+ i?^RIHt^RIHt^RIHt^RIHt^RIH t ^RI H t ^RI H t ^RIHtHtHtHtHtHt^R IHt^R IHt^R IHtHtHt^R IHt^R IHt^RI H!t!!RR]4t"Rt#R#))Rational)S)symbols)sign)sqrt)gcd) Complement)BasicTuplediffexpandEqInteger)ordered)_symbol)solveset nonlinsolve diophantine total_degree)Point)corecaa]tRt^toRtV3Rlt]R4t]R4t]R4t Rt Rt Rt R t R tR R ltR tVtV;t#)ImplicitRegiona- Represents an implicit region in space. Examples ======== >>> from sympy import Eq >>> from sympy.abc import x, y, z, t >>> from sympy.vector import ImplicitRegion >>> ImplicitRegion((x, y), x**2 + y**2 - 4) ImplicitRegion((x, y), x**2 + y**2 - 4) >>> ImplicitRegion((x, y), Eq(y*x, 1)) ImplicitRegion((x, y), x*y - 1) >>> parabola = ImplicitRegion((x, y), y**2 - 4*x) >>> parabola.degree 2 >>> parabola.equation -4*x + y**2 >>> parabola.rational_parametrization(t) (4/t**2, 4/t) >>> r = ImplicitRegion((x, y, z), Eq(z, x**2 + y**2)) >>> r.variables (x, y, z) >>> r.singular_points() EmptySet >>> r.regular_point() (-10, -10, 200) Parameters ========== variables : tuple to map variables in implicit equation to base scalars. equation : An expression or Eq denoting the implicit equation of the region. c<\V\4'g \V!p\V\4'dVPVP, p\ SV`WV4#N) isinstancer rlhsrhssuper__new__)cls variablesequation __class__s&&&Y/var/www/html/photoedit/myenv/lib/python3.14/site-packages/sympy/vector/implicitregion.pyr!ImplicitRegion.__new__9sJ)U++y)I h # #||hll2Hwsx88c(VP^,#)rargsselfs&r&r#ImplicitRegion.variablesByy|r(c(VP^,#)r*r,s&r&r$ImplicitRegion.equationFr/r(c,\VP4#r)rr$r,s&r&degreeImplicitRegion.degreeJsDMM**r(c VPp\VP4^8Xd>\\ WP^,\ P R74^,3#\VP4^8XdtVP^8Xdc\VPV4;pwr4rVrxV^,^V,V,8XdVP!V!wrW3#VP!V!wrW3#\VP4^8XdVPwrp \R^ 4Fp \R^ 4Fp \ VPWW/4VP^,\ P R7P'dKSW\\ VPWW/444^,3uu# K \VP44^8wd#\VP4,^,#\4h)a Returns a point on the implicit region. Examples ======== >>> from sympy.abc import x, y, z >>> from sympy.vector import ImplicitRegion >>> circle = ImplicitRegion((x, y), (x + 2)**2 + (y - 3)**2 - 16) >>> circle.regular_point() (-2, -1) >>> parabola = ImplicitRegion((x, y), x**2 - 4*y) >>> parabola.regular_point() (0, 0) >>> r = ImplicitRegion((x, y, z), (x + y + z)**4) >>> r.regular_point() (-10, -10, 20) References ========== - Erik Hillgarter, "Rational Points on Conics", Diploma Thesis, RISC-Linz, J. Kepler Universitat Linz, 1996. Available: https://www3.risc.jku.at/publications/download/risc_1355/Rational%20Points%20on%20Conics.pdf )domaini)r$lenr#listrrRealsr4 conic_coeff_regular_point_parabola_regular_point_ellipserangesubsis_emptysingular_pointsNotImplementedError)r-r$coeffsabcdefx_regy_regxyzs& r& regular_pointImplicitRegion.regular_pointNs6== t~~ ! #(NN1,=aggNOPQRT T  A %{{a,7,QQ)qQa41Q3q5=#'#?#?#HLE|#$(#>#>#GLE|# t~~ ! #nnGA!sB"3^E#HMM1Q2F$GXYIZcdcjcjkttt %d8HMM1UVJ^<_3`.abc.dee,( t##% &! +,,./2 2!##r(cW3R8g;'d7W53R8g;'d)V^,^V,V,8H;'dW3R8gpV'g \R4hV^8wdv^V,V,^V,V,, ^V,V,V^,, rV^8wd*V )V, p WBV ,,)^V,, p M~RpM{V^8wdu^V,V,^V,V,, ^V,V,V^,, rV^8wd*V )V, p WRV ,,)^V,, p MRpV'dX X 3#\R4h)r*Rational Point on the conic does not existF)rr) ValueError) r-rDrErFrGrHrIokd_dashf_dashrKrJs &&&&&&& r&r<&ImplicitRegion._regular_point_parabolas+6!]]qf&6]]1a41Q3q5=]]aVW]M]B !MNNAv"#A#a%!A#a%-1QAQ;#GFNEE'kNAaC0EBa"#A#a%!A#a%-1QAQ;#GFNEE'kNAaC0EBe|# !MNNr(c a.