Source code for sympy.geometry.ellipse

"""Elliptical geometrical entities.

Contains
* Ellipse
* Circle

"""

from __future__ import print_function, division

from sympy.core import S, C, sympify, pi, Dummy
from sympy.core.logic import fuzzy_bool
from sympy.core.numbers import oo, zoo, Rational
from sympy.simplify import simplify, trigsimp
from sympy.functions.elementary.miscellaneous import sqrt, Max, Min
from sympy.functions.elementary.complexes import im
from sympy.geometry.exceptions import GeometryError
from sympy.polys import Poly, PolynomialError, DomainError
from sympy.solvers import solve
from sympy.utilities.lambdify import lambdify
from sympy.utilities.iterables import uniq
from sympy.utilities.misc import filldedent
from .entity import GeometryEntity
from .point import Point
from .line import LinearEntity, Line
from .util import _symbol, idiff
from sympy.mpmath import findroot as nroot


import random

from sympy.utilities.decorator import doctest_depends_on


[docs]class Ellipse(GeometryEntity): """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(Point(3, 1), 3, 9/5) Plotting: >>> from sympy.plotting.pygletplot import PygletPlot as Plot >>> from sympy import Circle, Segment >>> c1 = Circle(Point(0,0), 1) >>> Plot(c1) # doctest: +SKIP [0]: cos(t), sin(t), 'mode=parametric' >>> p = Plot() # doctest: +SKIP >>> p[0] = c1 # doctest: +SKIP >>> radius = Segment(c1.center, c1.random_point()) >>> p[1] = radius # doctest: +SKIP >>> p # doctest: +SKIP [0]: cos(t), sin(t), 'mode=parametric' [1]: t*cos(1.546086215036205357975518382), t*sin(1.546086215036205357975518382), 'mode=parametric' """ def __new__( cls, center=None, hradius=None, vradius=None, eccentricity=None, **kwargs): hradius = sympify(hradius) vradius = sympify(vradius) eccentricity = sympify(eccentricity) if center is None: center = Point(0, 0) else: center = Point(center) if len(list(filter(None, (hradius, vradius, eccentricity)))) != 2: raise ValueError('Exactly two arguments of "hradius", ' '"vradius", and "eccentricity" must not be None."') if eccentricity is not None: if 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) return GeometryEntity.__new__(cls, center, hradius, vradius, **kwargs) @property
[docs] 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 Point(0, 0) """ return self.args[0]
@property
[docs] 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]
@property
[docs] 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]
@property
[docs] 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 """ rv = Min(*self.args[1:3]) if rv.func is Min: return self.vradius return rv
@property
[docs] 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 """ rv = Max(*self.args[1:3]) if rv.func is Max: return self.hradius return rv
@property
[docs] 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
[docs] 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*Integral(sqrt((-8*_x**2/9 + 1)/(-_x**2 + 1)), (_x, 0, 1)) """ if self.eccentricity == 1: return 2*pi*self.hradius else: x = C.Dummy('x', real=True) return 4*self.major*C.Integral( sqrt((1 - (self.eccentricity*x)**2)/(1 - x**2)), (x, 0, 1))
@property
[docs] 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
@property
[docs] 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 -2*sqrt(2) + 3 """ return self.major * (1 - self.eccentricity)
@property
[docs] 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)
@property
[docs] def focus_distance(self): """The focale 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
[docs] 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 (Point(-2*sqrt(2), 0), Point(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))
[docs] 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(Point(0, 1), 1, 2) >>> Ellipse((1, 0), 2, 1).rotate(pi) Ellipse(Point(-1, 0), 2, 1) """ if self.hradius == self.vradius: return self.func(*self.args) if (angle/S.Pi).is_integer: return super(Ellipse, self).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.')
