\h)SrSSKJr SSKJr SSKJrJr SSKJ r J r SSK J r J r SSKJrJrJrJrJr SSKJr SS KJr SS KJr SS KJr "S S \ 5rg)z4Parabolic geometrical entity. Contains * Parabola )S)ordered)_symbolsymbols)GeometryEntity GeometrySet)PointPoint2D)LineLine2DRay2D Segment2DLinearEntity3D)Ellipse)sign)simplify)solvec\rSrSrSrSSjr\S5r\S5r\S5r \S5r SS jr \S 5r \S 5r S r\S 5r\S5rSrg)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 U(a [USS9nO [SS5n[U5nURU5(a [S5e[R "XU40UD6$)N)dimrz*The focus must not be a point of directrix)r r contains ValueErrorr__new__)clsfocus directrixkwargss O/var/www/auris/envauris/lib/python3.13/site-packages/sympy/geometry/parabola.pyrParabola.__new__AsZ %Q'E!QKEO   e $ $IJ J%%c)FvFFcg)aReturns the ambient dimension of parabola. Returns ======= ambient_dimension : integer Examples ======== >>> from sympy import Parabola, Point, Line >>> f1 = Point(0, 0) >>> p1 = Parabola(f1, Line(Point(5, 8), Point(7, 8))) >>> p1.ambient_dimension 2 rselfs r!ambient_dimensionParabola.ambient_dimensionOs&r#cLURRUR5$)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 r!axis_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 r!rParabola.directrix~0yy|r#c"[R$)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#cp[USS9n[USS9nURRnU[RLaGSUR -XR R- -nX R R- S-nXE- $US:XaGSUR -X R R- -nXR R- S-nXE- $URupgURRSSupX- S-X'- S--nURRX5S-US-U S--- nXE- $)aThe equation of the parabola. Parameters ========== x : str, optional Label for the x-axis. Default value is 'x'. y : str, optional Label for the y-axis. Default value is 'y'. Returns ======= equation : SymPy expression Examples ======== >>> from sympy import Parabola, Point, Line >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8))) >>> p1.equation() -x**2 - 16*y + 64 >>> p1.equation('f') -f**2 - 16*y + 64 >>> p1.equation(y='z') -x**2 - 16*z + 64 TrealrrN) rrsloperInfinity p_parametervertexxyr coefficientsequation) r'r@rAmt1t2abcds r!rCParabola.equations'6 AD ! AD ! NN   ?d&&'1{{}}+<=Bkkmm#a'Bw!Vd&&'1{{}}+<=Bkkmm#a'B w ::DA>>..r2DA%!quqj(B((.11a4!Q$;?Bwr#cZURRUR5nUS- nU$)aThe focal length of the parabola. Returns ======= focal_lenght : number or symbolic expression Notes ===== The distance between the vertex and the focus (or the vertex and directrix), measured along the axis of symmetry, is the "focal length". See Also ======== https://en.wikipedia.org/wiki/Parabola Examples ======== >>> from sympy import Parabola, Point, Line >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8))) >>> p1.focal_length 4 r)rdistancer)r'rM focal_lengths r!rNParabola.focal_lengths+<>>**4::6z r#c URS$)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) rr0r&s r!rParabola.focus r3r#c [SSS9up#UR5n[U[5(aRX;aU/$[ [ [ XAR5/X#/SS9SVs/sHn[U5PM sn55$[U[5(aA[URX!RS4X1RS4/55S:XaU/$/$[U[[45(ax[ U[URSURS5R5/X#/SS9Sn[ [ UVs/sHoUU;dM [U5PM sn55$[U[[ 45(aJ[ [ [ XAR5/X#/SS9SVs/sHn[U5PM sn55$[U["5(a [%S5e[%S5es snfs snfs snf) 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 yTr9)setr/rz5Entity must be two dimensional, not three dimensionalzWrong type of argument were put)rrC isinstancerlistrrr r rsubs_argsrr r pointsrr TypeError)r'or@rA parabola_eqiresults r! intersectionParabola.intersection$s8u4(mmo a " "ys Gu **,/!T8CCD8F%G8F!U1X8F%GHII 7 # # ((1ggaj/Awwqz?)KLMQRRs  Iu- . .Kqxx{AHHQK099;=D""#%FV FVAvV FGH H FG, - -Ujjl+aV6??@6B!C6B6B!CDE E > * *ST T=> >%%G!G!Cs$G=  H H 2H cURRnU[RLa?URRSn[ UR RSU-5nOUS:Xa?URRSn[ UR RSU-5nOQURRUR 5n[ UR RUR- 5nX0R-$)aeP is a parameter of parabola. Returns ======= p : number or symbolic expression Notes ===== The absolute value of p is the focal length. The sign on p tells which way the parabola faces. Vertical parabolas that open up and horizontal that open right, give a positive value for p. Vertical parabolas that open down and horizontal that open left, give a negative value for p. See Also ======== https://www.sparknotes.com/math/precalc/conicsections/section2/ Examples ======== >>> from sympy import Parabola, Point, Line >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8))) >>> p1.p_parameter -4 rrr/) rr<rr=rBrrr1 projectionr@rN)r'rDr@prArJs r!r>Parabola.p_parameterZsB NN   ?++A.ATZZ__Q'!+,A !V++A.ATZZ__Q'!+,A))$**5ATZZ\\ACC'(A$$$$r#cURnURRnU[RLa5[ UR SUR- UR S5nU$US:Xa5[ UR SUR SUR- 5nU$URRU5SnU$)aThe vertex of the parabola. Returns ======= vertex : Point See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Parabola, Point, Line >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8))) >>> p1.vertex Point2D(0, 4) rr/) rrr<rr=r r1r>r,r^)r'rrDr?s r!r?Parabola.vertexs.  NN   ?5::a=4+;+;;UZZ]KF  !V5::a=%**Q-$:J:J*JKF **77=a@F r#r%)NN)r@rA)__name__ __module__ __qualname____firstlineno____doc__rpropertyr(r,rr6rCrNrr^r>r?__static_attributes__r%r#r!rrs*X G(==22  D*X  D24?l*%*%Xr#rN)rj sympy.corersympy.core.sortingrsympy.core.symbolrrsympy.geometry.entityrrsympy.geometry.pointr r sympy.geometry.liner r r rrsympy.geometry.ellipsersympy.functionsrsympy.simplify.simplifyrsympy.solvers.solversrrr%r#r!rws;&.=/NN* ,'R{Rr#