Source code for diofant.geometry.line

"""Line-like geometrical entities.

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

from ..core import Dummy, Eq, Integer, factor_terms, oo, pi, sympify
from ..core.compatibility import is_sequence
from ..functions import Piecewise, acos, sqrt, tan
from ..functions.elementary.trigonometric import _pi_coeff as pi_coeff
from ..logic import And, false, true
from ..simplify import simplify
from ..solvers import solve
from .entity import GeometryEntity, GeometrySet
from .exceptions import GeometryError
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(GeometrySet): """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 ======== diofant.geometry.entity.GeometryEntity """ def __new__(cls, p1, p2, **kwargs): p1 = Point(p1) p2 = Point(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) @property def ambient_dimension(self): return len(self.p1) @property def p1(self): """The first defining point of a linear entity. See Also ======== diofant.geometry.point.Point Examples ======== >>> 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 ======== diofant.geometry.point.Point Examples ======== >>> p1, p2 = Point(0, 0), Point(5, 3) >>> l = Line(p1, p2) >>> l.p2 Point2D(5, 3) """ return self.args[1] @property def coefficients(self): """The coefficients (`a`, `b`, `c`) for `ax + by + c = 0`. See Also ======== diofant.geometry.line.Line.equation Examples ======== >>> 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 Integer(1), Integer(0), -p1.x elif p1.y == p2.y: return Integer(0), Integer(1), -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] @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 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 ======== diofant.geometry.util.intersection Examples ======== >>> 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 """ # 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(self, other): """Are two linear entities parallel? Parameters ========== self : LinearEntity other : LinearEntity Returns ======= True : if self and other are parallel, False : otherwise. See Also ======== coefficients Examples ======== >>> 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 = self.coefficients a2, b2, c2 = other.coefficients return bool(simplify(a1*b2 - b1*a2) == 0) except AttributeError: return False
[docs] def is_perpendicular(self, other): """Are two linear entities perpendicular? Parameters ========== self : LinearEntity other : LinearEntity Returns ======= True : if self and other are perpendicular, False : otherwise. See Also ======== coefficients Examples ======== >>> 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 = self.coefficients a2, b2, c2 = other.coefficients return bool(simplify(a1*a2 + b1*b2) == 0) except AttributeError: return False
[docs] def angle_between(self, other): """The angle formed between the two linear entities. Parameters ========== self : LinearEntity other : 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 ======== >>> 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 = self.p2 - self.p1 v2 = other.p2 - other.p1 return 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 : diofant.geometry.point.Point Returns ======= line : Line See Also ======== is_parallel Examples ======== >>> 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 p = Point(p) 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 : diofant.geometry.point.Point Returns ======= line : Line See Also ======== is_perpendicular, perpendicular_segment Examples ======== >>> 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) 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 : diofant.geometry.point.Point Returns ======= segment : Segment Notes ===== Returns `p` itself if `p` is on this linear entity. See Also ======== perpendicular_line Examples ======== >>> 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(Point2D(2, 2), Point2D(4, 0)) """ p = Point(p) if p in self: return p a, b, c = self.coefficients if a == 0: # horizontal p2 = Point(p.x, self.p1.y) elif b == 0: # vertical p2 = Point(self.p1.x, p.y) else: # ax + by + c = 0 y = (-c - a*p.x)/b m = self.slope d2 = 1 + m**2 H = p.y - y dx = m*H/d2 dy = m*dx p2 = (p.x + dx, y + dy) return Segment(p, p2)
@property def length(self): """ The length of the line. Examples ======== >>> p1, p2 = Point(0, 0), Point(3, 5) >>> l1 = Line(p1, p2) >>> l1.length oo """ return oo @property def slope(self): """The slope of this linear entity, or infinity if vertical. Returns ======= slope : Expr See Also ======== coefficients Examples ======== >>> 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 oo return simplify(d2/d1) @property def points(self): """The two points used to define this linear entity. Returns ======= points : tuple of Points See Also ======== diofant.geometry.point.Point Examples ======== >>> 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, o): """Project a point, line, ray, or segment onto this linear entity. Parameters ========== other : diofant.geometry.point.Point or LinearEntity Returns ======= projection : diofant.geometry.point.Point or LinearEntity The return type matches the type of the parameter ``other``. Raises ====== diofant.geometry.exceptions.