Source code for sympy.geometry.plane

"""Geometrical Planes.

Contains
========
Plane

"""
from __future__ import print_function, division

from sympy.core import S, C, sympify, Dummy, nan, Eq, symbols, Symbol, Rational
from sympy.core.function import expand_mul
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.polys.polytools import cancel
from sympy.geometry.exceptions import GeometryError
from .entity import GeometryEntity
from .point3d import Point3D
from .point import Point
from .line3d import LinearEntity3D, Line3D, Segment3D, Ray3D
from .line import Line, Segment, Ray
from sympy.matrices import Matrix

[docs]class Plane(GeometryEntity): """ A plane is a flat, two-dimensional surface. A plane is the two-dimensional analogue of a point (zero-dimensions), a line (one-dimension) and a solid (three-dimensions). A plane can generally be constructed by two types of inputs. They are three non-collinear points and a point and the plane's normal vector. Attributes ========== p1 normal_vector Examples ======== >>> from sympy import Plane, Point3D >>> from sympy.abc import x >>> Plane(Point3D(1, 1, 1), Point3D(2, 3, 4), Point3D(2, 2, 2)) Plane(Point3D(1, 1, 1), (-1, 2, -1)) >>> Plane((1, 1, 1), (2, 3, 4), (2, 2, 2)) Plane(Point3D(1, 1, 1), (-1, 2, -1)) >>> Plane(Point3D(1, 1, 1), normal_vector=(1,4,7)) Plane(Point3D(1, 1, 1), (1, 4, 7)) """ def __new__(cls, p1, a=None, b=None, **kwargs): p1 = Point3D(p1) if not a and not b and kwargs.get('normal_vector', None): a = kwargs.pop('normal_vector') if not b and not isinstance(a, Point3D) and \ len(a) == 3: normal_vector = a elif a and b: p2 = Point3D(a) p3 = Point3D(b) if Point3D.are_collinear(p1, p2, p3): raise NotImplementedError('Enter three non-collinear points') a = p1.direction_ratio(p2) b = p1.direction_ratio(p3) normal_vector = tuple(Matrix(a).cross(Matrix(b))) else: raise ValueError('Either provide 3 3D points or a point with a ' 'normal vector') return GeometryEntity.__new__(cls, p1, normal_vector, **kwargs) @property
[docs] def p1(self): """The only defining point of the plane. Others can be obtained from the arbitrary_point method. See Also ======== sympy.geometry.point3d.Point3D Examples ======== >>> from sympy import Point3D, Plane >>> a = Plane(Point3D(1, 1, 1), Point3D(2, 3, 4), Point3D(2, 2, 2)) >>> a.p1 Point3D(1, 1, 1) """ return self.args[0]
@property
[docs] def normal_vector(self): """Normal vector of the given plane. Examples ======== >>> from sympy import Point3D, Plane >>> a = Plane(Point3D(1, 1, 1), Point3D(2, 3, 4), Point3D(2, 2, 2)) >>> a.normal_vector (-1, 2, -1) >>> a = Plane(Point3D(1, 1, 1), normal_vector=(1, 4, 7)) >>> a.normal_vector (1, 4, 7) """ return self.args[1]
[docs] def equation(self, x=None, y=None, z=None): """The equation of the Plane. Examples ======== >>> from sympy import Point3D, Plane >>> a = Plane(Point3D(1, 1, 2), Point3D(2, 4, 7), Point3D(3, 5, 1)) >>> a.equation() -23*x + 11*y - 2*z + 16 >>> a = Plane(Point3D(1, 4, 2), normal_vector=(6, 6, 6)) >>> a.equation() 6*x + 6*y + 6*z - 42 """ x, y, z = [i if i else Symbol(j, real=True) for i, j in zip((x, y, z), 'xyz')] a = Point3D(x, y, z) b = self.p1.direction_ratio(a) c = self.normal_vector return (sum(i*j for i, j in zip(b, c)))
