"""Geometrical Planes. Contains ======== Plane """ from sympy.core import Dummy, Rational, S, Symbol from sympy.core.symbol import _symbol from sympy.functions.elementary.trigonometric import cos, sin, acos, asin, sqrt from .entity import GeometryEntity from .line import (Line, Ray, Segment, Line3D, LinearEntity, LinearEntity3D, Ray3D, Segment3D) from .point import Point, Point3D from sympy.matrices import Matrix from sympy.polys.polytools import cancel from sympy.solvers import solve, linsolve from sympy.utilities.iterables import uniq, is_sequence from sympy.utilities.misc import filldedent, func_name, Undecidable from mpmath.libmp.libmpf import prec_to_dps import random x, y, z, t = [Dummy('plane_dummy') for i in range(4)] 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 >>> 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, dim=3) if a and b: p2 = Point(a, dim=3) p3 = Point(b, dim=3) if Point3D.are_collinear(p1, p2, p3): raise ValueError('Enter three non-collinear points') a = p1.direction_ratio(p2) b = p1.direction_ratio(p3) normal_vector = tuple(Matrix(a).cross(Matrix(b))) else: a = kwargs.pop('normal_vector', a) evaluate = kwargs.get('evaluate', True) if is_sequence(a) and len(a) == 3: normal_vector = Point3D(a).args if evaluate else a else: raise ValueError(filldedent(''' Either provide 3 3D points or a point with a normal vector expressed as a sequence of length 3''')) if all(coord.is_zero for coord in normal_vector): raise ValueError('Normal vector cannot be zero vector') return GeometryEntity.__new__(cls, p1, normal_vector, **kwargs) def __contains__(self, o): k = self.equation(x, y, z) if isinstance(o, (LinearEntity, LinearEntity3D)): d = Point3D(o.arbitrary_point(t)) e = k.subs([(x, d.x), (y, d.y), (z, d.z)]) return e.equals(0) try: o = Point(o, dim=3, strict=True) d = k.xreplace(dict(zip((x, y, z), o.args))) return d.equals(0) except TypeError: return False def _eval_evalf(self, prec=15, **options): pt, tup = self.args dps = prec_to_dps(prec) pt = pt.evalf(n=dps, **options) tup = tuple([i.evalf(n=dps, **options) for i in tup]) return self.func(pt, normal_vector=tup, evaluate=False) 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) """ 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 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 acos(c/(d*e)) def arbitrary_point(self, u=None, v=None): """ Returns an arbitrary point on the Plane. If given two parameters, the point ranges over the entire plane. If given 1 or no parameters, returns a point with one parameter which, when varying from 0 to 2*pi, moves the point in a circle of radius 1 about p1 of the Plane. Examples ======== >>> from sympy import Plane, Ray >>> from sympy.abc import u, v, t, r >>> p = Plane((1, 1, 1), normal_vector=(1, 0, 0)) >>> p.arbitrary_point(u, v) Point3D(1, u + 1, v + 1) >>> p.arbitrary_point(t) Point3D(1, cos(t) + 1, sin(t) + 1) While arbitrary values of u and v can move the point anywhere in the plane, the single-parameter point can be used to construct a ray whose arbitrary point can be located at angle t and radius r from p.p1: >>> Ray(p.p1, _).arbitrary_point(r) Point3D(1, r*cos(t) + 1, r*sin(t) + 1) Returns ======= Point3D """ circle = v is None if circle: u = _symbol(u or 't', real=True) else: u = _symbol(u or 'u', real=True) v = _symbol(v or 'v', real=True) x, y, z = self.normal_vector a, b, c = self.p1.args # x1, y1, z1 is a nonzero vector parallel to the plane if x.is_zero and y.is_zero: x1, y1, z1 = S.One, S.Zero, S.Zero else: x1, y1, z1 = -y, x, S.Zero # x2, y2, z2 is also parallel to the plane, and orthogonal to x1, y1, z1 x2, y2, z2 = tuple(Matrix((x, y, z)).cross(Matrix((x1, y1, z1)))) if circle: x1, y1, z1 = (w/sqrt(x1**2 + y1**2 + z1**2) for w in (x1, y1, z1)) x2, y2, z2 = (w/sqrt(x2**2 + y2**2 + z2**2) for w in (x2, y2, z2)) p = Point3D(a + x1*cos(u) + x2*sin(u), \ b + y1*cos(u) + y2*sin(u), \ c + z1*cos(u) + z2*sin(u)) else: p = Point3D(a + x1*u + x2*v, b + y1*u + y2*v, c + z1*u + z2*v) return p @staticmethod 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 = list(uniq(planes)) for i in planes: if not isinstance(i, Plane): raise ValueError('All objects should be Planes but got %s' % i.func) if len(planes) < 2: return False 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 l[0] not in line: return False return True def distance(self, o): """Distance between 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 Point3D, 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 """ if self.intersection(o) != []: return S.Zero if isinstance(o, (Segment3D, Ray3D)): a, b = o.p1, o.p2 pi, = self.intersection(Line3D(a, b)) if pi in o: return self.distance(pi) elif a in Segment3D(pi, b): return self.distance(a) else: assert isinstance(o, Segment3D) is True return self.distance(b) # following code handles `Point3D`, `LinearEntity3D`, `Plane` a = o if isinstance(o, Point3D) else o.p1 n = Point3D(self.normal_vector).unit d = (a - self.p1).dot(n) return abs(d) def equals(self, o): """ Returns True if self and o are the same mathematical entities. Examples ======== >>> from sympy import Plane, Point3D >>> a = Plane(Point3D(1, 2, 3), normal_vector=(1, 1, 1)) >>> b = Plane(Point3D(1, 2, 3), normal_vector=(2, 2, 2)) >>> c = Plane(Point3D(1, 2, 3), normal_vector=(-1, 4, 6)) >>> a.equals(a) True >>> a.equals(b) True >>> a.equals(c) False """ if isinstance(o, Plane): a = self.equation() b = o.equation() return cancel(a/b).is_constant() else: return False 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))) def