Source code for sympy.geometry.line3d

"""Line-like geometrical entities.

Contains
========
LinearEntity3D
Line3D
Ray3D
Segment3D

"""
from __future__ import print_function, division

from sympy.core import S, C, sympify, Dummy, nan
from sympy.functions.elementary.trigonometric import _pi_coeff as pi_coeff, \
    sqrt
from sympy.core.logic import fuzzy_and
from sympy.core.exprtools import factor_terms
from sympy.simplify.simplify import simplify
from sympy.solvers import solve
from sympy.geometry.exceptions import GeometryError
from .entity import GeometryEntity
from .point3d import Point3D
from .util import _symbol
from sympy.core.compatibility import is_sequence

[docs]class LinearEntity3D(GeometryEntity): """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) p2 = Point3D(p2) 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) @property
[docs] def p1(self): """The first defining point of a linear entity. See Also ======== sympy.geometry.point3d.Point3D Examples ======== >>> from sympy import Point3D, Line3D >>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1) >>> l = Line3D(p1, p2) >>> l.p1 Point3D(0, 0, 0) """ return self.args[0]
@property
[docs] def p2(self): """The second defining point of a linear entity. See Also ======== sympy.geometry.point3d.Point3D Examples ======== >>> from sympy import Point3D, Line3D >>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1) >>> l = Line3D(p1, p2) >>> l.p2 Point3D(5, 3, 1) """ return self.args[1]
@property
[docs] 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
[docs] 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)
@property
[docs] def length(self): """ The length of the line. Examples ======== >>> from sympy import Point3D, Line3D >>> p1, p2 = Point3D(0, 0, 0), Point3D(3, 5, 1) >>> l1 = Line3D(p1, p2) >>> l1.length oo """ return S.Infinity
@property
[docs] def points(self): """The two points used to define this linear entity. Returns ======= points : tuple of Points See Also ======== sympy.geometry.point3d.Point3D Examples ======== >>> from sympy import Point3D, Line3D >>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 11, 1) >>> l1 = Line3D(p1, p2) >>> l1.points (Point3D(0, 0, 0), Point3D(5, 11, 1)) """ return (self.p1, self.p2)
@staticmethod
[docs] 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 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 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 """ # 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. Examples ======== >>> 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 l1 == l2: ValueError('Enter two distinct lines') a = l1.direction_cosine b = l2.direction_cosine a = [abs(i) for i in a] b = [abs(i) for i in b] if a == b: return True else: 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 ======== direction_ratio Examples ======== >>> 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 """ a = sum([i*j for i, j in zip(l1.direction_ratio, l2.direction_ratio)]) if a == 0: return True else: 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 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) """ 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 : Point3D Returns ======= line : Line3D See Also ======== is_parallel Examples ======== >>> 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 """ d = self.direction_ratio return Line3D(p, direction_ratio=d)
[docs] def perpendicular_line(self, p): """Create a new Line perpendicular to this linear entity which passes through the point `p`. Parameters ========== p : Point3D Returns ======= line : Line3D See Also ======== is_perpendicular, perpendicular_segment Examples ======== >>> 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 """ from sympy.core import Symbol t = Symbol('t') if p in self: raise NotImplementedError("Given point should not be on the line") a = self.arbitrary_point(t) b = [i - j for i, j in zip(p.args, a.args)] c = sum([i*j for i, j in zip(b, self.direction_ratio)]) d = solve(c, t) e = a.subs(t, d[0]) return Line3D(p, e)