[docs] def projection(self, pt): """Project the given point onto the plane along the plane normal. Parameters ========== Point or Point3D Returns ======= Point3D Examples ======== >>> from sympy import Plane, Point, Point3D >>> A = Plane(Point3D(1, 1, 2), normal_vector=(1, 1, 1)) The projection is along the normal vector direction, not the z axis, so (1, 1) does not project to (1, 1, 2) on the plane A: >>> b = Point(1, 1) >>> A.projection(b) Point3D(5/3, 5/3, 2/3) >>> _ in A True But the point (1, 1, 2) projects to (1, 1) on the XY-plane: >>> XY = Plane((0, 0, 0), (0, 0, 1)) >>> XY.projection((1, 1, 2)) Point3D(1, 1, 0) """ rv = Point3D(pt) if rv in self: return rv return self.intersection(Line3D(rv, rv + Point3D(self.normal_vector)))[0]
[docs] def projection_line(self, line): """Project the given line onto the plane through the normal plane containing the line. Parameters ========== LinearEntity or LinearEntity3D Returns ======= Point3D, Line3D, Ray3D or Segment3D Notes ===== For the interaction between 2D and 3D lines(segments, rays), you should convert the line to 3D by using this method. For example for finding the intersection between a 2D and a 3D line, convert the 2D line to a 3D line by projecting it on a required plane and then proceed to find the intersection between those lines. Examples ======== >>> from sympy import Plane, Line, Line3D, Point, Point3D >>> a = Plane(Point3D(1, 1, 1), normal_vector=(1, 1, 1)) >>> b = Line(Point(1, 1), Point(2, 2)) >>> a.projection_line(b) Line3D(Point3D(4/3, 4/3, 1/3), Point3D(5/3, 5/3, -1/3)) >>> c = Line3D(Point3D(1, 1, 1), Point3D(2, 2, 2)) >>> a.projection_line(c) Point3D(1, 1, 1) """ from sympy.geometry.line import LinearEntity from sympy.geometry.line3d import LinearEntity3D if not isinstance(line, (LinearEntity, LinearEntity3D)): raise NotImplementedError('Enter a linear entity only') a, b = self.projection(line.p1), self.projection(line.p2) if a == b: # projection does not imply intersection so for # this case (line parallel to plane's normal) we # return the projection point return a if isinstance(line, (Line, Line3D)): return Line3D(a, b) if isinstance(line, (Ray, Ray3D)): return Ray3D(a, b) if isinstance(line, (Segment, Segment3D)): return Segment3D(a, b)
[docs] def is_parallel(self, l): """Is the given geometric entity parallel to the plane? Parameters ========== LinearEntity3D or Plane Returns ======= Boolean Examples ======== >>> from sympy import Plane, Point3D >>> a = Plane(Point3D(1,4,6), normal_vector=(2, 4, 6)) >>> b = Plane(Point3D(3,1,3), normal_vector=(4, 8, 12)) >>> a.is_parallel(b) True """ from sympy.geometry.line3d import LinearEntity3D if isinstance(l, LinearEntity3D): a = l.direction_ratio b = self.normal_vector c = sum([i*j for i, j in zip(a, b)]) if c == 0: return True else: return False elif isinstance(l, Plane): a = Matrix(l.normal_vector) b = Matrix(self.normal_vector) if a.cross(b).is_zero: return True else: return False
[docs] def is_perpendicular(self, l): """is the given geometric entity perpendicualar to the given plane? Parameters ========== LinearEntity3D or Plane Returns ======= Boolean Examples ======== >>> from sympy import Plane, Point3D >>> a = Plane(Point3D(1,4,6), normal_vector=(2, 4, 6)) >>> b = Plane(Point3D(2, 2, 2), normal_vector=(-1, 2, -1)) >>> a.is_perpendicular(b) True """ from sympy.geometry.line3d import LinearEntity3D if isinstance(l, LinearEntity3D): a = Matrix(l.direction_ratio) b = Matrix(self.normal_vector) if a.cross(b).is_zero: return True else: return False elif isinstance(l, Plane): a = Matrix(l.normal_vector) b = Matrix(self.normal_vector) if a.dot(b) == 0: return True else: return False else: return False
