] return [] elif isinstance(self, Segment): if isinstance(o, Ray): return o.intersection(self) elif isinstance(o, Segment): # A reminder that the points of Segments are ordered # in such a way that the following works. See # Segment.__new__ for details on the ordering. if self.p1 not in o: if self.p2 not in o: # Neither of the endpoints are in o so either # o is contained in this segment or it isn't if o in self: return [self] return [] else: # p1 not in o but p2 is. Either there is a # segment as an intersection, or they only # intersect at an endpoint if self.p2 == o.p1: return [o.p1] return [Segment(o.p1, self.p2)] elif self.p2 not in o: # p2 not in o but p1 is. Either there is a # segment as an intersection, or they only # intersect at an endpoint if self.p1 == o.p2: return [o.p2] return [Segment(o.p2, self.p1)] # Both points of self in o so the whole segment # is in o return [self] # Unknown linear entity return [] # Not parallel, so find the point of intersection px = simplify((b1*c2 - c1*b2) / t) py = simplify((a2*c1 - a1*c2) / t) inter = Point(px, py) # we do not use a simplistic 'inter in self and inter in o' # because that requires an equality test that is fragile; # instead we employ some diagnostics to see if the intersection # is valid def inseg(self): def _between(a, b, c): return c >= a and c <= b or c <= a and c >= b if _between(self.p1.x, self.p2.x, inter.x) and \ _between(self.p1.y, self.p2.y, inter.y): return True def inray(self): sray = Ray(self.p1, inter) if sray.xdirection == self.xdirection and \ sray.ydirection == self.ydirection: return True for i in range(2): if isinstance(self, Line): if isinstance(o, Line): return [inter] elif isinstance(o, Ray) and inray(o): return [inter] elif isinstance(o, Segment) and inseg(o): return [inter] elif isinstance(self, Ray) and inray(self): if isinstance(o, Ray) and inray(o): return [inter] elif isinstance(o, Segment) and inseg(o): return [inter] elif isinstance(self, Segment) and inseg(self): if isinstance(o, Segment) and inseg(o): return [inter] self, o = o, self return [] return o.intersection(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 Point(...) >>> # point should belong to the line >>> p3 in l1 True """ from random import randint # The lower and upper lower, upper = -2**32 - 1, 2**32 if self.slope is S.Infinity: if isinstance(self, Ray): if self.ydirection is S.Infinity: lower = self.p1.y else: upper = self.p1.y elif isinstance(self, Segment): lower = self.p1.y upper = self.p2.y x = self.p1.x y = randint(lower, upper) else: if isinstance(self, Ray): if self.xdirection is S.Infinity: lower = self.p1.x else: upper = self.p1.x elif isinstance(self, Segment): lower = self.p1.x upper = self.p2.x a, b, c = self.coefficients x = randint(lower, upper) y = simplify((-c - a*x) / b) return Point(x, y)
[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 """ def _norm(a, b, c): if a != 0: return 1, b/a, c/a elif b != 0: return a/b, 1, c/b else: return c return _norm(*self.coefficients) == _norm(*other.coefficients)
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))
[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()
def __eq__(self, other): """Subclasses should implement this method.""" raise NotImplementedError() def __hash__(self): return super(LinearEntity, self).__hash__()
[docs]class Line(LinearEntity): """An infinite line in space. A line is declared with two distinct points or a point and slope as defined using keyword `slope`. Notes ===== At the moment only lines in a 2D space can be declared, because Points can be defined only for 2D spaces. Parameters ========== p1 : Point pt : Point slope : sympy expression See Also ======== sympy.geometry.point.Point Examples ======== >>> import sympy >>> from sympy import Point >>> from sympy.abc import L >>> from sympy.geometry import Line, Segment >>> L = Line(Point(2,3), Point(3,5)) >>> L Line(Point(2, 3), Point(3, 5)) >>> L.points (Point(2, 3), Point(3, 5)) >>> L.equation() -2*x + y + 1 >>> L.coefficients (-2, 1, 1) Instantiate with keyword ``slope``: >>> Line(Point(0, 0), slope=0) Line(Point(0, 0), Point(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): p1, pt = p1.args else: p1 = Point(p1) if pt is not None and slope is None: try: p2 = Point(pt) except NotImplementedError: raise ValueError('The 2nd argument was not a valid Point; if it was meant to be a slope it should be given with keyword "slope".') if p1 == p2: raise ValueError('A line requires two distinct points.') elif slope is not None and pt is None: slope = sympify(slope) if slope.is_bounded is False: # when unbounded slope, don't change x p2 = p1 + Point(0, 1) else: # go over 1 up slope p2 = p1 + Point(1, slope) else: raise ValueError('A 2nd Point or keyword "slope" must be used.') return LinearEntity.__new__(cls, p1, p2, **kwargs)
[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'. 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() Point(4*t + 1, 3*t) """ t = _symbol(parameter) if t.name in (f.name for f in self.free_symbols): raise ValueError('Symbol %s already appears in object and cannot be used as a parameter.' % t.name) x = simplify(self.p1.x + t*(self.p2.x - self.p1.x)) y = simplify(self.p1.y + t*(self.p2.y - self.p1.y)) return Point(x, y)