[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 : Point3D Returns ======= segment : Segment3D Notes ===== Returns `p` itself if `p` is on this linear entity. See Also ======== perpendicular_line Examples ======== >>> 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)) """ from sympy.core import Symbol t = Symbol('t') if p in self: raise NotImplementedError("Given point should not be on the line") a = self.arbitrary_point(t) b = [i - j for i, j in zip(p.args, a.args)] c = sum([i*j for i, j in zip(b, self.direction_ratio)]) d = solve(c, t) e = a.subs(t, d[0]) return Segment3D(p, e)
[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 point is the intersection of L and the line perpendicular to L that passes through P. See Also ======== sympy.geometry.point3d.Point3D, perpendicular_line Examples ======== >>> from sympy import Point3D, Line3D, Segment3D, Rational >>> p1, p2, p3 = Point3D(0, 0, 1), Point3D(1, 1, 2), Point3D(2, 0, 1) >>> l1 = Line3D(p1, p2) >>> l1.projection(p3) Point3D(2/3, 2/3, 5/3) >>> p4, p5 = Point3D(10, 0, 1), Point3D(12, 1, 3) >>> s1 = Segment3D(p4, p5) >>> l1.projection(s1) [Segment3D(Point3D(10/3, 10/3, 13/3), Point3D(5, 5, 6))] """ tline = Line3D(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, Point3D): return _project(o) elif isinstance(o, LinearEntity3D): 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)
[docs] def intersection(self, o): """The intersection with another geometrical entity. Parameters ========== o : Point or LinearEntity3D Returns ======= intersection : list of geometrical entities See Also ======== sympy.geometry.point3d.Point3D Examples ======== >>> 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) [] """ from sympy.core import Symbol t1 = Symbol('t1') t2 = Symbol('t2') if isinstance(o, Point3D): if o in self: return [o] else: return [] elif isinstance(o, LinearEntity3D): a = self.direction_cosine b = o.direction_cosine a = [abs(i) for i in a] b = [abs(i) for i in b] if a == b: # assume they are parallel if isinstance(self, Line3D): if o.p1 in self: return [o] return [] elif isinstance(o, Line3D): if self.p1 in o: return [self] return [] elif isinstance(self, Ray3D): if isinstance(o, Ray3D): # case 1, rays in the same direction if self.xdirection == o.xdirection and \ self.ydirection == o.ydirection and \ self.zdirection == o.zdirection: return [self] if (self.source in o) else [o] # case 2, rays in the opposite directions else: if o.source in self: if self.source == o.source: return [self.source] return [Segment3D(o.source, self.source)] return [] elif isinstance(o, Segment3D): if o.p1 in self: if o.p2 in self: return [o] return [Segment3D(o.p1, self.source)] elif o.p2 in self: return [Segment3D(o.p2, self.source)] return [] elif isinstance(self, Segment3D): if isinstance(o, Ray3D): return o.intersection(self) elif isinstance(o, Segment3D): # A reminder that the points of Segments are ordered # in such a way that the following works. See # Segment3D.__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 [Segment3D(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 [Segment3D(o.p2, self.p1)] # Both points of self in o so the whole segment # is in o return [self] # Unknown linear entity return [] # If the lines are not parallel the solve their arbitrary points # to obtain the point of intersection a = self.arbitrary_point(t1) b = o.arbitrary_point(t2) c = solve([a.x - b.x, a.y - b.y], [t1, t2]) d = solve([a.x - b.x, a.z - b.z], [t1, t2]) if len(c) == 1 and len(d) == 1: return [] if c is {}: e = a.subs(t1, d[t1]) else: e = a.subs(t1, c[t1]) if e in self and e in o: return [e] else: return [] return o.intersection(self)
[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 : Point3D Raises ====== ValueError When ``parameter`` already appears in the Line's definition. See Also ======== sympy.geometry.point3d.Point3D Examples ======== >>> 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) 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)) z = simplify(self.p1.z + t*(self.p2.z - self.p1.z)) return Point3D(x, y, z)
