# Source code for sympy.geometry.point3d

"""Geometrical Points.

Contains
========
Point3D

"""

from __future__ import print_function, division

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.geometry.point import Point
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.complexes import im
from .entity import GeometryEntity
from sympy.matrices import Matrix
from sympy.core.numbers import Float
from sympy.core.evaluate import global_evaluate

[docs]class Point3D(GeometryEntity): """A point in a 3-dimensional Euclidean space. Parameters ========== coords : sequence of 3 coordinate values. Attributes ========== x y z length Raises ====== NotImplementedError When trying to create a point other than 2 or 3 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 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 elif isinstance(args[0], Point): 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]
@property
[docs] def length(self): """ Treating a Point as a Line, this returns 0 for the length of a Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 1) >>> p.length 0 """ return S.Zero
[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 """ # Coincident points are irrelevant and can confuse this algorithm. # Use only unique points. points = list(set(points)) if not all(isinstance(p, Point3D) for p in points): raise TypeError('Must pass only 3D Point objects') if len(points) < 2: return False if len(points) == 2: return True # two points always form a line if len(points) == 3: a = (points[0].direction_cosine(points[1])) b = (points[0].direction_cosine(points[2])) a = [abs(i) for i in a] b = [abs(i) for i in b] if a == b: return True else: return False # XXX Cross product is used now, # If the concept needs to extend to more than three # dimensions then another method would have to be used for i in range(len(points) - 2): pv1 = [j - k for j, k in zip(points[i].args, \ points[i + 1].args)] pv2 = [j - k for j, k in zip(points[i + 1].args, points[i + 2].args)] rank = Matrix([pv1, pv2]).rank() if (rank != 1): return False return True
@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 NotImplementedError: # XXX should be ValueError pass raise ValueError('At least 3 non-collinear points needed to define plane.')
[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 import Point3D >>> p1, p2 = Point3D(1, 1, 1), Point3D(4, 5, 0) >>> p1.distance(p2) sqrt(26) >>> from sympy.abc import x, y, z >>> p3 = Point3D(x, y, z) >>> p3.distance(Point3D(0, 0, 0)) sqrt(x**2 + y**2 + z**2) """ p = Point3D(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 import Point3D >>> p1, p2 = Point3D(1, 1, 1), Point3D(13, 5, 1) >>> p1.midpoint(p2) Point3D(7, 3, 1) """ p = Point3D(p) return Point3D([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 Point3D, Rational >>> p1 = Point3D(Rational(1, 2), Rational(3, 2), Rational(5, 2)) >>> p1 Point3D(1/2, 3/2, 5/2) >>> p1.evalf() Point3D(0.5, 1.5, 2.5) """ coords = [x.evalf(prec, **options) for x in self.args] return Point3D(*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 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 3x3 Matrix, matrix. See Also ======== geometry.entity.rotate geometry.entity.scale geometry.entity.translate """ x, y, z = self.args return Point3D(*(Matrix(1, 3, [x, y, z])*matrix).tolist()[0][:2])
[docs] def dot(self, p2): """Return dot product of self with another Point.""" p2 = Point3D(p2) x1, y1, z1 = self.args x2, y2, z2 = p2.args return x1*x2 + y1*y2 + z1*z2
def equals(self, other): if not isinstance(other, Point3D): return False return all(a.equals(b) for a, b in zip(self.args, other.args)) 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, Point3D): if len(other.args) == len(self.args): return Point3D(*[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.""" if isinstance(other, Point3D): if len(other.args) == len(self.args): return Point3D(*[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 subtract non-Point, %s, to a Point' % other) def __mul__(self, factor): """Multiply point's coordinates by a factor.""" factor = sympify(factor) return Point3D([x*factor for x in self.args]) def __div__(self, divisor): """Divide point's coordinates by a factor.""" divisor = sympify(divisor) return Point3D([x/divisor for x in self.args]) __truediv__ = __div__ def __neg__(self): """Negate the point.""" return Point3D([-x for x in self.args]) def __abs__(self): """Returns the distance between this point and the origin.""" origin = Point3D([0]*len(self.args)) return Point3D.distance(origin, self)