+ i)Rt^RIHt^RIHt^RIHtHt^RIH t H t ^RI H t H t ^RIHtHtHtHtHt^RIHt^RIHt^R IHt^R IHt!R R ] 4tR #)z4Parabolic geometrical entity. Contains * Parabola )S)ordered)_symbolsymbols)GeometryEntity GeometrySet)PointPoint2D)LineLine2DRay2D Segment2DLinearEntity3D)Ellipse)sign)simplify)solveca]tRt^toRtRRlt]R4t]R4t]R4t ]R4t RRlt ]R 4t ]R 4t R t]R 4t]R 4tRtVtR#)ParabolaaA parabolic GeometryEntity. A parabola is declared with a point, that is called 'focus', and a line, that is called 'directrix'. Only vertical or horizontal parabolas are currently supported. Parameters ========== focus : Point Default value is Point(0, 0) directrix : Line Attributes ========== focus directrix axis of symmetry focal length p parameter vertex eccentricity Raises ====== ValueError When `focus` is not a two dimensional point. When `focus` is a point of directrix. NotImplementedError When `directrix` is neither horizontal nor vertical. Examples ======== >>> from sympy import Parabola, Point, Line >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7,8))) >>> p1.focus Point2D(0, 0) >>> p1.directrix Line2D(Point2D(5, 8), Point2D(7, 8)) Nc V'd\V^R7pM \^^4p\V4pVPV4'd \R4h\P !WV3/VB#))dimz*The focus must not be a point of directrix)rr contains ValueErrorr__new__)clsfocus directrixkwargss&&&,w/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/geometry/parabola.pyrParabola.__new__AsZ %Q'E!QKEO   e $ $IJ J%%c)FvFFc ^#)aReturns the ambient dimension of parabola. Returns ======= ambient_dimension : integer Examples ======== >>> from sympy import Parabola, Point, Line >>> f1 = Point(0, 0) >>> p1 = Parabola(f1, Line(Point(5, 8), Point(7, 8))) >>> p1.ambient_dimension 2 selfs&rambient_dimensionParabola.ambient_dimensionOs&r!c LVPPVP4#)aReturn the axis of symmetry of the parabola: a line perpendicular to the directrix passing through the focus. Returns ======= axis_of_symmetry : Line See Also ======== sympy.geometry.line.Line Examples ======== >>> from sympy import Parabola, Point, Line >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8))) >>> p1.axis_of_symmetry Line2D(Point2D(0, 0), Point2D(0, 1)) )rperpendicular_linerr$s&raxis_of_symmetryParabola.axis_of_symmetryds0~~00<>> from sympy import Parabola, Point, Line >>> l1 = Line(Point(5, 8), Point(7, 8)) >>> p1 = Parabola(Point(0, 0), l1) >>> p1.directrix Line2D(Point2D(5, 8), Point2D(7, 8)) argsr$s&rrParabola.directrix~0yy|r!c "\P#)a7The eccentricity of the parabola. Returns ======= eccentricity : number A parabola may also be characterized as a conic section with an eccentricity of 1. As a consequence of this, all parabolas are similar, meaning that while they can be different sizes, they are all the same shape. See Also ======== https://en.wikipedia.org/wiki/Parabola Examples ======== >>> from sympy import Parabola, Point, Line >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8))) >>> p1.eccentricity 1 Notes ----- The eccentricity for every Parabola is 1 by definition. )rOner$s&r eccentricityParabola.eccentricitysBuu r!c @\VRR7p\VRR7pVPPpV\PJd`^VP ,WP P, ,pW P P, ^,pWE, #V^8Xd`^VP ,W P P, ,pWP P, ^,pWE, #VPwrgVPPR,wrW, ^,W', ^,,pVPPW4^,V^,V ^,,, pWE, #)aThe equation of the parabola. 