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 ======== diofant.geometry.point.Point, perpendicular_line Examples ======== >>> 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) Segment(Point2D(5, 5), Point2D(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 : diofant.geometry.point.Point or LinearEntity Returns ======= intersection : list of geometrical entities See Also ======== diofant.geometry.point.Point Examples ======== >>> 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) [] """ 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 and \ self.ydirection == o.ydirection: 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 [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): if self.p1 == inter: return True sray = Ray(self.p1, inter) if sray.xdirection == self.xdirection and \ sray.ydirection == self.ydirection: return True prec = (Line, Ray, Segment) expr = self if prec.index(expr.func) > prec.index(o.func): expr, o = o, expr rv = [inter] if isinstance(expr, Line): if isinstance(o, Line): return rv elif isinstance(o, Ray) and inray(o): return rv elif isinstance(o, Segment) and inseg(o): return rv elif isinstance(expr, Ray) and inray(expr): if isinstance(o, Ray) and inray(o): return rv elif isinstance(o, Segment) and inseg(o): return rv elif isinstance(expr, Segment) and inseg(expr): if isinstance(o, Segment) and inseg(o): return rv 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 : diofant.geometry.point.Point Raises ====== ValueError When ``parameter`` already appears in the Line's definition. See Also ======== diofant.geometry.point.Point Examples ======== >>> p1, p2 = Point(1, 0), Point(5, 3) >>> l1 = Line(p1, p2) >>> l1.arbitrary_point() Point2D(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) # multiply on the right so the variable gets # combined witht he coordinates of the point return self.p1 + (self.p2 - self.p1)*t
[docs] def random_point(self): """A random point on a LinearEntity. Returns ======= point : diofant.geometry.point.Point See Also ======== diofant.geometry.point.Point Examples ======== >>> 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 Point2D(...) >>> # 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 oo: if isinstance(self, Ray): if self.ydirection is oo: 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 oo: 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 = (-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 ======== >>> 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()
[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 : diofant.geometry.point.Point pt : diofant.geometry.point.Point slope : Expr See Also ======== diofant.geometry.point.Point Examples ======== >>> from diofant.abc import L >>> L = Line(Point(2, 3), Point(3, 5)) >>> L Line(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) Line(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): 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 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) else: raise ValueError('A 2nd Point or keyword "slope" must be used.') return LinearEntity.__new__(cls, p1, p2, **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 ======== >>> 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 : diofant expression See Also ======== LinearEntity.coefficients Examples ======== >>> 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 a*x + b*y + c
[docs] def contains(self, o): """ Return True if o is on this Line, or False otherwise. Examples ======== >>> 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 """ if is_sequence(o): o = Point(o) if isinstance(o, Point): o = o.func(*[simplify(i) for i in o.args]) x, y = Dummy(), Dummy() eq = self.equation(x, y) if not eq.has(y): return (solve(eq, x)[0][x] - o.x).equals(0) if not eq.has(x): return (solve(eq, y)[0][y] - o.y).equals(0) return (solve(eq.subs({x: o.x}), y)[0][y] - o.y).equals(0) elif not isinstance(o, LinearEntity): return False elif isinstance(o, Line): return self.equal(o) elif not self.is_similar(o): return False else: return o.p1 in self and o.p2 in self
[docs] def distance(self, o): """ Finds the shortest distance between a line and a point. Raises ====== NotImplementedError if o is not a Point Examples ======== >>> 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 """ if not isinstance(o, Point): if is_sequence(o): o = Point(o) a, b, c = self.coefficients if 0 in (a, b): return self.perpendicular_segment(o).length m = self.slope x = o.x y = m*x - c/b return abs(factor_terms(o.y - y))/sqrt(1 + m**2)