^V,V,V^,, pTpV'g \R4hV^8Xd(V^8Xd!Rp ^WE,W&,, ,p EMV^8wdTp ^V^,,V^,,^V,V,V,V,, ^V,V,V^,,,^V^,,V,V,,^V,V^,,V,, p MfTp ^V^,,V^,,^V,V,V,V,, ^V^,,V,V,,p V ^8g;'dV ^8;'dV ^8'*pV'g \R4h\V 4PR4p \V 4PR4p V PV PrV PV Pr\ W4pW,V, pW,V, pW,)V, p\ V4\\V4^4,p\VV, 4p\ V4\\V4^4,p\VV, 4p\ V4\\V4^4,p\VV, 4p\ \ VV4V4pVV, pVV, pVV, p\ VV4pVV, pVV, pVV,p\ VV4pVV, pVV,pVV, p\ VV4pVV,pVV, pVV, p\R4wpppVV^,,VV^,,,VV^,,,p\V4p \V 4^8Xd \R4hRp!V EFp"\V"!Pp#\P!V#^4o.V"^,p$V$^8XdRp!KA\#V$\$\&34'dK_V$Pp%\V%4^8Xdi\)\+V%44p&\-\.P0\3\5V$^4V&\.P044p'\)\+V'44S.V&&\V%4^8Xd\7\9V%44wp&p(\.P0Fp)V$P;V&V)4p*\-\.P0\3\5V*^4V(\.P044p+V+P<'dKfV)S.V&&\)\+V+44S.V(&M \V#4^8wd8\>;QJd.V.3RlV"4FNK 5M!V.3RlV"44wpppMV"wpppRp!M V!'d \R4hVV,V, pVV,V, pVV,V, pVV, pVV, pV^8XdVV^8XdOVV,^V,, ^V,, p,VV, ^V,, ^V,, p-V,V-3#V^8wdUV^V,V,, W%,,V , p,VVV,,, V, ^V,, p-V,V-3#V^V,V,, W$,,V , p-VVV-,, V, ^V,, p,V,V-3#)rRzx y zFTc3D<"TFqPS4xK R#5irr?.0sreps& r& 8ImplicitRegion._regular_point_ellipse..s'ASs S lJ)) rSrlimit_denominatorpqrrrabsrrrr8r free_symbolsdictfromkeysrintrnextiterr rIntegersrrr9rr?r@tuple)/r-rDrErFrGrHrIDrTKLk1k2l1l2ga1b1c1a2r1b2r2c2r3g1g2g3rLrMrNeq solutionsflagsolsymssol_zsyms_zrep_valuesrfi subs_sol_zq_valuesrJrKr_s/&&&&&&& @r&r=%ImplicitRegion._regular_point_ellipses!A1 AB !MNNAv!q&qsQSyMaadF1a4K!A#a%'!)+ac!eAqDj81QT6!8A:E1QPQT RS SadF1a4K!A#a%'!)+a1fQhqj8a00A!a%0B !MNN --f5A --f5ASS!##SS!##B A%B%B5!Bb$s2w**BbeBb$s2w**BbeBb$s2w**BbeBCBK$AABABABRBBBBBBBRBBBBBBBRBBBBBBBg&GAq!AqD2ad7"R1W,B#BI9~" !MNND c{//mmD!,AA:D!%#w88"//F6{a' f.#-ajj(2eQh'iH#+#4#4#4)*A)-d8n)=A %",4yA~"'%'AS'A%%'AS'A"A1a$'1a DE!H !MNN2r A2r A2r A!A!AAv!q&Q1qs+Q1qs+%< aQqSUQS!+QuWq1Q3/ %< QqSUQS!+QuWq1Q3/%< r(c VP.pVPF!pV\VPV4., pK# \V\ VP44#)aG Returns a set of singular points of the region. The singular points are those points on the region where all partial derivatives vanish. Examples ======== >>> from sympy.abc import x, y >>> from sympy.vector import ImplicitRegion >>> I = ImplicitRegion((x, y), (y-1)**2 -x**3 + 2*x**2 -x) >>> I.singular_points() {(1, 1)} )r$r#r rr9)r-eq_listvars& r&rAImplicitRegion.singular_pointssM"==/>>C T]]C01 1G"7D$899r(c ~\V\4'd VPpVPp\ VP 4F#wr4VP WDW,,4pK% \V4p\VP4^8wd!VPp\RV44pV#Tp\V4pV#)a Returns the multiplicity of a singular point on the region. A singular point (x,y) of region is said to be of multiplicity m if all the partial derivatives off to order m - 1 vanish there. Examples ======== >>> from sympy.abc import x, y, z >>> from sympy.vector import ImplicitRegion >>> I = ImplicitRegion((x, y, z), x**2 + y**3 - z**4) >>> I.singular_points() {(0, 0, 0)} >>> I.multiplicity((0, 0, 0)) 2 c38"TFp\V4xK R#5irr)r]terms& r&r`.ImplicitRegion.multiplicity..Ps954L&&5s) rrr+r$ enumerater#r?r r8minr)r-point modified_eqrrtermsms&& r& multiplicityImplicitRegion.multiplicity2s& eU # #JJEmm /FA%**3eh?K0[) { A %$$E9599A  EU#Ar(c 4 