[docs] 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(Point(0, 0), 4) >>> Ellipse((0, 0), 2, 1).scale(2) Ellipse(Point(0, 0), 4, 1) """ c = self.center if pt: pt = Point(pt) 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)
[docs] 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(Point(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. """ from .util import _uniquely_named_symbol 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) 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))))
[docs] 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) 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)
@doctest_depends_on(modules=('pyglet',))
[docs] 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)) [Line(Point(3, 0), Point(3, -12))] >>> # This will plot an ellipse together with a tangent line. >>> from sympy.plotting.pygletplot import PygletPlot as Plot >>> from sympy import Point, Ellipse >>> e = Ellipse(Point(0,0), 3, 2) >>> t = e.tangent_lines(e.random_point()) >>> p = Plot() >>> p[0] = e # doctest: +SKIP >>> p[1] = t # doctest: +SKIP """ p = Point(p) 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. x, y = Dummy('x'), Dummy('y') eq = self.equation(x, y) dydx = idiff(eq, y, x) slope = Line(p, Point(x, y)).slope tangent_points = solve([slope - dydx, eq], [x, y]) # handle horizontal and vertical tangent lines if len(tangent_points) == 1: assert 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))] # others return [Line(p, tangent_points[0]), Line(p, tangent_points[1])]
[docs] 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 """ inter = None if isinstance(o, Ellipse): inter = self.intersection(o) if isinstance(inter, Ellipse): return False return (inter is not None and len(inter) == 1 and isinstance(inter[0], Point)) elif isinstance(o, LinearEntity): inter = self._do_line_intersection(o) if inter is not None and len(inter) == 1: return inter[0] in o else: return False elif isinstance(o, Polygon): c = 0 for seg in o.sides: inter = self._do_line_intersection(seg) c += len([True for point in inter if point in seg]) return c == 1 else: raise NotImplementedError("Unknown argument type")
[docs] 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 Line, Point, Ellipse >>> e = Ellipse((0, 0), 2, 3) >>> c = e.center >>> e.normal_lines(c + Point(1, 0)) [Line(Point(0, 0), Point(1, 0))] >>> e.normal_lines(c) [Line(Point(0, 0), Point(0, 1)), Line(Point(0, 0), Point(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) [Line(Point(-38/47, -85/31), Point(9/47, -21/17)), Line(Point(19/13, -43/21), Point(32/13, -8/3))] Whereas the above solution has an operation count of 12, the exact solution has an operation count of 2020. """ p = Point(p) # 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 x, y = Dummy('x', real=True), Dummy('y', real=True) eq = self.equation(x, y) dydx = idiff(eq, y, x) norm = -1/dydx slope = Line(p, (x, y)).slope seq = slope - norm points = [] if prec is not None: yis = solve(seq, y)[0] xeq = eq.subs(y, yis).as_numer_denom()[0].expand() try: iv = list(zip(*Poly(xeq, x).intervals()))[0] # bisection is safest here since other methods may miss root xsol = [S(nroot(lambdify(x, xeq), i, solver="anderson")) for i in iv] points = [Point(i, solve(eq.subs(x, i), y)[0]).n(prec) for i in xsol] except (DomainError, PolynomialError): xvals = solve(xeq, x) points = [Point(xis, yis.xreplace({x: xis})) for xis in xvals] points = [pt.n(prec) if prec is not None else pt for pt in points] slopes = [norm.subs(zip((x, y), pt.args)) for pt in points] if prec is not None: slopes = [i.n(prec) if i not in (-oo, oo, zoo) else i for i in slopes] return [Line(pt, slope=s) for pt,s in zip(points, slopes)]
[docs] 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() Point(3*cos(t), 2*sin(t)) """ t = _symbol(parameter) 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*C.cos(t), self.center.y + self.vradius*C.sin(t))
[docs] 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) return [t, -S.Pi, S.Pi]