[docs] def distance(self, o): """Distance beteen the plane and another geometric entity. Parameters ========== Point3D, LinearEntity3D, Plane. Returns ======= distance Notes ===== This method accepts only 3D entities as it's parameter, but if you want to calculate the distance between a 2D entity and a plane you should first convert to a 3D entity by projecting onto a desired plane and then proceed to calculate the distance. Examples ======== >>> from sympy import Point, Point3D, Line, Line3D, Plane >>> a = Plane(Point3D(1, 1, 1), normal_vector=(1, 1, 1)) >>> b = Point3D(1, 2, 3) >>> a.distance(b) sqrt(3) >>> c = Line3D(Point3D(2, 3, 1), Point3D(1, 2, 2)) >>> a.distance(c) 0 """ from sympy.geometry.line3d import LinearEntity3D x, y, z = map(Dummy, 'xyz') if self.intersection(o) != []: return S.Zero if isinstance(o, Point3D): x, y, z = map(Dummy, 'xyz') k = self.equation(x, y, z) a, b, c = [k.coeff(i) for i in (x, y, z)] d = k.xreplace({x: o.args[0], y: o.args[1], z: o.args[2]}) t = abs(d/sqrt(a**2 + b**2 + c**2)) return t if isinstance(o, LinearEntity3D): a, b = o.p1, self.p1 c = Matrix(a.direction_ratio(b)) d = Matrix(self.normal_vector) e = c.dot(d) f = sqrt(sum([i**2 for i in self.normal_vector])) return abs(e / f) if isinstance(o, Plane): a, b = o.p1, self.p1 c = Matrix(a.direction_ratio(b)) d = Matrix(self.normal_vector) e = c.dot(d) f = sqrt(sum([i**2 for i in self.normal_vector])) return abs(e / f)
[docs] def angle_between(self, o): """Angle between the plane and other geometric entity. Parameters ========== LinearEntity3D, Plane. Returns ======= angle : angle in radians Notes ===== This method accepts only 3D entities as it's parameter, but if you want to calculate the angle between a 2D entity and a plane you should first convert to a 3D entity by projecting onto a desired plane and then proceed to calculate the angle. Examples ======== >>> from sympy import Point3D, Line3D, Plane >>> a = Plane(Point3D(1, 2, 2), normal_vector=(1, 2, 3)) >>> b = Line3D(Point3D(1, 3, 4), Point3D(2, 2, 2)) >>> a.angle_between(b) -asin(sqrt(21)/6) """ from sympy.geometry.line3d import LinearEntity3D if isinstance(o, LinearEntity3D): a = Matrix(self.normal_vector) b = Matrix(o.direction_ratio) c = a.dot(b) d = sqrt(sum([i**2 for i in self.normal_vector])) e = sqrt(sum([i**2 for i in o.direction_ratio])) return C.asin(c/(d*e)) if isinstance(o, Plane): a = Matrix(self.normal_vector) b = Matrix(o.normal_vector) c = a.dot(b) d = sqrt(sum([i**2 for i in self.normal_vector])) e = sqrt(sum([i**2 for i in o.normal_vector])) return C.acos(c/(d*e))
@staticmethod
[docs] def are_concurrent(*planes): """Is a sequence of Planes concurrent? Two or more Planes are concurrent if their intersections are a common line. Parameters ========== planes: list Returns ======= Boolean Examples ======== >>> from sympy import Plane, Point3D >>> a = Plane(Point3D(5, 0, 0), normal_vector=(1, -1, 1)) >>> b = Plane(Point3D(0, -2, 0), normal_vector=(3, 1, 1)) >>> c = Plane(Point3D(0, -1, 0), normal_vector=(5, -1, 9)) >>> Plane.are_concurrent(a, b) True >>> Plane.are_concurrent(a, b, c) False """ planes = set(planes) if len(planes) < 2: return False for i in planes: if not isinstance(i, Plane): raise ValueError('All objects should be Planes but got %s' % i.func) planes = list(planes) first = planes.pop(0) sol = first.intersection(planes[0]) if sol == []: return False else: line = sol[0] for i in planes[1:]: l = first.intersection(i) if not l or not l[0] in line: return False return True
