"""Elliptical geometrical entities. Contains * Ellipse * Circle """ from sympy.core.expr import Expr from sympy.core.relational import Eq from sympy.core import S, pi, sympify from sympy.core.evalf import N from sympy.core.parameters import global_parameters from sympy.core.logic import fuzzy_bool from sympy.core.numbers import Rational, oo from sympy.core.sorting import ordered from sympy.core.symbol import Dummy, uniquely_named_symbol, _symbol from sympy.simplify import simplify, trigsimp from sympy.functions.elementary.miscellaneous import sqrt, Max from sympy.functions.elementary.trigonometric import cos, sin from sympy.functions.special.elliptic_integrals import elliptic_e from .entity import GeometryEntity, GeometrySet from .exceptions import GeometryError from .line import Line, Segment, Ray2D, Segment2D, Line2D, LinearEntity3D from .point import Point, Point2D, Point3D from .util import idiff, find from sympy.polys import DomainError, Poly, PolynomialError from sympy.polys.polyutils import _not_a_coeff, _nsort from sympy.solvers import solve from sympy.solvers.solveset import linear_coeffs from sympy.utilities.misc import filldedent, func_name from mpmath.libmp.libmpf import prec_to_dps import random x, y = [Dummy('ellipse_dummy', real=True) for i in range(2)] class Ellipse(GeometrySet): """An elliptical GeometryEntity. Parameters ========== center : Point, optional Default value is Point(0, 0) hradius : number or SymPy expression, optional vradius : number or SymPy expression, optional eccentricity : number or SymPy expression, optional Two of `hradius`, `vradius` and `eccentricity` must be supplied to create an Ellipse. The third is derived from the two supplied. Attributes ========== center hradius vradius area circumference eccentricity periapsis apoapsis focus_distance foci Raises ====== GeometryError When `hradius`, `vradius` and `eccentricity` are incorrectly supplied as parameters. TypeError When `center` is not a Point. See Also ======== Circle Notes ----- Constructed from a center and two radii, the first being the horizontal radius (along the x-axis) and the second being the vertical radius (along the y-axis). When symbolic value for hradius and vradius are used, any calculation that refers to the foci or the major or minor axis will assume that the ellipse has its major radius on the x-axis. If this is not true then a manual rotation is necessary. Examples ======== >>> from sympy import Ellipse, Point, Rational >>> e1 = Ellipse(Point(0, 0), 5, 1) >>> e1.hradius, e1.vradius (5, 1) >>> e2 = Ellipse(Point(3, 1), hradius=3, eccentricity=Rational(4, 5)) >>> e2 Ellipse(Point2D(3, 1), 3, 9/5) """ def __contains__(self, o): if isinstance(o, Point): res = self.equation(x, y).subs({x: o.x, y: o.y}) return trigsimp(simplify(res)) is S.Zero elif isinstance(o, Ellipse): return self == o return False def __eq__(self, o): """Is the other GeometryEntity the same as this ellipse?""" return isinstance(o, Ellipse) and (self.center == o.center and self.hradius == o.hradius and self.vradius == o.vradius) def __hash__(self): return super().__hash__() def __new__( cls, center=None, hradius=None, vradius=None, eccentricity=None, **kwargs): hradius = sympify(hradius) vradius = sympify(vradius) if center is None: center = Point(0, 0) else: if len(center) != 2: raise ValueError('The center of "{}" must be a two dimensional point'.format(cls)) center = Point(center, dim=2) if len(list(filter(lambda x: x is not None, (hradius, vradius, eccentricity)))) != 2: raise ValueError(filldedent(''' Exactly two arguments of "hradius", "vradius", and "eccentricity" must not be None.''')) if eccentricity is not None: eccentricity = sympify(eccentricity) if eccentricity.is_negative: raise GeometryError("Eccentricity of ellipse/circle should lie between [0, 1)") elif hradius is None: hradius = vradius / sqrt(1 - eccentricity**2) elif vradius is None: vradius = hradius * sqrt(1 - eccentricity**2) if hradius == vradius: return Circle(center, hradius, **kwargs) if S.Zero in (hradius, vradius): return Segment(Point(center[0] - hradius, center[1] - vradius), Point(center[0] + hradius, center[1] + vradius)) if hradius.is_real is False or vradius.is_real is False: raise GeometryError("Invalid value encountered when computing hradius / vradius.") return GeometryEntity.