[docs] def random_point(self, seed=None): """A random point on the ellipse. Returns ======= point : Point See Also ======== sympy.geometry.point.Point arbitrary_point : Returns parameterized point on ellipse Notes ----- A random point may not appear to be on the ellipse, ie, `p in e` may return False. This is because the coordinates of the point will be floating point values, and when these values are substituted into the equation for the ellipse the result may not be zero because of floating point rounding error. Examples ======== >>> from sympy import Point, Ellipse, Segment >>> e1 = Ellipse(Point(0, 0), 3, 2) >>> e1.random_point() # gives some random point Point(...) >>> p1 = e1.random_point(seed=0); p1.n(2) Point(2.1, 1.4) The random_point method assures that the point will test as being in the ellipse: >>> p1 in e1 True Notes ===== An arbitrary_point with a random value of t substituted into it may not test as being on the ellipse because the expression tested that a point is on the ellipse doesn't simplify to zero and doesn't evaluate exactly to zero: >>> from sympy.abc import t >>> e1.arbitrary_point(t) Point(3*cos(t), 2*sin(t)) >>> p2 = _.subs(t, 0.1) >>> p2 in e1 False Note that arbitrary_point routine does not take this approach. A value for cos(t) and sin(t) (not t) is substituted into the arbitrary point. There is a small chance that this will give a point that will not test as being in the ellipse, so the process is repeated (up to 10 times) until a valid point is obtained. """ from sympy import sin, cos, Rational t = _symbol('t') 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 for i in range(10): # should be enough? # simplify this now or else the Float will turn s into a Float c = 2*Rational(rng.random()) - 1 s = sqrt(1 - c**2) p1 = Point(x.subs(cos(t), c), y.subs(sin(t), s)) if p1 in self: return p1 raise GeometryError( 'Having problems generating a point in the ellipse.')
[docs] def equation(self, x='x', y='y'): """The equation 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 See Also ======== arbitrary_point : Returns parameterized point on ellipse Examples ======== >>> from sympy import Point, Ellipse >>> e1 = Ellipse(Point(1, 0), 3, 2) >>> e1.equation() y**2/4 + (x/3 - 1/3)**2 - 1 """ x = _symbol(x) y = _symbol(y) t1 = ((x - self.center.x) / self.hradius)**2 t2 = ((y - self.center.y) / self.vradius)**2 return t1 + t2 - 1
def _do_line_intersection(self, o): """ Find the intersection of a LinearEntity and the ellipse. All LinearEntities are treated as a line and filtered at the end to see that they lie in o. """ hr_sq = self.hradius ** 2 vr_sq = self.vradius ** 2 lp = o.points ldir = lp[1] - lp[0] diff = lp[0] - self.center mdir = Point(ldir.x/hr_sq, ldir.y/vr_sq) mdiff = Point(diff.x/hr_sq, diff.y/vr_sq) a = ldir.dot(mdir) b = ldir.dot(mdiff) c = diff.dot(mdiff) - 1 det = simplify(b*b - a*c) result = [] if det == 0: t = -b / a result.append(lp[0] + (lp[1] - lp[0]) * t) # Definite and potential symbolic intersections are allowed. elif (det > 0) != False: root = sqrt(det) t_a = (-b - root) / a t_b = (-b + root) / a result.append( lp[0] + (lp[1] - lp[0]) * t_a ) result.append( lp[0] + (lp[1] - lp[0]) * t_b ) return [r for r in result if r in o] def _do_ellipse_intersection(self, o): """The intersection of an ellipse with another ellipse or a circle. Private helper method for `intersection`. """ x = Dummy('x', real=True) y = Dummy('y', real=True) seq = self.equation(x, y) oeq = o.equation(x, y) result = solve([seq, oeq], [x, y]) return [Point(*r) for r in list(uniq(result))]
[docs] 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, sqrt >>> e = Ellipse(Point(0, 0), 5, 7) >>> e.intersection(Point(0, 0)) [] >>> e.intersection(Point(5, 0)) [Point(5, 0)] >>> e.intersection(Line(Point(0,0), Point(0, 1))) [Point(0, -7), Point(0, 7)] >>> e.intersection(Line(Point(5,0), Point(5, 1))) [Point(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)) [Point(0, -3*sqrt(15)/4), Point(0, 3*sqrt(15)/4)] >>> e.intersection(Ellipse(Point(5, 0), 4, 3)) [Point(2, -3*sqrt(7)/4), Point(2, 3*sqrt(7)/4)] >>> e.intersection(Ellipse(Point(100500, 0), 4, 3)) [] >>> e.intersection(Ellipse(Point(0, 0), 3, 4)) [Point(-363/175, -48*sqrt(111)/175), Point(-363/175, 48*sqrt(111)/175), Point(3, 0)] >>> e.intersection(Ellipse(Point(-1, 0), 3, 4)) [Point(-17/5, -12/5), Point(-17/5, 12/5), Point(7/5, -12/5), Point(7/5, 12/5)] """ if isinstance(o, Point): if o in self: return [o] else: return [] elif isinstance(o, LinearEntity): # LinearEntity may be a ray/segment, so check the points # of intersection for coincidence first return self._do_line_intersection(o) elif isinstance(o, Circle): return self._do_ellipse_intersection(o) elif isinstance(o, Ellipse): if o == self: return self else: return self._do_ellipse_intersection(o) return o.intersection(self)