[docs] def perpendicular_line(self, pt): """A line perpendicular to the given plane. Parameters ========== pt: Point3D Returns ======= Line3D Examples ======== >>> from sympy import Plane, Point3D, Line3D >>> a = Plane(Point3D(1,4,6), normal_vector=(2, 4, 6)) >>> a.perpendicular_line(Point3D(9, 8, 7)) Line3D(Point3D(9, 8, 7), Point3D(11, 12, 13)) """ a = self.normal_vector return Line3D(pt, direction_ratio=a)
[docs] def parallel_plane(self, pt): """ Plane parallel to the given plane and passing through the point pt. Parameters ========== pt: Point3D Returns ======= Plane Examples ======== >>> from sympy import Plane, Point3D >>> a = Plane(Point3D(1, 4, 6), normal_vector=(2, 4, 6)) >>> a.parallel_plane(Point3D(2, 3, 5)) Plane(Point3D(2, 3, 5), (2, 4, 6)) """ a = self.normal_vector return Plane(pt, normal_vector=a)
[docs] def perpendicular_plane(self, *pts): """ Return a perpendicular passing through the given points. If the direction ratio between the points is the same as the Plane's normal vector then, to select from the infinite number of possible planes, a third point will be chosen on the z-axis (or the y-axis if the normal vector is already parallel to the z-axis). If less than two points are given they will be supplied as follows: if no point is given then pt1 will be self.p1; if a second point is not given it will be a point through pt1 on a line parallel to the z-axis (if the normal is not already the z-axis, otherwise on the line parallel to the y-axis). Parameters ========== pts: 0, 1 or 2 Point3D Returns ======= Plane Examples ======== >>> from sympy import Plane, Point3D, Line3D >>> a, b = Point3D(0, 0, 0), Point3D(0, 1, 0) >>> Z = (0, 0, 1) >>> p = Plane(a, normal_vector=Z) >>> p.perpendicular_plane(a, b) Plane(Point3D(0, 0, 0), (1, 0, 0)) """ if len(pts) > 2: raise ValueError('No more than 2 pts should be provided.') pts = list(pts) if len(pts) == 0: pts.append(self.p1) if len(pts) == 1: x, y, z = self.normal_vector if x == y == 0: dir = (0, 1, 0) else: dir = (0, 0, 1) pts.append(pts[0] + Point3D(*dir)) p1, p2 = [Point3D(i) for i in pts] l = Line3D(p1, p2) n = Line3D(p1, direction_ratio=self.normal_vector) if l in n: # XXX should an error be raised instead? # there are infinitely many perpendicular planes; x, y, z = self.normal_vector if x == y == 0: # the z axis is the normal so pick a pt on the y-axis p3 = Point3D(0, 1, 0) # case 1 else: # else pick a pt on the z axis p3 = Point3D(0, 0, 1) # case 2 # in case that point is already given, move it a bit if p3 in l: p3 *= 2 # case 3 else: p3 = p1 + Point3D(*self.normal_vector) # case 4 return Plane(p1, p2, p3)
[docs] def random_point(self, seed=None): """ Returns a random point on the Plane. Returns ======= Point3D """ import random if seed is not None: rng = random.Random(seed) else: rng = random t = Dummy('t') return self.arbitrary_point(t).subs(t, Rational(rng.random()))
[docs] def arbitrary_point(self, t=None): """ Returns an arbitrary point on the Plane; varying `t` from 0 to 2*pi will move the point in a circle of radius 1 about p1 of the Plane. Examples ======== >>> from sympy.geometry.plane import Plane >>> from sympy.abc import t >>> p = Plane((0, 0, 0), (0, 0, 1), (0, 1, 0)) >>> p.arbitrary_point(t) Point3D(0, cos(t), sin(t)) >>> _.distance(p.p1).simplify() 1 Returns ======= Point3D """ from sympy import cos, sin t = t or Dummy('t') x, y, z = self.normal_vector a, b, c = self.p1.args if x == y == 0: return Point3D(a + cos(t), b + sin(t), c) elif x == z == 0: return Point3D(a + cos(t), b, c + sin(t)) elif y == z == 0: return Point3D(a, b + cos(t), c + sin(t)) m = Dummy() p = self.projection(Point3D(self.p1.x + cos(t), self.p1.y + sin(t), 0)*m) return p.xreplace({m: solve(p.distance(self.p1) - 1, m)[0]})