__new__(cls, center, hradius, vradius, **kwargs) def _svg(self, scale_factor=1., fill_color="#66cc99"): """Returns SVG ellipse element for the Ellipse. Parameters ========== scale_factor : float Multiplication factor for the SVG stroke-width. Default is 1. fill_color : str, optional Hex string for fill color. Default is "#66cc99". """ c = N(self.center) h, v = N(self.hradius), N(self.vradius) return ( '' ).format(2. * scale_factor, fill_color, c.x, c.y, h, v) @property def ambient_dimension(self): return 2 @property def apoapsis(self): """The apoapsis of the ellipse. The greatest distance between the focus and the contour. Returns ======= apoapsis : number See Also ======== periapsis : Returns shortest distance between foci and contour Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.apoapsis 2*sqrt(2) + 3 """ return self.major * (1 + self.eccentricity) def arbitrary_point(self, parameter='t'): """A parameterized point on the ellipse. Parameters ========== parameter : str, optional Default value is 't'. Returns ======= arbitrary_point : Point Raises ====== ValueError When `parameter` already appears in the functions. See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Point, Ellipse >>> e1 = Ellipse(Point(0, 0), 3, 2) >>> e1.arbitrary_point() Point2D(3*cos(t), 2*sin(t)) """ t = _symbol(parameter, real=True) if t.name in (f.name for f in self.free_symbols): raise ValueError(filldedent('Symbol %s already appears in object ' 'and cannot be used as a parameter.' % t.name)) return Point(self.center.x + self.hradius*cos(t), self.center.y + self.vradius*sin(t)) @property def area(self): """The area of the ellipse. Returns ======= area : number Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.area 3*pi """ return simplify(S.Pi * self.hradius * self.vradius) @property def bounds(self): """Return a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure. """ h, v = self.hradius, self.vradius return (self.center.x - h, self.center.y - v, self.center.x + h, self.center.y + v) @property def center(self): """The center of the ellipse. Returns ======= center : number See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.center Point2D(0, 0) """ return self.args[0] @property def circumference(self): """The circumference of the ellipse. Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.circumference 12*elliptic_e(8/9) """ if self.eccentricity == 1: # degenerate return 4*self.major elif self.eccentricity == 0: # circle return 2*pi*self.hradius else: return 4*self.major*elliptic_e(self.eccentricity**2) @property def eccentricity(self): """The eccentricity of the ellipse. Returns ======= eccentricity : number Examples ======== >>> from sympy import Point, Ellipse, sqrt >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, sqrt(2)) >>> e1.eccentricity sqrt(7)/3 """ return self.focus_distance / self.major def encloses_point(self, p): """ Return True if p is enclosed by (is inside of) self. Notes ----- Being on the border of self is considered False. Parameters ========== p : Point Returns ======= encloses_point : True, False or None See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Ellipse, S >>> from sympy.abc import t >>> e = Ellipse((0, 0), 3, 2) >>> e.encloses_point((0, 0)) True >>> e.encloses_point(e.arbitrary_point(t).subs(t, S.Half)) False >>> e.encloses_point((4, 0)) False """ p = Point(p, dim=2) if p in self: return False if len(self.foci) == 2: # if the combined distance from the foci to p (h1 + h2) is less # than the combined distance from the foci to the minor axis # (which is the same as the major axis length) then p is inside # the ellipse h1, h2 = [f.distance(p) for f in self.foci] test = 2*self.major - (h1 + h2) else: test = self.radius - self.center.distance(p) return fuzzy_bool(test.is_positive) def equation(self, x='x', y='y', _slope=None): """ Returns the equation of an ellipse aligned with the x and y axes; when slope is given, the equation returned corresponds to an ellipse with a major axis having that slope. Parameters ========== x : str, optional Label for the x-axis. Default value is 'x'. y : str, optional Label for the y-axis. Default value is 'y'. _slope : Expr, optional The slope of the major axis. Ignored when 'None'. Returns ======= equation : SymPy expression See Also ======== arbitrary_point : Returns parameterized point on ellipse Examples ======== >>> from sympy import Point, Ellipse, pi >>> from sympy.abc import x, y >>> e1 = Ellipse(Point(1, 0), 3, 2) >>> eq1 = e1.equation(x, y); eq1 y**2/4 + (x/3 - 1/3)**2 - 1 >>> eq2 = e1.equation(x, y, _slope=1); eq2 (-x + y + 1)**2/8 + (x + y - 1)**2/18 - 1 A point on e1 satisfies eq1. Let's use one on the x-axis: >>> p1 = e1.center + Point(e1.major, 0) >>> assert eq1.subs(x, p1.x).subs(y, p1.y) == 0 When rotated the same as the rotated ellipse, about the center point of the ellipse, it will satisfy the rotated ellipse's equation, too: >>> r1 = p1.rotate(pi/4, e1.center) >>> assert eq2.subs(x, r1.x).subs(y, r1.y) == 0 References ========== .. [1] https://math.stackexchange.com/questions/108270/what-is-the-equation-of-an-ellipse-that-is-not-aligned-with-the-axis .. [2] https://en.wikipedia.org/wiki/Ellipse#Shifted_ellipse """ x = _symbol(x, real=True) y = _symbol(y, real=True) dx = x - self.center.x dy = y - self.center.y if _slope is not None: L = (dy - _slope*dx)**2 l = (_slope*dy + dx)**2 h = 1 + _slope**2 b = h*self.major**2 a = h*self.minor**2 return l/b + L/a - 1 else: t1 = (dx/self.hradius)**2 t2 = (dy/self.vradius)**2 return t1 + t2 - 1 def evolute(self, x='x', y='y'): """The equation of evolute of the ellipse. Parameters ========== x : str, optional Label for the x-axis. Default value is 'x'. y : str, optional Label for the y-axis. Default value is 'y'. Returns ======= equation : SymPy expression Examples ======== >>> from sympy import Point, Ellipse >>> e1 = Ellipse(Point(1, 0), 3, 2) >>> e1.evolute() 2**(2/3)*y**(2/3) + (3*x - 3)**(2/3) - 5**(2/3) """ if len(self.args) != 3: raise NotImplementedError('Evolute of arbitrary Ellipse is not supported.') x = _symbol(x, real=True) y = _symbol(y, real=True) t1 = (self.hradius*(x - self.center.x))**Rational(2, 3) t2 = (self.vradius*(y - self.center.y))**Rational(2, 3) return t1 + t2 - (self.hradius**2 - self.vradius**2)**Rational(2, 3) @property def foci(self): """The foci of the ellipse. Notes ----- The foci can only be calculated if the major/minor axes are known. Raises ====== ValueError When the major and minor axis cannot be determined. See Also ======== sympy.geometry.point.Point focus_distance : Returns the distance between focus and center Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.foci (Point2D(-2*sqrt(2), 0), Point2D(2*sqrt(2), 0)) """ c = self.center hr, vr = self.hradius, self.vradius if hr == vr: return (c, c) # calculate focus distance manually, since focus_distance calls this # routine fd = sqrt(self.major**2 - self.minor**2) if hr == self.minor: # foci on the y-axis return (c + Point(0, -fd), c + Point(0, fd)) elif hr == self.major: # foci on the x-axis return (c + Point(-fd, 0), c + Point(fd, 0)) @property def focus_distance(self): """The focal distance of the ellipse. The distance between the center and one focus. Returns ======= focus_distance : number See Also ======== foci Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.focus_distance 2*sqrt(2) """ return Point.distance(self.center, self.foci[0]) @property def hradius(self): """The horizontal radius of the ellipse. Returns ======= hradius : number See Also ======== vradius, major, minor Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.hradius 3 """ return self.args[1] def intersection(self, o): """The intersection of this ellipse and another geometrical entity `o`. Parameters ========== o : GeometryEntity Returns ======= intersection : list of GeometryEntity objects Notes ----- Currently supports intersections with Point, Line, Segment, Ray, Circle and Ellipse types. See Also ======== sympy.geometry.entity.GeometryEntity Examples ======== >>> from sympy import Ellipse, Point, Line >>> e = Ellipse(Point(0, 0), 5, 7) >>> e.intersection(Point(0, 0)) [] >>> e.intersection(Point(5, 0)) [Point2D(5, 0)] >>> e.intersection(Line(Point(0,0), Point(0, 1))) [Point2D(0, -7), Point2D(0, 7)] >>> e.intersection(Line(Point(5,0), Point(5, 1))) [Point2D(5, 0)] >>> e.intersection(Line(Point(6,0), Point(6, 1))) [] >>> e = Ellipse(Point(-1, 0), 4, 3) >>> e.intersection(Ellipse(Point(1, 0), 4, 3)) [Point2D(0, -3*sqrt(15)/4), Point2D(0, 3*sqrt(15)/4)] >>> e.intersection(Ellipse(Point(5, 0), 4, 3)) [Point2D(2, -3*sqrt(7)/4), Point2D(2, 3*sqrt(7)/4)] >>> e.intersection(Ellipse(Point(100500, 0), 4, 3)) [] >>> e.intersection(Ellipse(Point(0, 0), 3, 4)) [Point2D(3, 0), Point2D(-363/175, -48*sqrt(111)/175), Point2D(-363/175, 48*sqrt(111)/175)] >>> e.intersection(Ellipse(Point(-1, 0), 3, 4)) [Point2D(-17/5, -12/5), Point2D(-17/5, 12/5), Point2D(7/5, -12/5), Point2D(7/5, 12/5)] """ # TODO: Replace solve with nonlinsolve, when nonlinsolve will be able to solve in real domain if isinstance(o, Point): if o in self: return [o] else: return [] elif isinstance(o, (Segment2D, Ray2D)): ellipse_equation = self.equation(x, y) result = solve([ellipse_equation, Line( o.points[0], o.points[1]).equation(x, y)], [x, y], set=True)[1] return list(ordered([Point(i) for i in result if i in o])) elif isinstance(o, Polygon): return o.intersection(self) elif isinstance(o, (Ellipse, Line2D)): if o == self: return self else: ellipse_equation = self.equation(x, y) return list(ordered([Point(i) for i in solve( [ellipse_equation, o.equation(x, y)], [x, y], set=True)[1]])) elif isinstance(o, LinearEntity3D): raise TypeError('Entity must be two dimensional, not three dimensional') else: raise TypeError('Intersection not handled for %s' % func_name(o)) def is_tangent(self, o): """Is `o` tangent to the ellipse? Parameters ========== o : GeometryEntity An Ellipse, LinearEntity or Polygon Raises ====== NotImplementedError When the wrong type of argument is supplied. Returns ======= is_tangent: boolean True if o is tangent to the ellipse, False otherwise. See Also ======== tangent_lines Examples ======== >>> from sympy import Point, Ellipse, Line >>> p0, p1, p2 = Point(0, 0), Point(3, 0), Point(3, 3) >>> e1 = Ellipse(p0, 3, 2) >>> l1 = Line(p1, p2) >>> e1.is_tangent(l1) True """ if isinstance(o, Point2D): return False elif isinstance(o, Ellipse): intersect = self.intersection(o) if isinstance(intersect, Ellipse): return True elif intersect: return all((self.tangent_lines(i)[0]).equals(o.tangent_lines(i)[0]) for i in intersect) else: return False elif isinstance(o, Line2D): hit = self.intersection(o) if not hit: return False if len(hit) == 1: return True # might return None if it can't decide return hit[0].equals(hit[1]) elif isinstance(o, Ray2D): intersect = self.intersection(o) if len(intersect) == 1: return intersect[0] != o.source and not self.encloses_point(o.source) else: return False elif isinstance(o, (Segment2D, Polygon)): all_tangents = False segments = o.sides if isinstance(o, Polygon) else [o] for segment in segments: intersect = self.intersection(segment) if len(intersect) == 1: if not any(intersect[0] in i for i in segment.points) \ and not any(self.encloses_point(i) for i in segment.points): all_tangents = True continue else: return False else: return all_tangents return all_tangents elif isinstance(o, (LinearEntity3D, Point3D)): raise TypeError('Entity must be two dimensional, not three dimensional') else: raise TypeError('Is_tangent not handled for %s' % func_name(o)) @property def major(self): """Longer axis of the ellipse (if it can be determined) else hradius. Returns ======= major : number or expression See Also ======== hradius, vradius, minor Examples ======== >>> from sympy import Point, Ellipse, Symbol >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.major 3 >>> a = Symbol('a') >>> b = Symbol('b') >>> Ellipse(p1, a, b).major a >>> Ellipse(p1, b, a).major b >>> m = Symbol('m') >>> M = m + 1 >>> Ellipse(p1, m, M).major m + 1 """ ab = self.args[1:3] if len(ab) == 1: return ab[0] a, b = ab o = b - a < 0 if o == True: return a elif o == False: return b return self.hradius @property def minor(self): """Shorter axis of the ellipse (if it can be determined) else vradius. Returns ======= minor : number or expression See Also ======== hradius, vradius, major Examples ======== >>> from sympy import Point, Ellipse, Symbol >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.minor 1 >>> a = Symbol('a') >>> b = Symbol('b') >>> Ellipse(p1, a, b).minor b >>> Ellipse(p1, b, a).minor a >>> m = Symbol('m') >>> M = m + 1 >>> Ellipse(p1, m, M).minor m """ ab = self.args[1:3] if len(ab) == 1: return ab[0] a, b = ab o = a - b < 0 if o == True: return a elif o == False: return b return self.vradius def normal_lines(self, p, prec=None): """Normal lines between `p` and the ellipse. Parameters ========== p : Point Returns ======= normal_lines : list with 1, 2 or 4 Lines Examples ======== >>> from sympy import Point, Ellipse >>> e = Ellipse((0, 0), 2, 3) >>> c = e.center >>> e.normal_lines(c + Point(1, 0)) [Line2D(Point2D(0, 0), Point2D(1, 0))] >>> e.normal_lines(c) [Line2D(Point2D(0, 0), Point2D(0, 1)), Line2D(Point2D(0, 0), Point2D(1, 0))] Off-axis points require the solution of a quartic equation. This often leads to very large expressions that may be of little practical use. An approximate solution of `prec` digits can be obtained by passing in the desired value: >>> e.normal_lines((3, 3), prec=2) [Line2D(Point2D(-0.81, -2.7), Point2D(0.19, -1.2)), Line2D(Point2D(1.5, -2.0), Point2D(2.5, -2.7))] Whereas the above solution has an operation count of 12, the exact solution has an operation count of 2020. """ p = Point(p, dim=2) # XXX change True to something like self.angle == 0 if the arbitrarily # rotated ellipse is introduced. # https://github.com/sympy/sympy/issues/2815) if True: rv = [] if p.x == self.center.x: rv.append(Line(self.center, slope=oo)) if p.y == self.center.y: rv.append(Line(self.center, slope=0)) if rv: # at these special orientations of p either 1 or 2 normals # exist and we are done return rv # find the 4 normal points and construct lines through them with # the corresponding slope eq = self.equation(x, y) dydx = idiff(eq, y, x) norm = -1/dydx slope = Line(p, (x, y)).slope seq = slope - norm # TODO: Replace solve with solveset, when this line is tested yis = solve(seq, y)[0] xeq = eq.subs(y, yis).as_numer_denom()[0].expand() if len(xeq.free_symbols) == 1: try: # this is so much faster, it's worth a try xsol = Poly(xeq, x).real_roots() except (DomainError, PolynomialError, NotImplementedError): # TODO: Replace solve with solveset, when these lines are tested xsol = _nsort(solve(xeq, x), separated=True)[0] points = [Point(i, solve(eq.subs(x, i), y)[0]) for i in xsol] else: raise NotImplementedError( 'intersections for the general ellipse are not supported') slopes = [norm.subs(zip((x, y), pt.args)) for pt in points] if prec is not None: points = [pt.n(prec) for pt in points] slopes = [i if _not_a_coeff(i) else i.n(prec) for i in slopes] return [Line(pt, slope=s) for pt, s in zip(points, slopes)] @property def periapsis(self): """The periapsis of the ellipse. The shortest distance between the focus and the contour. Returns ======= periapsis : number See Also ======== apoapsis : Returns greatest distance between focus and contour Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.periapsis 3 - 2*sqrt(2) """ return self.major * (1 - self.eccentricity) @property def semilatus_rectum(self): """ Calculates the semi-latus rectum of the Ellipse. Semi-latus rectum is defined as one half of the chord through a focus parallel to the conic section directrix of a conic section. Returns ======= semilatus_rectum : number See Also ======== apoapsis : Returns greatest distance between focus and contour periapsis : The shortest distance between the focus and the contour Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.semilatus_rectum 1/3 References ========== .. [1] https://mathworld.wolfram.com/SemilatusRectum.html .. [2] https://en.wikipedia.org/wiki/Ellipse#Semi-latus_rectum """ return self.major * (1 - self.eccentricity ** 2) def auxiliary_circle(self): """Returns a Circle whose diameter is the major axis of the ellipse. Examples ======== >>> from sympy import Ellipse, Point, symbols >>> c = Point(1, 2) >>> Ellipse(c, 8, 7).auxiliary_circle() Circle(Point2D(1, 2), 8) >>> a, b = symbols('a b') >>> Ellipse(c, a, b).auxiliary_circle() Circle(Point2D(1, 2), Max(a, b)) """ return Circle(self.center, Max(self.hradius, self.vradius)) def director_circle(self): """ Returns a Circle consisting of all points where two perpendicular tangent lines to the ellipse cross each other. Returns ======= Circle A director circle returned as a geometric object. Examples ======== >>> from sympy import Ellipse, Point, symbols >>> c = Point(3,8) >>> Ellipse(c, 7, 9).director_circle() Circle(Point2D(3, 8), sqrt(130)) >>> a, b = symbols('a b') >>> Ellipse(c, a, b).director_circle() Circle(Point2D(3, 8), sqrt(a**2 + b**2)) References ========== .. [1] https://en.wikipedia.org/wiki/Director_circle """ return Circle(self.center, sqrt(self.hradius**2 + self.vradius**2)) def plot_interval(self, parameter='t'): """The plot interval for the default geometric plot of the Ellipse. Parameters ========== parameter : str, optional Default value is 't'. Returns ======= plot_interval : list [parameter, lower_bound, upper_bound] Examples ======== >>> from sympy import Point, Ellipse >>> e1 = Ellipse(Point(0, 0), 3, 2) >>> e1.plot_interval() [t, -pi, pi] """ t = _symbol(parameter, real=True) return [t, -S.Pi, S.Pi] def random_point(self, seed=None): """A random point on the ellipse. Returns ======= point : Point Examples ======== >>> from sympy import Point, Ellipse >>> e1 = Ellipse(Point(0, 0), 3, 2) >>> e1.random_point() # gives some random point Point2D(...) >>> p1 = e1.random_point(seed=0); p1.n(2) Point2D(2.1, 1.4) Notes ===== When creating a random point, one may simply replace the parameter with a random number. When doing so, however, the random number should be made a Rational or else the point may not test as being in the ellipse: >>> from sympy.abc import t >>> from sympy import Rational >>> arb = e1.arbitrary_point(t); arb Point2D(3*cos(t), 2*sin(t)) >>> arb.subs(t, .1) in e1 False >>> arb.subs(t, Rational(.1)) in e1 True >>> arb.subs(t, Rational('.1')) in e1 True See Also ======== sympy.geometry.point.Point arbitrary_point : Returns parameterized point on ellipse """ t = _symbol('t', real=True) x, y = self.arbitrary_point(t).args # get a random value in [-1, 1) corresponding to cos(t) # and confirm that it will test as being in the ellipse if seed is not None: rng = random.Random(seed) else: rng = random # simplify this now or else the Float will turn s into a Float r = Rational(rng.random()) c = 2*r - 1 s = sqrt(1 - c**2) return Point(x.subs(cos(t), c), y.subs(sin(t), s)) def reflect(self, line): """Override GeometryEntity.reflect since the radius is not a GeometryEntity. Examples ======== >>> from sympy import Circle, Line >>> Circle((0, 1), 1).reflect(Line((0, 0), (1, 1))) Circle(Point2D(1, 0), -1) >>> from sympy import Ellipse, Line, Point >>> Ellipse(Point(3, 4), 1, 3).reflect(Line(Point(0, -4), Point(5, 