[docs] def is_similar(self, other): """ Return True if self and other are contained in the same line. Examples ======== >>> from sympy import Point3D, Line3D >>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(2, 2, 2) >>> l1 = Line3D(p1, p2) >>> l2 = Line3D(p1, p3) >>> l1.is_similar(l2) True """ if isinstance(other, Line3D): if self.direction_cosine == other.direction_cosine and other.p1 in self: return True else: return False raise NotImplementedError()
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()
[docs]class Line3D(LinearEntity3D): """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.point3d.Point3D Examples ======== >>> import sympy >>> from sympy import Point3D >>> from sympy.abc import L >>> 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): p1, pt = p1.args else: p1 = Point3D(p1) if pt is not None and len(direction_ratio) == 0: try: pt = Point3D(pt) except NotImplementedError: raise ValueError('The 2nd argument was not a valid Point. ' 'If it was the direction_ratio of the desired line, enter it ' 'with keyword "direction_ratio".') 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 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 Point3D, Line3D >>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1) >>> l1 = Line3D(p1, p2) >>> l1.plot_interval() [t, -5, 5] """ t = _symbol(parameter) return [t, -5, 5]
[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(x), _symbol(y), _symbol(z), _symbol(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] def contains(self, o): """Return True if o is on this Line, or False otherwise. Examples ======== >>> from sympy import Line3D >>> a = (0, 0, 0) >>> b = (1, 1, 1) >>> c = (2, 2, 2) >>> l1 = Line3D(a, b) >>> l2 = Line3D(b, a) >>> l1 == l2 False >>> l1 in l2 True """ if is_sequence(o): o = Point3D(o) if isinstance(o, Point3D): sym = list(map(Dummy, 'xyz')) eq = self.equation(*sym) a = [eq[0].subs(sym[0], o.args[0]), eq[1].subs(sym[1], o.args[1]), eq[2].subs(sym[2], o.args[2])] a = [i for i in a if i != nan] if len(a) == 1: return True first = a.pop(0) for i in a: rv = first.equals(i) if not rv: return rv return True elif not isinstance(o, LinearEntity3D): return False elif isinstance(o, Line3D): return all(i in self for i in o.points)
[docs] def distance(self, o): """ Finds the shortest distance between a line and a point. Raises ====== NotImplementedError is raised if o is not an instance of Point3D Examples ======== >>> from sympy import Point3D, Line3D >>> p1, p2 = Point3D(0, 0, 0), Point3D(1, 1, 1) >>> s = Line3D(p1, p2) >>> s.distance(Point3D(-1, 1, 1)) 2*sqrt(6)/3 >>> s.distance((-1, 1, 1)) 2*sqrt(6)/3 """ if not isinstance(o, Point3D): if is_sequence(o): o = Point3D(o) if o in self: return S.Zero a = self.perpendicular_segment(o).length return a
[docs] def equals(self, other): """Returns True if self and other are the same mathematical entities""" if not isinstance(other, Line3D): return False return Point3D.are_collinear(self.p1, other.p1, self.p2, other.p2)
[docs]class Ray3D(LinearEntity3D): """ 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.point3d.Point3D, Line3D Examples ======== >>> import sympy >>> from sympy import Point3D, pi >>> from sympy.abc import r >>> 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): if isinstance(p1, LinearEntity3D): p1, pt = p1.args else: p1 = Point3D(p1) if pt is not None and len(direction_ratio) == 0: try: pt = Point3D(pt) except NotImplementedError: raise ValueError('The 2nd argument was not a valid Point. ' 'If it was the direction_ratio of the desired line, enter it ' 'with keyword "direction_ratio".') 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) @property
[docs] def source(self): """The point from which the ray emanates. See Also ======== sympy.geometry.point3d.Point3D Examples ======== >>> from sympy import Point3D, Ray3D >>> p1, p2 = Point3D(0, 0, 0), Point3D(4, 1, 5) >>> r1 = Ray3D(p1, p2) >>> r1.source Point3D(0, 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 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
[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 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