[docs] def plot_interval(self, parameter='t'): """The plot interval for the default geometric plot of 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) return [t, -5, 5]
[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, y = _symbol(x), _symbol(y) 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 simplify(a*x + b*y + c)
[docs] def contains(self, o): """Return True if o is on this Line, or False otherwise.""" if isinstance(o, Point): x, y = Dummy(), Dummy() eq = self.equation(x, y) if not eq.has(y): return (solve(eq, x)[0] - o.x).equals(0) if not eq.has(x): return (solve(eq, y)[0] - o.y).equals(0) return (solve(eq.subs(x, o.x), y)[0] - o.y).equals(0) elif not isinstance(o, LinearEntity): return False elif isinstance(o, Line): return self.__eq__(o) elif not self.is_similar(o): return False else: return o.p1 in self and o.p2 in self
def __eq__(self, other): """Return True if other is equal to this Line, or False otherwise.""" if not isinstance(other, Line): return False return Point.is_collinear(self.p1, self.p2, other.p1, other.p2) def __hash__(self): return super(Line, self).__hash__()
[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 xdirection ydirection See Also ======== sympy.geometry.point.Point, Line Notes ===== At the moment only rays in a 2D space can be declared, because Points can be defined only for 2D spaces. Examples ======== >>> import sympy >>> from sympy import Point, pi >>> from sympy.abc import r >>> from sympy.geometry import Ray >>> r = Ray(Point(2, 3), Point(3, 5)) >>> r = Ray(Point(2, 3), Point(3, 5)) >>> r Ray(Point(2, 3), Point(3, 5)) >>> r.points (Point(2, 3), Point(3, 5)) >>> r.source Point(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) if pt is not None and angle is None: try: p2 = Point(pt) except NotImplementedError: raise ValueError('The 2nd argument was not a valid Point;\nif 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 if not p2: p2 = p1 + Point(1, C.tan(c)) else: raise ValueError('A 2nd point or keyword "angle" must be used.') return LinearEntity.__new__(cls, p1, p2, **kwargs) @property
[docs] 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 Point(0, 0) """ return self.p1
@property
[docs] 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
[docs] 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 arbitrary_point(self, parameter='t'): """A parameterized point on the Ray. Parameters ========== parameter : str, optional The name of the parameter which will be used for the parametric point. The default value is 't'. Returns ======= point : Point Raises ====== ValueError When ``parameter`` already appears in the Ray's definition. See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Ray, Point, Segment, S, simplify, solve >>> from sympy.abc import t >>> r = Ray(Point(0, 0), Point(2, 3)) >>> p = r.arbitrary_point(t) The parameter `t` used in the arbitrary point maps 0 to the origin of the ray and 1 to the end of the ray at infinity (which will show up as NaN). >>> p.subs(t, 0), p.subs(t, 1) (Point(0, 0), Point(oo, oo)) The unit that `t` moves you is based on the spacing of the points used to define the ray. >>> p.subs(t, 1/(S(1) + 1)) # one unit Point(2, 3) >>> p.subs(t, 2/(S(1) + 2)) # two units out Point(4, 6) >>> p.subs(t, S.Half/(S(1) + S.Half)) # half a unit out Point(1, 3/2) If you want to be located a distance of 1 from the origin of the ray, what value of `t` is needed? a) Find the unit length and pick `t` accordingly. >>> u = Segment(r.p1, p.subs(t, S.Half)).length # S.Half = 1/(1 + 1) >>> want = 1 >>> t_need = want/u >>> p_want = p.subs(t, t_need/(1 + t_need)) >>> simplify(Segment(r.p1, p_want).length) 1 b) Find the `t` that makes the length from origin to `p` equal to 1. >>> l = Segment(r.p1, p).length >>> t_need = solve(l**2 - want**2, t) # use the square to remove abs() if it is there >>> t_need = [w for w in t_need if w.n() > 0][0] # take positive t >>> p_want = p.subs(t, t_need) >>> simplify(Segment(r.p1, p_want).length) 1 """ t = _symbol(parameter) if t.name in (f.name for f in self.free_symbols): raise ValueError('Symbol %s already appears in object and cannot be used as a parameter.' % t.name) x = simplify(self.p1.x + t/(1 - t)*(self.p2.x - self.p1.x)) y = simplify(self.p1.y + t/(1 - t)*(self.p2.y - self.p1.y)) return Point(x, y)
[docs] def plot_interval(self, parameter='t'): """The plot interval for the default geometric plot of the Ray. Parameters ========== parameter : str, optional Default value is 't'. Returns ======= plot_interval : list [parameter, lower_bound, upper_bound] Examples ======== >>> from sympy import Point, Ray, pi >>> r = Ray((0, 0), angle=pi/4) >>> r.plot_interval() [t, 0, 5*sqrt(2)/(1 + 5*sqrt(2))] """ t = _symbol(parameter) p = self.arbitrary_point(t) # get a t corresponding to length of 10 want = 10 u = Segment(self.p1, p.subs(t, S.Half)).length # gives unit length t_need = want/u return [t, 0, t_need/(1 + t_need)]