0))) Traceback (most recent call last): ... NotImplementedError: General Ellipse is not supported but the equation of the reflected Ellipse is given by the zeros of: f(x, y) = (9*x/41 + 40*y/41 + 37/41)**2 + (40*x/123 - 3*y/41 - 364/123)**2 - 1 Notes ===== Until the general ellipse (with no axis parallel to the x-axis) is supported a NotImplemented error is raised and the equation whose zeros define the rotated ellipse is given. """ if line.slope in (0, oo): c = self.center c = c.reflect(line) return self.func(c, -self.hradius, self.vradius) else: x, y = [uniquely_named_symbol( name, (self, line), modify=lambda s: '_' + s, real=True) for name in 'xy'] expr = self.equation(x, y) p = Point(x, y).reflect(line) result = expr.subs(zip((x, y), p.args ), simultaneous=True) raise NotImplementedError(filldedent( 'General Ellipse is not supported but the equation ' 'of the reflected Ellipse is given by the zeros of: ' + "f(%s, %s) = %s" % (str(x), str(y), str(result)))) def rotate(self, angle=0, pt=None): """Rotate ``angle`` radians counterclockwise about Point ``pt``. Note: since the general ellipse is not supported, only rotations that are integer multiples of pi/2 are allowed. Examples ======== >>> from sympy import Ellipse, pi >>> Ellipse((1, 0), 2, 1).rotate(pi/2) Ellipse(Point2D(0, 1), 1, 2) >>> Ellipse((1, 0), 2, 1).rotate(pi) Ellipse(Point2D(-1, 0), 2, 1) """ if self.hradius == self.vradius: return self.func(self.center.rotate(angle, pt), self.hradius) if (angle/S.Pi).is_integer: return super().rotate(angle, pt) if (2*angle/S.Pi).is_integer: return self.func(self.center.rotate(angle, pt), self.vradius, self.hradius) # XXX see https://github.com/sympy/sympy/issues/2815 for general ellipes raise NotImplementedError('Only rotations of pi/2 are currently supported for Ellipse.') def scale(self, x=1, y=1, pt=None): """Override GeometryEntity.scale since it is the major and minor axes which must be scaled and they are not GeometryEntities. Examples ======== >>> from sympy import Ellipse >>> Ellipse((0, 0), 2, 1).scale(2, 4) Circle(Point2D(0, 0), 4) >>> Ellipse((0, 0), 2, 1).scale(2) Ellipse(Point2D(0, 0), 4, 1) """ c = self.center if pt: pt = Point(pt, dim=2) return self.translate(*(-pt).args).scale(x, y).translate(*pt.args) h = self.hradius v = self.vradius return self.func(c.scale(x, y), hradius=h*x, vradius=v*y) def tangent_lines(self, p): """Tangent lines between `p` and the ellipse. If `p` is on the ellipse, returns the tangent line through point `p`. Otherwise, returns the tangent line(s) from `p` to the ellipse, or None if no tangent line is possible (e.g., `p` inside ellipse). Parameters ========== p : Point Returns ======= tangent_lines : list with 1 or 2 Lines Raises ====== NotImplementedError Can only find tangent lines for a point, `p`, on the ellipse. See Also ======== sympy.geometry.point.Point, sympy.geometry.line.Line Examples ======== >>> from sympy import Point, Ellipse >>> e1 = Ellipse(Point(0, 0), 3, 2) >>> e1.tangent_lines(Point(3, 0)) [Line2D(Point2D(3, 0), Point2D(3, -12))] """ p = Point(p, dim=2) if self.encloses_point(p): return [] if p in self: delta = self.center - p rise = (self.vradius**2)*delta.x run = -(self.hradius**2)*delta.y p2 = Point(simplify(p.x + run), simplify(p.y + rise)) return [Line(p, p2)] else: if len(self.foci) == 2: f1, f2 = self.foci maj = self.hradius test = (2*maj - Point.distance(f1, p) - Point.distance(f2, p)) else: test = self.radius - Point.distance(self.center, p) if test.is_number and test.is_positive: return [] # else p is outside the ellipse or we can't tell. In case of the # latter, the solutions returned will only be valid if # the point is not inside the ellipse; if it is, nan will result. eq = self.equation(x, y) dydx = idiff(eq, y, x) slope = Line(p, Point(x, y)).slope # TODO: Replace solve with solveset, when this line is tested tangent_points = solve([slope - dydx, eq], [x, y]) # handle horizontal and vertical tangent lines if len(tangent_points) == 1: if tangent_points[0][ 0] == p.x or tangent_points[0][1] == p.y: return [Line(p, p + Point(1, 0)), Line(p, p + Point(0, 1))] else: return [Line(p, p + Point(0, 1)), Line(p, tangent_points[0])] # others return [Line(p, tangent_points[0]), Line(p, tangent_points[1])] @property def vradius(self): """The vertical radius of the ellipse. Returns ======= vradius : number See Also ======== hradius, major, minor Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.vradius 1 """ return self.args[2] def second_moment_of_area(self, point=None): """Returns the second moment and product moment area of an ellipse. Parameters ========== point : Point, two-tuple of sympifiable objects, or None(default=None) point is the point about which second moment of