[docs] def equal(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]class Ray(LinearEntity): """ A Ray is a semi-line in the space with a source point and a direction. Parameters ========== p1 : diofant.geometry.point.Point The source of the Ray p2 : diofant.geometry.point.Point or Expr 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 ======== diofant.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 ======== >>> from diofant.abc import r >>> r = Ray(Point(2, 3), Point(3, 5)) >>> r = Ray(Point(2, 3), Point(3, 5)) >>> r Ray(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) if pt is not None and angle is None: try: p2 = Point(pt) except ValueError: from ...utilities 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.denominator == 2: if c.numerator == 1: p2 = p1 + Point(0, 1) elif c.numerator == 3: p2 = p1 + Point(0, -1) elif c.denominator == 1: if c.numerator == 0: p2 = p1 + Point(1, 0) elif c.numerator == 1: p2 = p1 + Point(-1, 0) if p2 is None: c *= pi else: c = angle % (2*pi) if not p2: m = 2*c/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 LinearEntity.__new__(cls, p1, p2, **kwargs) @property def source(self): """The point from which the ray emanates. See Also ======== diofant.geometry.point.Point Examples ======== >>> p1, p2 = Point(0, 0), Point(4, 1) >>> r1 = Ray(p1, p2) >>> r1.source Point2D(0, 0) """ return self.p1 @property def direction(self): """The direction in which the ray emanates. See Also ======== diofant.geometry.point.Point Examples ======== >>> p1, p2 = Point(0, 0), Point(4, 1) >>> r1 = Ray(p1, p2) >>> r1.direction Point2D(4, 1) """ return self.p2 - self.p1 @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 ======== >>> 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 oo elif self.p1.x == self.p2.x: return Integer(0) else: return -oo @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 ======== >>> 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 oo elif self.p1.y == self.p2.y: return Integer(0) else: return -oo
[docs] def distance(self, o): """ Finds the shortest distance between the ray and a point. Raises ====== NotImplementedError if o is not a Point Examples ======== >>> 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 """ if not isinstance(o, Point): if is_sequence(o): o = Point(o) s = self.perpendicular_segment(o) if isinstance(s, Point): if self.contains(s): return Integer(0) else: # since arg-order is arbitrary, find the non-o point non_o = s.p1 if s.p1 != o else s.p2 if self.contains(non_o): return Line(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 ======== >>> r = Ray((0, 0), angle=pi/4) >>> r.plot_interval() [t, 0, 10] """ t = _symbol(parameter) return [t, 0, 10]
[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 contains(self, o): """ Is other GeometryEntity contained in this Ray? Examples ======== >>> 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 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 is_sequence(o): o = Point(o) if isinstance(o, Point): if Point.is_collinear(self.p1, self.p2, o): if self.xdirection is oo: rv = o.x >= self.source.x elif self.xdirection == -oo: rv = o.x <= self.source.x elif self.ydirection is oo: rv = o.y >= self.source.y else: rv = o.y <= self.source.y if rv in (true, 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]class Segment(LinearEntity): """A undirected line segment in space. Parameters ========== p1 : diofant.geometry.point.Point p2 : diofant.geometry.point.Point Attributes ========== length : Expr midpoint : diofant.geometry.point.Point See Also ======== diofant.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 ======== >>> from diofant.abc import s >>> Segment((1, 0), (1, 1)) # tuples are interpreted as pts Segment(Point2D(1, 0), Point2D(1, 1)) >>> s = Segment(Point(4, 3), Point(1, 1)) >>> s Segment(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) p2 = Point(p2) if p1 == p2: return Point(p1) if (p1.x - p2.x).is_positive: p1, p2 = p2, p1 elif (p1.x == p2.x) and (p1.y - p2.y).is_positive: p1, p2 = p2, p1 return LinearEntity.__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 ======== >>> 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 : diofant.geometry.point.Point Returns ======= bisector : Line or Segment See Also ======== LinearEntity.perpendicular_segment Examples ======== >>> p1, p2, p3 = Point(0, 0), Point(6, 6), Point(5, 1) >>> s1 = Segment(p1, p2) >>> s1.perpendicular_bisector() Line(Point2D(3, 3), Point2D(9, -3)) >>> s1.perpendicular_bisector(p3) Segment(Point2D(3, 3), Point2D(5, 1)) """ l = LinearEntity.perpendicular_line(self, self.midpoint) if p is None or Point(p) not in l: return l else: return Segment(self.midpoint, p)
@property def length(self): """The length of the line segment. See Also ======== diofant.geometry.point.Point.distance Examples ======== >>> p1, p2 = Point(0, 0), Point(4, 3) >>> s1 = Segment(p1, p2) >>> s1.length 5 """ return Point.distance(self.p1, self.p2) @property def midpoint(self): """The midpoint of the line segment. See Also ======== diofant.geometry.point.Point.midpoint Examples ======== >>> p1, p2 = Point(0, 0), Point(4, 3) >>> s1 = Segment(p1, p2) >>> s1.midpoint Point2D(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 if o is not a Point Examples ======== >>> 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) """ if is_sequence(o): o = Point(o) if isinstance(o, Point): seg_vector = self.p2 - self.p1 pt_vector = o - self.p1 t = seg_vector.dot(pt_vector)/self.length**2 if t >= 1: distance = Point.distance(self.p2, o) elif t <= 0: distance = Point.distance(self.p1, o) else: distance = Point.distance( self.p1 + Point(t*seg_vector.x, t*seg_vector.y), o) return distance raise NotImplementedError()
[docs] def contains(self, other): """ Is the other GeometryEntity contained within this Segment? Examples ======== >>> 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][t] else: ti = solve(y - other.y, t)[0][t] if ti.is_number: return 0 <= ti <= 1 return return False