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 Parabola, Point, Line >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8))) >>> p1.equation() -x**2 - 16*y + 64 >>> p1.equation('f') -f**2 - 16*y + 64 >>> p1.equation(y='z') -x**2 - 16*z + 64 Treal:NrN) rrsloperInfinity p_parametervertexxyr coefficientsequation) r%r<r=mt1t2abcds &&& rr?Parabola.equations%6 AD ! AD ! NN   ?d&&'1{{}}+<=Bkkmm#a'Bw!Vd&&'1{{}}+<=Bkkmm#a'B w ::DA>>..r2DA%!quqj(B((.11a4!Q$;?Bwr!c bVPPVP4pV^, pV#)aThe focal length of the parabola. Returns ======= focal_lenght : number or symbolic expression Notes ===== The distance between the vertex and the focus (or the vertex and directrix), measured along the axis of symmetry, is the "focal length". See Also ======== https://en.wikipedia.org/wiki/Parabola Examples ======== >>> from sympy import Parabola, Point, Line >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8))) >>> p1.focal_length 4 )rdistancer)r%rI focal_lengths& rrJParabola.focal_lengths+<>>**4::6z r!c (VP^,#)aThe focus of the parabola. Returns ======= focus : Point See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Parabola, Point, Line >>> f1 = Point(0, 0) >>> p1 = Parabola(f1, Line(Point(5, 8), Point(7, 8))) >>> p1.focus Point2D(0, 0) r-r$s&rrParabola.focus r0r!c  r\RRR7wr#VP4p\V\4'dZW9dV.#\ \ \ WAP4.W#.RR7^,Uu.uFp\V4NK up44#\V\4'dK\VPW!P^,3W1P^,3.44^8XdV.#.#\V\\34'd\ V\VP^,VP^,4P4.W#.RR7^,p\ \ VUu.uFqUV9gK \V4NK up44#\V\\ 34'dQ\ \ \ WAP4.W#.RR7^,Uu.uFp\V4NK up44#\V\"4'd \%R4h\%R4huupiuupiuupi)abThe intersection of the parabola and another geometrical entity `o`. Parameters ========== o : GeometryEntity, LinearEntity Returns ======= intersection : list of GeometryEntity objects Examples ======== >>> from sympy import Parabola, Point, Ellipse, Line, Segment >>> p1 = Point(0,0) >>> l1 = Line(Point(1, -2), Point(-1,-2)) >>> parabola1 = Parabola(p1, l1) >>> parabola1.intersection(Ellipse(Point(0, 0), 2, 5)) [Point2D(-2, 0), Point2D(2, 0)] >>> parabola1.intersection(Line(Point(-7, 3), Point(12, 3))) [Point2D(-4, 3), Point2D(4, 3)] >>> parabola1.intersection(Segment((-12, -65), (14, -68))) [] zx yTr6)setz5Entity must be two dimensional, not three dimensionalzWrong type of argument were put)rr? isinstancerlistrrrr rsubs_argsr r r pointsrr TypeError)r%or<r= parabola_eqiresults&& r intersectionParabola.intersection$s8u4(mmo a " "ys Gu **,/!T8CCD8F%G8F!U1X8F%GHII 7 # # ((1ggaj/Awwqz?)KLMQRRs  Iu- . .Kqxx{AHHQK099;=D""#%FV FVAvV FGH H FG, - -Ujjl+aV6??@6B!C6B6B!CDE E > * *ST T=> >%%G!G!Cs0H* +H/ 8H/ H4 c ^VPPpV\PJdLVPP^,p\ VP P^,V,4pMV^8XdLVPP^,p\ VP P^,V,4pMUVPPVP 4p\ VP PVP, 4pW0P,#)aeP is a parameter of parabola. Returns ======= p : number or symbolic expression Notes ===== The absolute value of p is the focal length. The sign on p tells which way the parabola faces. Vertical parabolas that open up and horizontal that open right, give a positive value for p. Vertical parabolas that open down and horizontal that open left, give a negative value for p. See Also ======== https://www.sparknotes.com/math/precalc/conicsections/section2/ Examples ======== >>> from sympy import Parabola, Point, Line >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8))) >>> p1.p_parameter -4 ) rr8rr9r>rrr. projectionr<rJ)r%r@r<pr=rFs& rr:Parabola.p_parameterZsB NN   ?++A.ATZZ__Q'!+,A !V++A.ATZZ__Q'!+,A))$**5ATZZ\\ACC'(A$$$$r!c VPpVPPpV\PJdB\ VP ^,VP, VP ^,4pV#V^8XdB\ VP ^,VP ^,VP, 4pV#VPPV4^,pV#)aThe vertex of the parabola. Returns ======= vertex : Point See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Parabola, Point, Line >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8))) >>> p1.vertex Point2D(0, 4) ) rrr8rr9rr.r:r*rZ)r%rr@r;s& rr;Parabola.vertexs.  NN   ?5::a=4+;+;;UZZ]KF  !V5::a=%**Q-$:J:J*JKF **77=a@F r!r#)NN)r<r=)__name__ __module__ __qualname____firstlineno____doc__rpropertyr&r*rr3r?rJrrZr:r;__static_attributes____classdictcell__) __classdict__s@rrrs*X G(==22  D*X  D24?l*%*%Xr!rN)rf sympy.corersympy.core.sortingrsympy.core.symbolrrsympy.geometry.entityrrsympy.geometry.pointrr sympy.geometry.liner r r r rsympy.geometry.ellipsersympy.functionsrsympy.simplify.simplifyrsympy.solvers.solversrrr#r!rrus;&.=/NN* ,'R{Rr!