area is to be found. If "point=None" it will be calculated about the axis passing through the centroid of the ellipse. Returns ======= I_xx, I_yy, I_xy : number or SymPy expression I_xx, I_yy are second moment of area of an ellise. I_xy is product moment of area of an ellipse. Examples ======== >>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.second_moment_of_area() (3*pi/4, 27*pi/4, 0) References ========== .. [1] https://en.wikipedia.org/wiki/List_of_second_moments_of_area """ I_xx = (S.Pi*(self.hradius)*(self.vradius**3))/4 I_yy = (S.Pi*(self.hradius**3)*(self.vradius))/4 I_xy = 0 if point is None: return I_xx, I_yy, I_xy # parallel axis theorem I_xx = I_xx + self.area*((point[1] - self.center.y)**2) I_yy = I_yy + self.area*((point[0] - self.center.x)**2) I_xy = I_xy + self.area*(point[0] - self.center.x)*(point[1] - self.center.y) return I_xx, I_yy, I_xy def polar_second_moment_of_area(self): """Returns the polar second moment of area of an Ellipse It is a constituent of the second moment of area, linked through the perpendicular axis theorem. While the planar second moment of area describes an object's resistance to deflection (bending) when subjected to a force applied to a plane parallel to the central axis, the polar second moment of area describes an object's resistance to deflection when subjected to a moment applied in a plane perpendicular to the object's central axis (i.e. parallel to the cross-section) Examples ======== >>> from sympy import symbols, Circle, Ellipse >>> c = Circle((5, 5), 4) >>> c.polar_second_moment_of_area() 128*pi >>> a, b = symbols('a, b') >>> e = Ellipse((0, 0), a, b) >>> e.polar_second_moment_of_area() pi*a**3*b/4 + pi*a*b**3/4 References ========== .. [1] https://en.wikipedia.org/wiki/Polar_moment_of_inertia """ second_moment = self.second_moment_of_area() return second_moment[0] + second_moment[1] def section_modulus(self, point=None): """Returns a tuple with the section modulus of an ellipse Section modulus is a geometric property of an ellipse defined as the ratio of second moment of area to the distance of the extreme end of the ellipse from the centroidal axis. Parameters ========== point : Point, two-tuple of sympifyable objects, or None(default=None) point is the point at which section modulus is to be found. If "point=None" section modulus will be calculated for the point farthest from the centroidal axis of the ellipse. Returns ======= S_x, S_y: numbers or SymPy expressions S_x is the section modulus with respect to the x-axis S_y is the section modulus with respect to the y-axis A negative sign indicates that the section modulus is determined for a point below the centroidal axis. Examples ======== >>> from sympy import Symbol, Ellipse, Circle, Point2D >>> d = Symbol('d', positive=True) >>> c = Circle((0, 0), d/2) >>> c.section_modulus() (pi*d**3/32, pi*d**3/32) >>> e = Ellipse(Point2D(0, 0), 2, 4) >>> e.section_modulus() (8*pi, 4*pi) >>> e.section_modulus((2, 2)) (16*pi, 4*pi) References ========== .. [1] https://en.wikipedia.org/wiki/Section_modulus """ x_c, y_c = self.center if point is None: # taking x and y as maximum distances from centroid x_min, y_min, x_max, y_max = self.bounds y = max(y_c - y_min, y_max - y_c) x = max(x_c - x_min, x_max - x_c) else: # taking x and y as distances of the given point from the center point = Point2D(point) y = point.y - y_c x = point.x - x_c second_moment = self.second_moment_of_area() S_x = second_moment[0]/y S_y = second_moment[1]/x return S_x, S_y class Circle(Ellipse): """A circle in space. Constructed simply from a center and a radius, from three non-collinear points, or the equation of a circle. Parameters ========== center : Point radius : number or SymPy expression points : sequence of three Points equation : equation of a circle Attributes ========== radius (synonymous with hradius, vradius, major and minor) circumference equation Raises ====== GeometryError When the given equation is not that of a circle. When trying to construct circle from incorrect parameters. See Also ======== Ellipse, sympy.geometry.point.Point Examples ======== >>> from sympy import Point, Circle, Eq >>> from sympy.abc import x, y, a, b A circle constructed from a center and radius: >>> c1 = Circle(Point(0, 0), 5) >>> c1.hradius, c1.vradius, c1.radius (5, 5, 5) A circle constructed from three points: >>> c2 = Circle(Point(0, 0), Point(1, 1), Point(1, 0)) >>> c2.hradius, c2.vradius, c2.radius, c2.center (sqrt(2)/2, sqrt(2)/2, sqrt(2)/2, Point2D(1/2, 1/2)) A circle can be constructed from an equation in the form `a*x**2 + by**2 + gx + hy + c = 0`, too: >>> Circle(x**2 + y**2 - 25) Circle(Point2D(0, 0), 5) If the variables corresponding to x and y are named something