intersection(self, o): """ The intersection with other geometrical entity. Parameters ========== Point, Point3D, LinearEntity, LinearEntity3D, Plane Returns ======= List Examples ======== >>> from sympy import Point3D, 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))] """ if not isinstance(o, GeometryEntity): o = Point(o, dim=3) if isinstance(o, Point): if o in self: return [o] else: return [] if isinstance(o, (LinearEntity, LinearEntity3D)): # recast to 3D 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) if o in self: return [o] else: a = Point3D(o.arbitrary_point(t)) p1, n = self.p1, Point3D(self.normal_vector) # TODO: Replace solve with solveset, when this line is tested c = solve((a - p1).dot(n), t) if not c: return [] else: c = [i for i in c if i.is_real is not False] if len(c) > 1: c = [i for i in c if i.is_real] if len(c) != 1: raise Undecidable("not sure which point is real") p = a.subs(t, c[0]) if p not in o: return [] # e.g. a segment might not intersect a plane return [p] if isinstance(o, Plane): if self.equals(o): 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) result = list(linsolve([d, e], x, y, z))[0] for i in (x, y, z): result = result.subs(i, 0) return [Line3D(Point3D(result), direction_ratio=c)] def is_coplanar(self, o): """ Returns True if `o` is coplanar with self, else False. Examples ======== >>> from sympy import Plane >>> 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): 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]) 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 """ 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_matrix: return True else: return False 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 """ if isinstance(l, LinearEntity3D): a = Matrix(l.direction_ratio) b = Matrix(self.normal_vector) if a.cross(b).is_zero_matrix: 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 @property 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] @property def p1(self): """The only defining point of the plane. Others can be obtained from the arbitrary_point method. See Also ======== sympy.geometry.point.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] 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) def perpendicular_line(self, pt): """A line perpendicular to the given plane. Parameters ========== pt: Point3D Returns ======= Line3D Examples ======== >>> from sympy import Plane, Point3D >>> 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) 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 >>> 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 = [Point(i, dim=3) 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) 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, Point3D >>> a = Plane(Point3D(1, 1, 1), normal_vector=(1, 1, 1)) >>> b = Line(Point3D(1, 1), Point3D(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) """ 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) 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, 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 = Point3D(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 = Point(pt, dim=3) if rv in self: return rv return self.intersection(Line3D(rv, rv + Point3D(self.normal_vector)))[0] def random_point(self, seed=None): """ Returns a random point on the Plane. Returns ======= Point3D Examples ======== >>> from sympy import Plane >>> p = Plane((1, 0, 0), normal_vector=(0, 1, 0)) >>> r = p.random_point(seed=42) # seed value is optional >>> r.n(3) Point3D(2.29, 0, -1.35) The random point can be moved to lie on the circle of radius 1 centered on p1: >>> c = p.p1 + (r - p.p1).unit >>> c.distance(p.p1).equals(1) True """ if seed is not None: rng = random.Random(seed) else: rng = random params = { x: 2*Rational(rng.gauss(0, 1)) - 1, y: 2*Rational(rng.gauss(0, 1)) - 1} return self.arbitrary_point(x, y).subs(params) def parameter_value(self, other, u, v=None): """Return the parameter(s) corresponding to the given point. Examples ======== >>> from sympy import pi, Plane >>> from sympy.abc import t, u, v >>> p = Plane((2, 0, 0), (0, 0, 1), (0, 1, 0)) By default, the parameter value returned defines a point that is a distance of 1 from the Plane's p1 value and in line with the given point: >>> on_circle = p.arbitrary_point(t).subs(t, pi/4) >>> on_circle.distance(p.p1) 1 >>> p.parameter_value(on_circle, t) {t: pi/4} Moving the point twice as far from p1 does not change the parameter value: >>> off_circle = p.p1 + (on_circle - p.p1)*2 >>> off_circle.distance(p.p1) 2 >>> p.parameter_value(off_circle, t) {t: pi/4} If the 2-value parameter is desired, supply the two parameter symbols and a replacement dictionary will be returned: >>> p.parameter_value(on_circle, u, v) {u: sqrt(10)/10, v: sqrt(10)/30} >>> p.parameter_value(off_circle, u, v) {u: sqrt(10)/5, v: sqrt(10)/15} """ if not isinstance(other, GeometryEntity): other = Point(other, dim=self.ambient_dimension) if not isinstance(other, Point): raise ValueError("other must be a point") if other == self.p1: return other if isinstance(u, Symbol) and v is None: delta = self.arbitrary_point(u) - self.p1 eq = delta - (other - self.p1).unit sol = solve(eq, u, dict=True) elif isinstance(u, Symbol) and isinstance(v, Symbol): pt = self.arbitrary_point(u, v) sol = solve(pt - other, (u, v), dict=True) else: raise ValueError('expecting 1 or 2 symbols') if not sol: raise ValueError("Given point is not on %s" % func_name(self)) return sol[0] # {t: tval} or {u: uval, v: vval} @property def ambient_dimension(self): return self.p1.ambient_dimension