[docs] def intersection(self, o): """ The intersection with other geometrical entity. Parameters ========== Point, Point3D, LinearEntity, LinearEntity3D, Plane Returns ======= List Examples ======== >>> from sympy import Point, Point3D, Line, Line3D, Plane >>> a = Plane(Point3D(1, 2, 3), normal_vector=(1, 1, 1)) >>> b = Point3D(1, 2, 3) >>> a.intersection(b) [Point3D(1, 2, 3)] >>> c = Line3D(Point3D(1, 4, 7), Point3D(2, 2, 2)) >>> a.intersection(c) [Point3D(2, 2, 2)] >>> d = Plane(Point3D(6, 0, 0), normal_vector=(2, -5, 3)) >>> e = Plane(Point3D(2, 0, 0), normal_vector=(3, 4, -3)) >>> d.intersection(e) [Line3D(Point3D(78/23, -24/23, 0), Point3D(147/23, 321/23, 23))] """ from sympy.geometry.line3d import LinearEntity3D from sympy.geometry.line import LinearEntity if isinstance(o, (Point, Point3D)): if o in self: return [Point3D(o)] else: return [] if isinstance(o, (LinearEntity, LinearEntity3D)): if o in self: p1, p2 = o.p1, o.p2 if isinstance(o, Segment): o = Segment3D(p1, p2) elif isinstance(o, Ray): o = Ray3D(p1, p2) elif isinstance(o, Line): o = Line3D(p1, p2) else: raise ValueError('unhandled linear entity: %s' % o.func) return [o] else: x, y, z = map(Dummy, 'xyz') t = Dummy() # unnamed else it may clash with a symbol in o a = Point3D(o.arbitrary_point(t)) b = self.equation(x, y, z) c = solve(b.subs(list(zip((x, y, z), a.args))), t) if not c: return [] else: p = a.subs(t, c[0]) if p not in self: return [] # e.g. a segment might not intersect a plane return [p] if isinstance(o, Plane): if o == self: return [self] if self.is_parallel(o): return [] else: x, y, z = map(Dummy, 'xyz') a, b = Matrix([self.normal_vector]), Matrix([o.normal_vector]) c = list(a.cross(b)) d = self.equation(x, y, z) e = o.equation(x, y, z) f = solve((d.subs(z, 0), e.subs(z, 0)), [x, y]) if len(f) == 2: return [Line3D(Point3D(f[x], f[y], 0), direction_ratio=c)] g = solve((d.subs(y, 0), e.subs(y, 0)),[x, z]) if len(g) == 2: return [Line3D(Point3D(g[x], 0, g[z]), direction_ratio=c)] h = solve((d.subs(x, 0), e.subs(x, 0)),[y, z]) if len(h) == 2: return [Line3D(Point3D(0, h[y], h[z]), direction_ratio=c)]
def __contains__(self, o): from sympy.geometry.line3d import LinearEntity3D from sympy.geometry.line import LinearEntity x, y, z = map(Dummy, 'xyz') k = self.equation(x, y, z) if isinstance(o, Point): o = Point3D(o) if isinstance(o, Point3D): d = k.xreplace(dict(zip((x, y, z), o.args))) return d.equals(0) elif isinstance(o, (LinearEntity, LinearEntity3D)): t = Dummy() d = Point3D(o.arbitrary_point(t)) e = k.subs([(x, d.x), (y, d.y), (z, d.z)]) return e.equals(0) else: return False
[docs] def is_coplanar(self, o): """ Returns True if `o` is coplanar with self, else False. Examples ======== >>> from sympy import Plane, Point3D >>> o = (0, 0, 0) >>> p = Plane(o, (1, 1, 1)) >>> p2 = Plane(o, (2, 2, 2)) >>> p == p2 False >>> p.is_coplanar(p2) True """ if isinstance(o, Plane): x, y, z = map(Dummy, 'xyz') return not cancel(self.equation(x, y, z)/o.equation(x, y, z)).has(x, y, z) if isinstance(o, Point3D): return o in self elif isinstance(o, LinearEntity3D): return all(i in self for i in self) elif isinstance(o, GeometryEntity): # XXX should only be handling 2D objects now return all(i == 0 for i in self.normal_vector[:2])