def __eq__(self, other): """Is the other GeometryEntity equal to this Ray?""" if not isinstance(other, Ray): return False return (self.source == other.source) and (other.p2 in self) def __hash__(self): return super(Ray, self).__hash__()
[docs] def contains(self, o): """Is other GeometryEntity contained in this Ray?""" if isinstance(o, Ray): return (Point.is_collinear(self.p1, self.p2, o.p1, o.p2) and self.xdirection == o.xdirection and self.ydirection == o.ydirection) elif isinstance(o, Segment): return o.p1 in self and o.p2 in self elif isinstance(o, Point): if Point.is_collinear(self.p1, self.p2, o): if self.xdirection is S.Infinity: rv = o.x >= self.source.x elif self.xdirection is S.NegativeInfinity: rv = o.x <= self.source.x elif self.ydirection is S.Infinity: rv = o.y >= self.source.y else: rv = o.y <= self.source.y if isinstance(rv, bool): return rv raise Undecidable('Cannot determine if %s is in %s' % (o, self)) else: # Points are not collinear, so the rays are not parallel # and hence it is impossible for self to contain o return False # No other known entity can be contained in a Ray return False
[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.point.Point, Line Notes ===== At the moment only segments in a 2D space can be declared, because Points can be defined only for 2D spaces. Examples ======== >>> import sympy >>> from sympy import Point >>> from sympy.abc import s >>> from sympy.geometry import Segment >>> Segment((1, 0), (1, 1)) # tuples are interpreted as pts Segment(Point(1, 0), Point(1, 1)) >>> s = Segment(Point(4, 3), Point(1, 1)) >>> s Segment(Point(1, 1), Point(4, 3)) >>> s.points (Point(1, 1), Point(4, 3)) >>> s.slope 2/3 >>> s.length sqrt(13) >>> s.midpoint Point(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) p2 = Point(p2) if p1 == p2: return Point(p1) if p1.x > p2.x: p1, p2 = p2, p1 elif p1.x == p2.x and p1.y > p2.y: p1, p2 = p2, p1 return LinearEntity.__new__(cls, p1, p2, **kwargs)
[docs] def arbitrary_point(self, parameter='t'): """A parameterized point on the Segment. Parameters ========== parameter : str, optional The name of the parameter which will be used for the parametric point. The default value is 't'. Returns ======= point : Point Parameters ========== parameter : str, optional The name of the parameter which will be used for the parametric point. The default value is 't'. Returns ======= point : Point Raises ====== ValueError When ``parameter`` already appears in the Segment's definition. See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Point, Segment >>> p1, p2 = Point(1, 0), Point(5, 3) >>> s1 = Segment(p1, p2) >>> s1.arbitrary_point() Point(4*t + 1, 3*t) """ t = _symbol(parameter) if t.name in (f.name for f in self.free_symbols): raise ValueError('Symbol %s already appears in object and cannot be used as a parameter.' % t.name) x = simplify(self.p1.x + t*(self.p2.x - self.p1.x)) y = simplify(self.p1.y + t*(self.p2.y - self.p1.y)) return Point(x, y)
[docs] def plot_interval(self, parameter='t'): """The plot interval for the default geometric plot of the Segment. 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) return [t, 0, 1]
[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() Line(Point(3, 3), Point(9, -3)) >>> s1.perpendicular_bisector(p3) Segment(Point(3, 3), Point(5, 1)) """ l = LinearEntity.perpendicular_line(self, self.midpoint) if p is None or p not in l: return l else: return Segment(self.midpoint, p)
@property
[docs] 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 """ return Point.distance(self.p1, self.p2)
@property
[docs] 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 Point(2, 3/2) """ return Point.midpoint(self.p1, self.p2)
[docs] def distance(self, o): """ Finds the shortest distance between a line segment and a point. Raises ====== NotImplementedError is raised if o 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) """ if isinstance(o, Point): return self._do_point_distance(o) raise NotImplementedError()
def _do_point_distance(self, pt): """Calculates the distance between a point and a line segment.""" seg_vector = self.p2 - self.p1 pt_vector = pt - self.p1 t = seg_vector.dot(pt_vector)/self.length**2 if t >= 1: distance = Point.distance(self.p2, pt) elif t <= 0: distance = Point.distance(self.p1, pt) else: distance = Point.distance(self.p1 + Point(t*seg_vector.x, t*seg_vector.y), pt) return distance def __eq__(self, other): """Is the other GeometryEntity equal to this Ray?""" if not isinstance(other, Segment): return False return (self.p1 == other.p1) and (self.p2 == other.p2) def __hash__(self): return super(Segment, self).__hash__()
[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 """ if isinstance(other, Segment): return other.p1 in self and other.p2 in self elif isinstance(other, Point): if Point.is_collinear(self.p1, self.p2, other): t = Dummy('t') x, y = self.arbitrary_point(t).args if self.p1.x != self.p2.x: ti = solve(x - other.x, t)[0] else: ti = solve(y - other.y, t)[0] if ti.is_number: return 0 <= ti <= 1 return None # No other known entity can be contained in a Ray return False