[docs] 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] def distance(self, o): """ Finds the shortest distance between the ray and a point. Raises ====== NotImplementedError is raised if o is not a Point Examples ======== >>> from sympy import Point3D, Ray3D >>> p1, p2 = Point3D(0, 0, 0), Point3D(1, 1, 2) >>> s = Ray3D(p1, p2) >>> s.distance(Point3D(-1, -1, 2)) sqrt(6) >>> s.distance((-1, -1, 2)) sqrt(6) """ if not isinstance(o, Point3D): if is_sequence(o): o = Point3D(o) if o in self: return S.Zero s = self.perpendicular_segment(o) if not isinstance(s, Point3D): non_o = s.p1 if s.p1 == o else s.p2 if self.contains(non_o): return Line3D(self).distance(o) # = s.length but simpler # the following applies when neither of the above apply return self.source.distance(o)
[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 Point3D, Ray3D, pi >>> r = Ray3D(Point3D(0, 0, 0), Point3D(1, 1, 1)) >>> r.plot_interval() [t, 0, 10] """ t = _symbol(parameter) return [t, 0, 10]
[docs] def contains(self, o): """Is other GeometryEntity contained in this Ray?""" if isinstance(o, Ray3D): return (Point3D.are_collinear(self.p1, self.p2, o.p1, o.p2) and self.xdirection == o.xdirection and self.ydirection == o.ydirection and self.zdirection == o.zdirection) elif isinstance(o, Segment3D): return o.p1 in self and o.p2 in self elif is_sequence(o): o = Point3D(o) if isinstance(o, Point3D): if Point3D.are_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 elif self.ydirection is S.NegativeInfinity: rv = o.y <= self.source.y elif self.zdirection is S.Infinity: rv = o.z <= self.source.z else: rv = o.z <= self.source.z if rv == True or rv == False: return bool(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] def equals(self, other): """Returns True if self and other are the same mathematical entities""" if not isinstance(other, Ray3D): return False return self.source == other.source and other.p2 in self
[docs]class Segment3D(LinearEntity3D): """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 ======== >>> import sympy >>> from sympy import Point3D >>> from sympy.abc import s >>> 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 = Point3D(p1) p2 = Point3D(p2) if p1 == p2: return Point3D(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)
[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 Point3D, Segment3D >>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 0) >>> s1 = Segment3D(p1, p2) >>> s1.plot_interval() [t, 0, 1] """ t = _symbol(parameter) return [t, 0, 1]
@property
[docs] def length(self): """The length of the line segment. See Also ======== sympy.geometry.point3d.Point3D.distance Examples ======== >>> from sympy import Point3D, Segment3D >>> p1, p2 = Point3D(0, 0, 0), Point3D(4, 3, 3) >>> s1 = Segment3D(p1, p2) >>> s1.length sqrt(34) """ return Point3D.distance(self.p1, self.p2)
@property
[docs] def midpoint(self): """The midpoint of the line segment. See Also ======== sympy.geometry.point3d.Point3D.midpoint Examples ======== >>> 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 Point3D.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 Point3D Examples ======== >>> 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 is_sequence(o): o = Point3D(o) if isinstance(o, Point3D): seg_vector = self.p2 - self.p1 pt_vector = o - self.p1 t = seg_vector.dot(pt_vector)/self.length**2 if t >= 1: distance = Point3D.distance(self.p2, o) elif t <= 0: distance = Point3D.distance(self.p1, o) else: distance = Point3D.distance( self.p1 + Point3D(t*seg_vector.x, t*seg_vector.y, t*seg_vector.y), o) return distance raise NotImplementedError()
[docs] def contains(self, other): """ Is the other GeometryEntity contained within this Segment? Examples ======== >>> 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 """ if is_sequence(other): other = Point3D(other) if isinstance(other, Segment3D): return other.p1 in self and other.p2 in self elif isinstance(other, Point3D): if Point3D.are_collinear(self.p1, self.p2, other): if other.distance(self.p1) + other.distance(self.p2) == self.length: return True else: return False return False