[docs] 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) y = _symbol(y) 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)
def __eq__(self, o): """Is the other GeometryEntity the same as this ellipse?""" return isinstance(o, GeometryEntity) and (self.center == o.center and self.hradius == o.hradius and self.vradius == o.vradius) def __hash__(self): return super(Ellipse, self).__hash__() def __contains__(self, o): if isinstance(o, Point): x = C.Dummy('x', real=True) y = C.Dummy('y', real=True) 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
[docs]class Circle(Ellipse): """A circle in space. Constructed simply from a center and a radius, or from three non-collinear points. Parameters ========== center : Point radius : number or sympy expression points : sequence of three Points Attributes ========== radius (synonymous with hradius, vradius, major and minor) circumference equation Raises ====== GeometryError When trying to construct circle from three collinear points. When trying to construct circle from incorrect parameters. See Also ======== Ellipse, sympy.geometry.point.Point Examples ======== >>> from sympy.geometry import Point, Circle >>> # 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 costructed 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, Point(1/2, 1/2)) """ def __new__(cls, *args, **kwargs): c, r = None, None if len(args) == 3: args = [Point(a) for a in args] if Point.is_collinear(*args): raise GeometryError( "Cannot construct a circle from three collinear points") from .polygon import Triangle t = Triangle(*args) c = t.circumcenter r = t.circumradius elif len(args) == 2: # Assume (center, radius) pair c = Point(args[0]) r = sympify(args[1]) if not (c is None or r is None): return GeometryEntity.__new__(cls, c, r, **kwargs) raise GeometryError("Circle.__new__ received unknown arguments") @property
[docs] 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]
@property
[docs] 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)
@property
[docs] 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
[docs] 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) y = _symbol(y) t1 = (x - self.center.x)**2 t2 = (y - self.center.y)**2 return t1 + t2 - self.major**2
[docs] 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) [Point(5, 0)] >>> c1.intersection(Ray(p1, p2)) [Point(5*sqrt(2)/2, 5*sqrt(2)/2)] >>> c1.intersection(Line(p2, p3)) [] """ if isinstance(o, Circle): if o.center == self.center: if o.radius == self.radius: return o return [] dx, dy = (o.center - self.center).args d = sqrt(simplify(dy**2 + dx**2)) R = o.radius + self.radius if d > R or d < abs(self.radius - o.radius): return [] a = simplify((self.radius**2 - o.radius**2 + d**2) / (2*d)) x2 = self.center.x + (dx * a/d) y2 = self.center.y + (dy * a/d) h = sqrt(simplify(self.radius**2 - a**2)) rx = -dy * (h/d) ry = dx * (h/d) xi_1 = simplify(x2 + rx) xi_2 = simplify(x2 - rx) yi_1 = simplify(y2 + ry) yi_2 = simplify(y2 - ry) ret = [Point(xi_1, yi_1)] if xi_1 != xi_2 or yi_1 != yi_2: ret.append(Point(xi_2, yi_2)) return ret return Ellipse.intersection(self, o)
[docs] 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(Point(0, 0), 2) >>> Circle((0, 0), 1).scale(2, 4) Ellipse(Point(0, 0), 2, 4) """ c = self.center if pt: pt = Point(pt) 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)
[docs] 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(Point(1, 0), -1) """ c = self.center c = c.reflect(line) return self.func(c, -self.radius)
from .polygon import Polygon