else, their name or symbol can be supplied: >>> Circle(Eq(a**2 + b**2, 25), x='a', y=b) Circle(Point2D(0, 0), 5) """ def __new__(cls, *args, **kwargs): evaluate = kwargs.get('evaluate', global_parameters.evaluate) if len(args) == 1 and isinstance(args[0], (Expr, Eq)): x = kwargs.get('x', 'x') y = kwargs.get('y', 'y') equation = args[0].expand() if isinstance(equation, Eq): equation = equation.lhs - equation.rhs x = find(x, equation) y = find(y, equation) try: a, b, c, d, e = linear_coeffs(equation, x**2, y**2, x, y) except ValueError: raise GeometryError("The given equation is not that of a circle.") if S.Zero in (a, b) or a != b: raise GeometryError("The given equation is not that of a circle.") center_x = -c/a/2 center_y = -d/b/2 r2 = (center_x**2) + (center_y**2) - e/a return Circle((center_x, center_y), sqrt(r2), evaluate=evaluate) else: c, r = None, None if len(args) == 3: args = [Point(a, dim=2, evaluate=evaluate) for a in args] t = Triangle(*args) if not isinstance(t, Triangle): return t c = t.circumcenter r = t.circumradius elif len(args) == 2: # Assume (center, radius) pair c = Point(args[0], dim=2, evaluate=evaluate) r = args[1] # this will prohibit imaginary radius try: r = Point(r, 0, evaluate=evaluate).x except ValueError: raise GeometryError("Circle with imaginary radius is not permitted") if not (c is None or r is None): if r == 0: return c return GeometryEntity.__new__(cls, c, r, **kwargs) raise GeometryError("Circle.__new__ received unknown arguments") def _eval_evalf(self, prec=15, **options): pt, r = self.args dps = prec_to_dps(prec) pt = pt.evalf(n=dps, **options) r = r.evalf(n=dps, **options) return self.func(pt, r, evaluate=False) @property def circumference(self): """The circumference of the circle. Returns ======= circumference : number or SymPy expression Examples ======== >>> from sympy import Point, Circle >>> c1 = Circle(Point(3, 4), 6) >>> c1.circumference 12*pi """ return 2 * S.Pi * self.radius def equation(self, x='x', y='y'): """The equation of the circle. Parameters ========== x : str or Symbol, optional Default value is 'x'. y : str or Symbol, optional Default value is 'y'. Returns ======= equation : SymPy expression Examples ======== >>> from sympy import Point, Circle >>> c1 = Circle(Point(0, 0), 5) >>> c1.equation() x**2 + y**2 - 25 """ x = _symbol(x, real=True) y = _symbol(y, real=True) t1 = (x - self.center.x)**2 t2 = (y - self.center.y)**2 return t1 + t2 - self.major**2 def intersection(self, o): """The intersection of this circle with another geometrical entity. Parameters ========== o : GeometryEntity Returns ======= intersection : list of GeometryEntities Examples ======== >>> from sympy import Point, Circle, Line, Ray >>> p1, p2, p3 = Point(0, 0), Point(5, 5), Point(6, 0) >>> p4 = Point(5, 0) >>> c1 = Circle(p1, 5) >>> c1.intersection(p2) [] >>> c1.intersection(p4) [Point2D(5, 0)] >>> c1.intersection(Ray(p1, p2)) [Point2D(5*sqrt(2)/2, 5*sqrt(2)/2)] >>> c1.intersection(Line(p2, p3)) [] """ return Ellipse.intersection(self, o) @property def radius(self): """The radius of the circle. Returns ======= radius : number or SymPy expression See Also ======== Ellipse.major, Ellipse.minor, Ellipse.hradius, Ellipse.vradius Examples ======== >>> from sympy import Point, Circle >>> c1 = Circle(Point(3, 4), 6) >>> c1.radius 6 """ return self.args[1] def reflect(self, line): """Override GeometryEntity.reflect since the radius is not a GeometryEntity. Examples ======== >>> from sympy import Circle, Line >>> Circle((0, 1), 1).reflect(Line((0, 0), (1, 1))) Circle(Point2D(1, 0), -1) """ c = self.center c = c.reflect(line) return self.func(c, -self.radius) def scale(self, x=1, y=1, pt=None): """Override GeometryEntity.scale since the radius is not a GeometryEntity. Examples ======== >>> from sympy import Circle >>> Circle((0, 0), 1).scale(2, 2) Circle(Point2D(0, 0), 2) >>> Circle((0, 0), 1).scale(2, 4) Ellipse(Point2D(0, 0), 2, 4) """ c = self.center if pt: pt = Point(pt, dim=2) return self.translate(*(-pt).args).scale(x, y).translate(*pt.args) c = c.scale(x, y) x, y = [abs(i) for i in (x, y)] if x == y: return self.func(c, x*self.radius) h = v = self.radius return Ellipse(c, hradius=h*x, vradius=v*y) @property def vradius(self): """ This Ellipse property is an alias for the Circle's radius. Whereas hradius, major and minor can use Ellipse's conventions, the vradius does not exist for a circle. It is always a positive value in order that the Circle, like Polygons, will have an area that can be positive or negative as determined by the sign of the hradius. Examples ======== >>> from sympy import Point, Circle >>> c1 = Circle(Point(3, 4), 6) >>> c1.vradius 6 """ return abs(self.radius) from .polygon import Polygon, Triangle