Source code for sympy.geometry.point

"""Geometrical Points.



from sympy.core import S, sympify
from sympy.core.compatibility import iterable
from sympy.core.containers import Tuple
from sympy.simplify import simplify, nsimplify
from sympy.geometry.exceptions import GeometryError
from sympy.functions.elementary.miscellaneous import sqrt
from entity import GeometryEntity
from sympy.matrices import Matrix
from sympy.core.numbers import Float

[docs]class Point(GeometryEntity): """A point in a 2-dimensional Euclidean space. Parameters ========== coords : sequence of 2 coordinate values. Attributes ========== x y length Raises ====== NotImplementedError When trying to create a point with more than two dimensions. When `intersection` is called with object other than a Point. TypeError When trying to add or subtract points with different dimensions. Notes ===== Currently only 2-dimensional points are supported. See Also ======== sympy.geometry.line.Segment : Connects two Points Examples ======== >>> from sympy.geometry import Point >>> from import x >>> Point(1, 2) Point(1, 2) >>> Point([1, 2]) Point(1, 2) >>> Point(0, x) Point(0, x) Floats are automatically converted to Rational unless the evaluate flag is False: >>> Point(0.5, 0.25) Point(1/2, 1/4) >>> Point(0.5, 0.25, evaluate=False) Point(0.5, 0.25) """ def __new__(cls, *args, **kwargs): if iterable(args[0]): coords = Tuple(*args[0]) elif isinstance(args[0], Point): coords = args[0].args else: coords = Tuple(*args) if len(coords) != 2: raise NotImplementedError( "Only two dimensional points currently supported") if kwargs.get('evaluate', True): coords = [simplify(nsimplify(c, rational=True)) for c in coords] return GeometryEntity.__new__(cls, *coords) def __hash__(self): return super(Point, self).__hash__() def __eq__(self, other): ts, to = type(self), type(other) if ts is not to: return False return self.args == other.args def __lt__(self, other): return self.args < other.args def __contains__(self, item): return item == self @property
[docs] def x(self): """ Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point >>> p = Point(0, 1) >>> p.x 0 """ return self.args[0]
[docs] def y(self): """ Returns the Y coordinate of the Point. Examples ======== >>> from sympy import Point >>> p = Point(0, 1) >>> p.y 1 """ return self.args[1]
[docs] def length(self): """ Treating a Point as a Line, this returns 0 for the length of a Point. Examples ======== >>> from sympy import Point >>> p = Point(0, 1) >>> p.length 0 """ return S.Zero
[docs] def is_collinear(*points): """Is a sequence of points collinear? Test whether or not a set of points are collinear. Returns True if the set of points are collinear, or False otherwise. Parameters ========== points : sequence of Point Returns ======= is_collinear : boolean Notes ===== Slope is preserved everywhere on a line, so the slope between any two points on the line should be the same. Take the first two points, p1 and p2, and create a translated point v1 with p1 as the origin. Now for every other point we create a translated point, vi with p1 also as the origin. Note that these translations preserve slope since everything is consistently translated to a new origin of p1. Since slope is preserved then we have the following equality: * v1_slope = vi_slope * v1.y/v1.x = vi.y/vi.x (due to translation) * v1.y*vi.x = vi.y*v1.x * v1.y*vi.x - vi.y*v1.x = 0 (*) Hence, if we have a vi such that the equality in (*) is False then the points are not collinear. We do this test for every point in the list, and if all pass then they are collinear. See Also ======== sympy.geometry.line.Line Examples ======== >>> from sympy import Point >>> from import x >>> p1, p2 = Point(0, 0), Point(1, 1) >>> p3, p4, p5 = Point(2, 2), Point(x, x), Point(1, 2) >>> Point.is_collinear(p1, p2, p3, p4) True >>> Point.is_collinear(p1, p2, p3, p5) False """ # Coincident points are irrelevant and can confuse this algorithm. # Use only unique points. points = list(set(points)) if len(points) == 0: return False if len(points) <= 2: return True # two points always form a line points = [Point(a) for a in points] # XXX Cross product is used now, but that only extends to three # dimensions. If the concept needs to extend to greater # dimensions then another method would have to be used p1 = points[0] p2 = points[1] v1 = p2 - p1 x1, y1 = v1.args rv = True for p3 in points[2:]: x2, y2 = (p3 - p1).args test = simplify(x1*y2 - y1*x2).equals(0) if test is False: return False if rv and not test: rv = test return rv
[docs] def is_concyclic(*points): """Is a sequence of points concyclic? Test whether or not a sequence of points are concyclic (i.e., they lie on a circle). Parameters ========== points : sequence of Points Returns ======= is_concyclic : boolean True if points are concyclic, False otherwise. See Also ======== sympy.geometry.ellipse.Circle Notes ===== No points are not considered to be concyclic. One or two points are definitely concyclic and three points are conyclic iff they are not collinear. For more than three points, create a circle from the first three points. If the circle cannot be created (i.e., they are collinear) then all of the points cannot be concyclic. If the circle is created successfully then simply check the remaining points for containment in the circle. Examples ======== >>> from sympy.geometry import Point >>> p1, p2 = Point(-1, 0), Point(1, 0) >>> p3, p4 = Point(0, 1), Point(-1, 2) >>> Point.is_concyclic(p1, p2, p3) True >>> Point.is_concyclic(p1, p2, p3, p4) False """ if len(points) == 0: return False if len(points) <= 2: return True points = [Point(p) for p in points] if len(points) == 3: return (not Point.is_collinear(*points)) try: from ellipse import Circle c = Circle(points[0], points[1], points[2]) for point in points[3:]: if point not in c: return False return True except GeometryError: # Circle could not be created, because of collinearity of the # three points passed in, hence they are not concyclic. return False # """ # # This code is from Maple # def f(u): # dd = u[0]**2 + u[1]**2 + 1 # u1 = 2*u[0] / dd # u2 = 2*u[1] / dd # u3 = (dd - 2) / dd # return u1,u2,u3 # u1,u2,u3 = f(points[0]) # v1,v2,v3 = f(points[1]) # w1,w2,w3 = f(points[2]) # p = [v1 - u1, v2 - u2, v3 - u3] # q = [w1 - u1, w2 - u2, w3 - u3] # r = [p[1]*q[2] - p[2]*q[1], p[2]*q[0] - p[0]*q[2], p[0]*q[1] - p[1]*q[0]] # for ind in xrange(3, len(points)): # s1,s2,s3 = f(points[ind]) # test = simplify(r[0]*(s1-u1) + r[1]*(s2-u2) + r[2]*(s3-u3)) # if test != 0: # return False # return True # """
[docs] def distance(self, p): """The Euclidean distance from self to point p. Parameters ========== p : Point Returns ======= distance : number or symbolic expression. See Also ======== sympy.geometry.line.Segment.length Examples ======== >>> from sympy.geometry import Point >>> p1, p2 = Point(1, 1), Point(4, 5) >>> p1.distance(p2) 5 >>> from import x, y >>> p3 = Point(x, y) >>> p3.distance(Point(0, 0)) sqrt(x**2 + y**2) """ p = Point(p) return sqrt(sum([(a - b)**2 for a, b in zip(self.args, p.args)]))
[docs] def midpoint(self, p): """The midpoint between self and point p. Parameters ========== p : Point Returns ======= midpoint : Point See Also ======== sympy.geometry.line.Segment.midpoint Examples ======== >>> from sympy.geometry import Point >>> p1, p2 = Point(1, 1), Point(13, 5) >>> p1.midpoint(p2) Point(7, 3) """ return Point([simplify((a + b)*S.Half) for a, b in zip(self.args, p.args)])
