# Source code for sympy.geometry.point

"""Geometrical Points.

Contains
========
Point
Point2D
Point3D

"""

from __future__ import division, print_function

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

from .entity import GeometryEntity

[docs]class Point(GeometryEntity): """A point in a n-dimensional Euclidean space. Parameters ========== coords : sequence of n-coordinate values. In the special case where n=2 or 3, a Point2D or Point3D will be created as appropriate. Attributes ========== length origin: A Point representing the origin of the appropriately-dimensioned space. Raises ====== TypeError When trying to add or subtract points with different dimensions. When intersection is called with object other than a Point. See Also ======== sympy.geometry.line.Segment : Connects two Points Examples ======== >>> from sympy.geometry import Point >>> from sympy.abc import x >>> Point(1, 2, 3) Point3D(1, 2, 3) >>> Point([1, 2]) Point2D(1, 2) >>> Point(0, x) Point2D(0, x) Floats are automatically converted to Rational unless the evaluate flag is False: >>> Point(0.5, 0.25) Point2D(1/2, 1/4) >>> Point(0.5, 0.25, evaluate=False) Point2D(0.5, 0.25) """ def __new__(cls, *args, **kwargs): evaluate = kwargs.get('evaluate', global_evaluate[0]) if iterable(args[0]): if isinstance(args[0], Point) and not evaluate: return args[0] args = args[0] # unpack the arguments into a friendly Tuple # if we were already a Point, we're doing an excess # iteration, but we'll worry about efficiency later coords = Tuple(*args) if any(a.is_number and im(a) for a in coords): raise ValueError('Imaginary coordinates not permitted.') # Turn any Floats into rationals and simplify # any expressions before we instantiate if evaluate: coords = coords.xreplace(dict( [(f, simplify(nsimplify(f, rational=True))) for f in coords.atoms(Float)])) if len(coords) == 2: return Point2D(coords, **kwargs) if len(coords) == 3: return Point3D(coords, **kwargs) return GeometryEntity.__new__(cls, *coords) is_Point = True def __contains__(self, item): return item == self def is_concyclic(*args): # Coincident points are irrelevant and can confuse this algorithm. # Use only unique points. args = list(set(args)) if not all(isinstance(p, Point) for p in args): raise TypeError('Must pass only Point objects') return args[0].is_concyclic(*args[1:])
[docs] def is_collinear(*args): """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 sympy.abc 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; use only unique points. args = list(set(args)) if not all(isinstance(p, Point) for p in args): raise TypeError('Must pass only Point objects') if len(args) == 0: return False if len(args) <= 2: return True # translate our points points = [p - args[0] for p in args[1:]] for p in points[1:]: if not Point.is_scalar_multiple(points[0], p): return False return True
[docs] def is_scalar_multiple(p1, p2): """Returns whether p1 and p2 are scalar multiples of eachother. """ # if the vectors p1 and p2 are linearly dependent, then they must # be scalar multiples of eachother m = Matrix([p1.args, p2.args]) # XXX: issue #9480 we need simplify=True otherwise the # rank may be computed incorrectly return m.rank(simplify=True) < 2
@property
[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
@property
[docs] def origin(self): """A point of all zeros of the same ambient dimension as the current point""" return Point([0]*len(self))
@property
[docs] def is_zero(self): """True if every coordinate is zero, otherwise False.""" return all(x == S.Zero for x in self.args)
@property
[docs] def ambient_dimension(self): """The dimension of the ambient space the point is in. I.e., if the point is in R^n, the ambient dimension will be n""" return len(self)
[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 sympy.abc import x, y >>> p3 = Point(x, y) >>> p3.distance(Point(0, 0)) sqrt(x**2 + y**2) """ return sqrt(sum([(a - b)**2 for a, b in zip( self.args, p.args if isinstance(p, Point) else p)]))
[docs] def taxicab_distance(self, p): """The Taxicab Distance from self to point p. Returns the sum of the horizontal and vertical distances to point p. Parameters ========== p : Point Returns ======= taxicab_distance : The sum of the horizontal and vertical distances to point p. See Also ======== sympy.geometry.Point.distance Examples ======== >>> from sympy.geometry import Point >>> p1, p2 = Point(1, 1), Point(4, 5) >>> p1.taxicab_distance(p2) 7 """ p = Point(p) return sum(abs(a - b) 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) Point2D(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 Point2D(1/2, 3/2) >>> p1.evalf() Point2D(0.5, 1.5) """ coords = [x.evalf(prec, **options) for x in self.args] return Point(*coords, 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) [Point2D(0, 0)] """ if isinstance(o, Point): if len(self) != len(o): raise ValueError("Points must be of the same dimension to intersect") if self == o: return [self] return [] return o.intersection(self)
[docs] def dot(self, p2): """Return dot product of self with another Point.""" p2 = Point(p2) return Add(*[a*b for a,b in zip(self, p2)])
