\h?SSKJr SSKJr SSKJr SSKJr SSKJ r SSK J r SSK J r SSKJrJrJrJrJrJr SS KJr SS KJr SS KJrJrJr SS KJr SS KJr SSK J!r! "SS\5r"Sr#g))Rational)S)symbols)sign)sqrt)gcd) Complement)BasicTuplediffexpandEqInteger)ordered)_symbol)solveset nonlinsolve diophantine total_degree)Point)corec^\rSrSrSrU4Sjr\S5r\S5r\S5r Sr Sr S r S r S rSS jrS rU=r$)ImplicitRegiona- Represents an implicit region in space. Examples ======== >>> from sympy import Eq >>> from sympy.abc import x, y, z, t >>> from sympy.vector import ImplicitRegion >>> ImplicitRegion((x, y), x**2 + y**2 - 4) ImplicitRegion((x, y), x**2 + y**2 - 4) >>> ImplicitRegion((x, y), Eq(y*x, 1)) ImplicitRegion((x, y), x*y - 1) >>> parabola = ImplicitRegion((x, y), y**2 - 4*x) >>> parabola.degree 2 >>> parabola.equation -4*x + y**2 >>> parabola.rational_parametrization(t) (4/t**2, 4/t) >>> r = ImplicitRegion((x, y, z), Eq(z, x**2 + y**2)) >>> r.variables (x, y, z) >>> r.singular_points() EmptySet >>> r.regular_point() (-10, -10, 200) Parameters ========== variables : tuple to map variables in implicit equation to base scalars. equation : An expression or Eq denoting the implicit equation of the region. c>[U[5(d[U6n[U[5(aURUR- n[ TU]XU5$N) isinstancer rlhsrhssuper__new__)cls variablesequation __class__s S/var/www/auris/envauris/lib/python3.13/site-packages/sympy/vector/implicitregion.pyr"ImplicitRegion.__new__9sJ)U++y)I h # #||hll2Hwsx88c URS$)Nrargsselfs r'r$ImplicitRegion.variablesByy|r)c URS$)Nr+r-s r'r%ImplicitRegion.equationFr0r)c,[UR5$r)rr%r-s r'degreeImplicitRegion.degreeJsDMM**r)cURn[UR5S:Xa2[[ XRS[ R S95S4$[UR5S:XacURS:XaS[URU5=nup4pVpxUS-SU-U-:XaUR"U6upX4$UR"U6upX4$[UR5S:XaURupn [SS5Hn [SS5Hwn [ URXX05URS[ R S9R(aMKX[[ URXX0555S4s s $ M [UR55S:wa[UR5S$[5e) a Returns a point on the implicit region. Examples ======== >>> from sympy.abc import x, y, z >>> from sympy.vector import ImplicitRegion >>> circle = ImplicitRegion((x, y), (x + 2)**2 + (y - 3)**2 - 16) >>> circle.regular_point() (-2, -1) >>> parabola = ImplicitRegion((x, y), x**2 - 4*y) >>> parabola.regular_point() (0, 0) >>> r = ImplicitRegion((x, y, z), (x + y + z)**4) >>> r.regular_point() (-10, -10, 20) References ========== - Erik Hillgarter, "Rational Points on Conics", Diploma Thesis, RISC-Linz, J. Kepler Universitat Linz, 1996. Available: https://www3.risc.jku.at/publications/download/risc_1355/Rational%20Points%20on%20Conics.pdf r2r)domaini )r%lenr$listrrRealsr5 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)cX4S:g=(a( X54S:g=(a US-SU-U-:H=(a X4S:gnU(d [S5eUS:wa<SU-U-SU-U-- SU-U-US-- pUS:waU *U- n XBU --*SU-- n ODSnOAUS:wa;SU-U-SU-U-- SU-U-US-- pUS:waU *U- n XRU --*SU-- n OSnU(aW W 4$[S5e)N)rrr9r:*Rational Point on the conic does not existrF) ValueError) r.rIrJrKrLrMrNokd_dashf_dashrPrOs r'rA&ImplicitRegion._regular_point_parabolas"6!]qf&6]1a41Q3q5=]aVW]M]B !MNNAv"#A#a%!A#a%-1QAQ;#GFNEE'kNAaC0EBa"#A#a%!A#a%-1QAQ;#GFNEE'kNAaC0EBe|# !MNNr)c  ^.SU-U-US-- nUnU(d [S5eUS:XaUS:XaSn