aVPpVPpV^8Xdn\VP4^8XdV3#\VP4^8Xd-VPwrV\ \ W644^,pWW3#\ 4hRpV^8XdWVeTpMO\VP44^8wd"\ VP44^,pMVP4p\VP44^8wdVP4p V Fp \V !Pp \PV ^4o\V 4^8wd3\;QJd.V3RlV 4FNK 5M!V3RlV 44p VPV 4V^, 8XgKT pM \V4^8Xd \ 4hTp \VP4F#wrV P!WW,,4p K% \#V 4p ^;ppV P$F'p\'V4V8Xd VV, pKVV, pK) R V,p\)V\4'gV3p\VP4^8XEdV^,pVR8Xd\+RRR7pM \+RRR7p\+VRR7pVP!VP^,VVP^,V/4pVP!VP^,VVP^,V/4pVVV, ,P!V^4V^,,pVVV, ,P!V^4V^,,pVV3#\VP4^8XEdeVwppRV9d\+RRR7pM \+RRR7p\+VRR7p\+VRR7pVP!VP^,VVP^,VVP^,V/4pVP!VP^,VVP^,VVP^,V/4pVVV, ,P!V^4V^,,pVVV, ,P!V^4V^,,pVVV, ,P!V^4V^,,pVVV3#\ 4h) a Returns the rational parametrization of implicit region. Examples ======== >>> from sympy import Eq >>> from sympy.abc import x, y, z, s, t >>> from sympy.vector import ImplicitRegion >>> parabola = ImplicitRegion((x, y), y**2 - 4*x) >>> parabola.rational_parametrization() (4/t**2, 4/t) >>> circle = ImplicitRegion((x, y), Eq(x**2 + y**2, 4)) >>> circle.rational_parametrization() (4*t/(t**2 + 1), 4*t**2/(t**2 + 1) - 2) >>> I = ImplicitRegion((x, y), x**3 + x**2 - y**2) >>> I.rational_parametrization() (t**2 - 1, t*(t**2 - 1)) >>> cubic_curve = ImplicitRegion((x, y), x**3 + x**2 - y**2) >>> cubic_curve.rational_parametrization(parameters=(t)) (t**2 - 1, t*(t**2 - 1)) >>> sphere = ImplicitRegion((x, y, z), x**2 + y**2 + z**2 - 4) >>> sphere.rational_parametrization(parameters=(t, s)) (-2 + 4/(s**2 + t**2 + 1), 4*s/(s**2 + t**2 + 1), 4*t/(s**2 + t**2 + 1)) For some conics, regular_points() is unable to find a point on curve. To calulcate the parametric representation in such cases, user need to determine a point on the region and pass it using reg_point. >>> c = ImplicitRegion((x, y), (x - 1/2)**2 + (y)**2 - (1/4)**2) >>> c.rational_parametrization(reg_point=(3/4, 0)) (0.75 - 0.5/(t**2 + 1), -0.5*t/(t**2 + 1)) References ========== - Christoph M. Hoffmann, "Conversion Methods between Parametric and Implicit Curves and Surfaces", Purdue e-Pubs, 1990. Available: https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1827&context=cstech c3D<"TFqPS4xK R#5irr[r\s& r&r`:ImplicitRegion.rational_parametrization..s"?166#;;rbr^s_T)realrr_rc)r$r4r8r#r9rrBrArOr rhrirjrorrr?r r+rrr)r- parameters reg_pointr$r4rLrMy_parrrAspointrrrrhnhn_1r parameter1r^tx_par parameter2rz_parr_s&&& @r&rational_parametrization'ImplicitRegion.rational_parametrizationWs5^== Q;4>>"a' {"T^^$)~~Xh23A6x)++ Q;$!t++-.!3 !5!5!78;E ..0E t##% &! +"224O)f~22mmD!,t9>"U"?"?UU"?"??F$$V, :"E* u:?%' '  /FA%**3eh?K0[)  T$$DD!V+d   % $w*e,,$J t~~ ! ##AJS Dt,Cd+ .A$..+Qq0A1EFB99dnnQ/DNN14EqIJDR[&&q!,uQx7ER[&&q!,uQx7E%<   A %%/ "J j Dt,Cd+ .A .A$..+Qq0A1dnnUVFWYZ[\B99dnnQ/DNN14Eq$..YZJ[]^_`DR[&&q!,uQx7ER[&&q!,uQx7ER[&&q!,uQx7E%& &!##r(r))rr^N)__name__ __module__ __qualname____firstlineno____doc__r!propertyr#r$r4rOr<r=rArr__static_attributes____classdictcell__ __classcell__)r% __classdict__s@@r&rrsv&N9++5$nO4z x:.#JT$T$r(rc\V4^8wd \4hV^,pV^,p\V4pVPV^,4pVPW#,4pVPV^,4pVPV^4PV^4pVPV^4PV^4pVPV^4PV^4p WEWgW3#))rrSr coeff) r#r$rLrMrDrErFrGrHrIs && r&r;r;sH"l! 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