[docs] def equals(self, other): """Returns whether the coordinates of self and other agree.""" # a point is equal to another point if all its components are equal if not isinstance(other, Point) or len(self.args) != len(other.args): return False return all(a.equals(b) for a,b in zip(self.args, other.args))
def __len__(self): return len(self.args) def __iter__(self): return self.args.__iter__() def __eq__(self, other): if not isinstance(other, Point) or len(self.args) != len(other.args): return False return self.args == other.args def __hash__(self): return hash(self.args) def __getitem__(self, key): return self.args[key] def __add__(self, other): """Add other to self by incrementing self's coordinates by those of other. See Also ======== sympy.geometry.entity.translate """ if iterable(other) and len(other) == len(self): return Point([simplify(a + b) for a, b in zip(self, other)]) else: raise ValueError( "Points must have the same number of dimensions") 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([simplify(x*factor) for x in self.args]) def __div__(self, divisor): """Divide point's coordinates by a factor.""" divisor = sympify(divisor) return Point([simplify(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)) return Point.distance(origin, self)
[docs]class Point2D(Point): """A point in a 2-dimensional Euclidean space. Parameters ========== coords : sequence of 2 coordinate values. Attributes ========== x y length Raises ====== TypeError When trying to add or subtract points with different dimensions. When trying to create a point with more than two dimensions. When intersection is called with object other than a Point. See Also ======== sympy.geometry.line.Segment : Connects two Points Examples ======== >>> from sympy.geometry import Point2D >>> from sympy.abc import x >>> Point2D(1, 2) Point2D(1, 2) >>> Point2D([1, 2]) Point2D(1, 2) >>> Point2D(0, x) Point2D(0, x) Floats are automatically converted to Rational unless the evaluate flag is False: >>> Point2D(0.5, 0.25) Point2D(1/2, 1/4) >>> Point2D(0.5, 0.25, evaluate=False) Point2D(0.5, 0.25) """ def __new__(cls, *args, **kwargs): eval = kwargs.get('evaluate', global_evaluate[0]) check = True if isinstance(args[0], Point2D): if not eval: return args[0] args = args[0].args check = False else: if iterable(args[0]): args = args[0] if len(args) != 2: raise ValueError( "Only two dimensional points currently supported") coords = Tuple(*args) if check: if any(a.is_number and im(a) for a in coords): raise ValueError('Imaginary args not permitted.') if eval: coords = coords.xreplace(dict( [(f, simplify(nsimplify(f, rational=True))) for f in coords.atoms(Float)])) return GeometryEntity.__new__(cls, *coords) def __contains__(self, item): return item == self @property
[docs] def x(self): """ Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.x 0 """ return self.args[0]
@property
[docs] def y(self): """ Returns the Y coordinate of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.y 1 """ return self.args[1]
@property
[docs] def bounds(self): """Return a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure. """ return (self.x, self.y, self.x, self.y)
[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
[docs] def rotate(self, angle, pt=None): """Rotate angle radians counterclockwise about Point pt. See Also ======== rotate, scale Examples ======== >>> from sympy import Point2D, pi >>> t = Point2D(1, 0) >>> t.rotate(pi/2) Point2D(0, 1) >>> t.rotate(pi/2, (2, 0)) Point2D(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 Point2D >>> t = Point2D(1, 1) >>> t.scale(2) Point2D(2, 1) >>> t.scale(2, 2) Point2D(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 Point2D >>> t = Point2D(0, 1) >>> t.translate(2) Point2D(2, 1) >>> t.translate(2, 2) Point2D(2, 3) >>> t + Point2D(2, 2) Point2D(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 """ try: col, row = matrix.shape valid_matrix = matrix.is_square and col == 3 except AttributeError: # We hit this block if matrix argument is not actually a Matrix. valid_matrix = False if not valid_matrix: raise ValueError("The argument to the transform function must be " \ + "a 3x3 matrix") x, y = self.args return Point(*(Matrix(1, 3, [x, y, 1])*matrix).tolist()[0][:2])
[docs]class Point3D(Point): """A point in a 3-dimensional Euclidean space. Parameters ========== coords : sequence of 3 coordinate values. Attributes ========== x y z length Raises ====== TypeError When trying to add or subtract points with different dimensions. When intersection is called with object other than a Point. Notes ===== Currently only 2-dimensional and 3-dimensional points are supported. Examples ======== >>> from sympy import Point3D >>> from sympy.abc import x >>> Point3D(1, 2, 3) Point3D(1, 2, 3) >>> Point3D([1, 2, 3]) Point3D(1, 2, 3) >>> Point3D(0, x, 3) Point3D(0, x, 3) Floats are automatically converted to Rational unless the evaluate flag is False: >>> Point3D(0.5, 0.25, 2) Point3D(1/2, 1/4, 2) >>> Point3D(0.5, 0.25, 3, evaluate=False) Point3D(0.5, 0.25, 3) """ def __new__(cls, *args, **kwargs): eval = kwargs.get('evaluate', global_evaluate[0]) if isinstance(args[0], (Point, Point3D)): if not eval: return args[0] args = args[0].args else: if iterable(args[0]): args = args[0] if len(args) not in (2, 3): raise TypeError( "Enter a 2 or 3 dimensional point") coords = Tuple(*args) if len(coords) == 2: coords += (S.Zero,) if eval: coords = coords.xreplace(dict( [(f, simplify(nsimplify(f, rational=True))) for f in coords.atoms(Float)])) return GeometryEntity.__new__(cls, *coords) def __contains__(self, item): return item == self @property