SXE-X&-- -n OUS:waMUn SUS--US--SU-U-U-U-- SU-U-US---SUS--U-U--SU-US--U-- n O.Un SUS--US--SU-U-U-U-- SUS--U-U--n U S:g=(a U S:=(a U S:(+nU(d [S5e[U 5RS5n [U 5RS5n U RU RpU RU Rp[ X5nX-U- nX-U- nX-*U- n[ U5[[U5S5-n[UU- 5n[ U5[[U5S5-n[UU- 5n[ U5[[U5S5-n[UU- 5n[ [ UU5U5nUU- nUU- nUU- n[ UU5nUU- nUU- nUU-n[ UU5nUU- nUU-nUU- n[ UU5nUU-nUU- nUU- n[S5unnnUUS--UUS---UUS---n[U5n [U 5S:Xa [S5eS n!U GHn"[U"6Rn#[R!U#S 5m.U"Sn$U$S:XaS n!M;[#U$[$[&45(aMXU$Rn%[U%5S :Xah[)[+U%55n&[-[.R0[3[5U$S5U&[.R055n'[)[+U'55T.U&'[U%5S:Xa[7[9U%55un&n([.R0Hn)U$R;U&U)5n*[-[.R0[3[5U*S5U([.R055n+U+R<(aMeU)T.U&'[)[+U+55T.U(' O [U#5S:wa[?U.4S jU"55unnnOU"unnnS n! O U!(a [S5eUU-U- nUU-U- nUU-U- nUU- nUU- nUS:Xa,US:Xa&UU-SU-- SU-- n,UU- SU-- SU-- n-U,U-4$US:wa(USU-U-- X%--U - n,UUU,-- U- SU-- n-U,U-4$USU-U-- X$--U - n-UUU--- U- SU-- n,U,U-4$)Nr:r9rWrlJ)zx y zFr;Tr2c3D># UHoRT5v M g7frrD.0sreps r' 8ImplicitRegion._regular_point_ellipse..s'ASs S ) rXrlimit_denominatorpqrrrabsrrrr=r free_symbolsdictfromkeysrintrnextiterr rIntegersrrr>rrDrEtuple)/r.rIrJrKrLrMrNDrYKLk1k2l1l2ga1b1c1a2r1b2r2c2r3g1g2g3rQrRrSeq solutionsflagsolsymssol_zsyms_zrjp_valuesrki subs_sol_zq_valuesrOrPres/ @r'rB%ImplicitRegion._regular_point_ellipses!A1 AB !MNNAv!q&qsQSyMaadF1a4K!A#a%'!)+ac!eAqDj81QT6!8A:E1QPQT RS SadF1a4K!A#a%'!)+a1fQhqj8a0A!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"A1a$'1a DE!H !MNN2r A2r A2r A!A!AAv!q&Q1qs+Q1qs+%< aQqSUQS!+QuWq1Q3/ %< QqSUQS!+QuWq1Q3/%< r)cUR/nURHnU[URU5/- nM [U[ UR55$)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 rr>)r.eq_listvars r'rFImplicitRegion.singular_pointssM"==/>>C T]]C01 1G"7D$899r)cj[U[5(a URnURn[ UR 5Hup4UR XDX-5nM [U5n[UR5S:wa URn[SU55nU$Un[U5nU$)a Returns the multiplicity of a singular point on the region. A singular point (x,y) of region is said to be of multiplicity m if all the partial derivatives off to order m - 1 vanish there. Examples ======== >>> from sympy.abc import x, y, z >>> from sympy.vector import ImplicitRegion >>> I = ImplicitRegion((x, y, z), x**2 + y**3 - z**4) >>> I.singular_points() {(0, 0, 0)} >>> I.multiplicity((0, 0, 0)) 2 rc38# UHn[U5v M g7frr)rcterms r'rf.ImplicitRegion.multiplicity..Ps954L&&5s) rrr,r% enumerater$rDr r=minr)r.point modified_eqrrtermsms r' multiplicityImplicitRegion.multiplicity2s& eU # #JJEmm /FA%**3eh?K0[) { A %$$E9599A  EU#Ar)ct^URnURnUS:Xag[UR5S:XaU4$[UR5S:Xa(URupV[ [ X655SnXW4$[ 5eSnUS:XaPUbUnOJ[UR55S:wa[ UR55SnOUR5n[UR55S:waUR5n U Hkn [U 6Rn [RU S5m[U 5S:wa[U4SjU 55n URU 5US- :XdMiU n O [U5S:Xa [ 5eUn [UR5HupU R!XX-5n M [#U 5n S=nnU R$Hn['U5U:XaUU- nMUU- nM SU-n[)U[5(dU4n[UR5S:XaUSnUS:Xa [+SS S 9nO [+SS S 9n[+US S 9nUR!URSUURSU05nUR!URSUURSU05nUUU- -R!US5US-nUUU- -R!US5US-nUU4$[UR5S :XGaUunnS U;a [+S S S 9nO [+S S S 9n[+US S 9n[+US S 9nUR!URSUURSUURSU05nUR!URSUURSUURSU05nUUU- -R!US5US-nUUU- -R!US5US-nUUU- -R!US5US-nUUU4$[ 5e)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. 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