[docs] def evalf(self, prec=None, **options): """Evaluate the coordinates of the point. This method will, where possible, create and return a new Point where the coordinates are evaluated as floating point numbers to the precision indicated (default=15). Returns ======= point : Point Examples ======== >>> from sympy import Point, Rational >>> p1 = Point(Rational(1, 2), Rational(3, 2)) >>> p1 Point(1/2, 3/2) >>> p1.evalf() Point(0.5, 1.5) """ if prec is None: coords = [x.evalf(**options) for x in self.args] else: coords = [x.evalf(prec, **options) for x in self.args] return Point(*coords, **dict(evaluate=False))
n = evalf
[docs] def intersection(self, o): """The intersection between this point and another point. Parameters ========== other : Point Returns ======= intersection : list of Points Notes ===== The return value will either be an empty list if there is no intersection, otherwise it will contain this point. Examples ======== >>> from sympy import Point >>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 0) >>> p1.intersection(p2) [] >>> p1.intersection(p3) [Point(0, 0)] """ if isinstance(o, Point): if self == o: return [self] return [] return o.intersection(self)
[docs] def rotate(self, angle, pt=None): """Rotate ``angle`` radians counterclockwise about Point ``pt``. See Also ======== rotate, scale Examples ======== >>> from sympy import Point, pi >>> t = Point(1, 0) >>> t.rotate(pi/2) Point(0, 1) >>> t.rotate(pi/2, (2, 0)) Point(2, -1) """ from sympy import cos, sin, Point c = cos(angle) s = sin(angle) rv = self if pt is not None: pt = Point(pt) rv -= pt x, y = rv.args rv = Point(c*x - s*y, s*x + c*y) if pt is not None: rv += pt return rv
[docs] def scale(self, x=1, y=1, pt=None): """Scale the coordinates of the Point by multiplying by ``x`` and ``y`` after subtracting ``pt`` -- default is (0, 0) -- and then adding ``pt`` back again (i.e. ``pt`` is the point of reference for the scaling). See Also ======== rotate, translate Examples ======== >>> from sympy import Point >>> t = Point(1, 1) >>> t.scale(2) Point(2, 1) >>> t.scale(2, 2) Point(2, 2) """ if pt: pt = Point(pt) return self.translate(*(-pt).args).scale(x, y).translate(*pt.args) return Point(self.x*x, self.y*y)
[docs] def translate(self, x=0, y=0): """Shift the Point by adding x and y to the coordinates of the Point. See Also ======== rotate, scale Examples ======== >>> from sympy import Point >>> t = Point(0, 1) >>> t.translate(2) Point(2, 1) >>> t.translate(2, 2) Point(2, 3) >>> t + Point(2, 2) Point(2, 3) """ return Point(self.x + x, self.y + y)
[docs] def transform(self, matrix): """Return the point after applying the transformation described by the 3x3 Matrix, ``matrix``. See Also ======== geometry.entity.rotate geometry.entity.scale geometry.entity.translate """ x, y = self.args return Point(*(Matrix(1, 3, [x, y, 1])*matrix).tolist()[0][:2])
[docs] def dot(self, p2): """Return dot product of self with another Point.""" p2 = Point(p2) x1, y1 = self.args x2, y2 = p2.args return x1*x2 + y1*y2
def __add__(self, other): """Add other to self by incrementing self's coordinates by those of other. See Also ======== sympy.geometry.entity.translate """ if isinstance(other, Point): if len(other.args) == len(self.args): return Point(*[simplify(a + b) for a, b in zip(self.args, other.args)]) else: raise TypeError( "Points must have the same number of dimensions") else: raise ValueError('Cannot add non-Point, %s, to a Point' % other) def __sub__(self, other): """Subtract two points, or subtract a factor from this point's coordinates.""" return self + (-other) def __mul__(self, factor): """Multiply point's coordinates by a factor.""" factor = sympify(factor) return Point([x*factor for x in self.args]) def __div__(self, divisor): """Divide point's coordinates by a factor.""" divisor = sympify(divisor) return Point([x/divisor for x in self.args]) __truediv__ = __div__ def __neg__(self): """Negate the point.""" return Point([-x for x in self.args]) def __abs__(self): """Returns the distance between this point and the origin.""" origin = Point([0]*len(self.args)) return Point.distance(origin, self)