[docs] def x(self): """ Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 3) >>> p.x 0 """ return self.args[0]
@property
[docs] def y(self): """ Returns the Y coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 2) >>> p.y 1 """ return self.args[1]
@property
[docs] def z(self): """ Returns the Z coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 1) >>> p.z 1 """ return self.args[2]
[docs] def direction_ratio(self, point): """ Gives the direction ratio between 2 points Parameters ========== p : Point3D Returns ======= list Examples ======== >>> from sympy import Point3D >>> p1 = Point3D(1, 2, 3) >>> p1.direction_ratio(Point3D(2, 3, 5)) [1, 1, 2] """ return [(point.x - self.x),(point.y - self.y),(point.z - self.z)]
[docs] def direction_cosine(self, point): """ Gives the direction cosine between 2 points Parameters ========== p : Point3D Returns ======= list Examples ======== >>> from sympy import Point3D >>> p1 = Point3D(1, 2, 3) >>> p1.direction_cosine(Point3D(2, 3, 5)) [sqrt(6)/6, sqrt(6)/6, sqrt(6)/3] """ a = self.direction_ratio(point) b = sqrt(sum(i**2 for i in a)) return [(point.x - self.x) / b,(point.y - self.y) / b, (point.z - self.z) / b]
@staticmethod
[docs] def are_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 ======= are_collinear : boolean See Also ======== sympy.geometry.line3d.Line3D Examples ======== >>> from sympy import Point3D, Matrix >>> from sympy.abc import x >>> p1, p2 = Point3D(0, 0, 0), Point3D(1, 1, 1) >>> p3, p4, p5 = Point3D(2, 2, 2), Point3D(x, x, x), Point3D(1, 2, 6) >>> Point3D.are_collinear(p1, p2, p3, p4) True >>> Point3D.are_collinear(p1, p2, p3, p5) False """ return Point.is_collinear(*points)
@staticmethod
[docs] def are_coplanar(*points): """ This function tests whether passed points are coplanar or not. It uses the fact that the triple scalar product of three vectors vanishes if the vectors are coplanar. Which means that the volume of the solid described by them will have to be zero for coplanarity. Parameters ========== A set of points 3D points Returns ======= boolean Examples ======== >>> from sympy import Point3D >>> p1 = Point3D(1, 2, 2) >>> p2 = Point3D(2, 7, 2) >>> p3 = Point3D(0, 0, 2) >>> p4 = Point3D(1, 1, 2) >>> Point3D.are_coplanar(p1, p2, p3, p4) True >>> p5 = Point3D(0, 1, 3) >>> Point3D.are_coplanar(p1, p2, p3, p5) False """ from sympy.geometry.plane import Plane points = list(set(points)) if len(points) < 3: raise ValueError('At least 3 points are needed to define a plane.') a, b = points[:2] for i, c in enumerate(points[2:]): try: p = Plane(a, b, c) for j in (0, 1, i): points.pop(j) return all(p.is_coplanar(i) for i in points) except ValueError: pass raise ValueError('At least 3 non-collinear points needed to define plane.')
[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 Point3D >>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, 0, 0) >>> p1.intersection(p2) [] >>> p1.intersection(p3) [Point3D(0, 0, 0)] """ if isinstance(o, Point3D): if self == o: return [self] return [] return o.intersection(self)
[docs] def scale(self, x=1, y=1, z=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 ======== translate Examples ======== >>> from sympy import Point3D >>> t = Point3D(1, 1, 1) >>> t.scale(2) Point3D(2, 1, 1) >>> t.scale(2, 2) Point3D(2, 2, 1) """ if pt: pt = Point3D(pt) return self.translate(*(-pt).args).scale(x, y, z).translate(*pt.args) return Point3D(self.x*x, self.y*y, self.z*z)
[docs] def translate(self, x=0, y=0, z=0): """Shift the Point by adding x and y to the coordinates of the Point. See Also ======== rotate, scale Examples ======== >>> from sympy import Point3D >>> t = Point3D(0, 1, 1) >>> t.translate(2) Point3D(2, 1, 1) >>> t.translate(2, 2) Point3D(2, 3, 1) >>> t + Point3D(2, 2, 2) Point3D(2, 3, 3) """ return Point3D(self.x + x, self.y + y, self.z + z)
[docs] def transform(self, matrix): """Return the point after applying the transformation described by the 4x4 Matrix, matrix. See Also ======== geometry.entity.rotate geometry.entity.scale geometry.entity.translate """ try: col, row = matrix.shape valid_matrix = matrix.is_square and col == 4 except AttributeError: # We hit this block if matrix argument is not actually a Matrix. valid_matrix = False if not valid_matrix: raise ValueError("The argument to the transform function must be " \ + "a 4x4 matrix") from sympy.matrices.expressions import Transpose x, y, z = self.args m = Transpose(matrix) return Point3D(*(Matrix(1, 4, [x, y, z, 1])*m).tolist()[0][:3])