Ed?ddlmZddlmZddlmZddlmZddlm Z ddl m Z ddl m Z ddlmZmZmZmZmZmZdd lmZdd lmZdd lmZmZmZdd lmZdd lmZddl m!Z!GddeZ"dZ#dS))Rational)S)symbols)sign)sqrt)gcd) Complement)BasicTuplediffexpandEqInteger)ordered)_symbol)solveset nonlinsolve diophantine total_degree)Point)coreceZdZdZfdZedZedZedZdZ dZ dZ d Z d Z dd ZxZS)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. ct|ts t|}t|tr|j|jz }t |||SN) isinstancer rlhsrhssuper__new__)cls variablesequation __class__s ;lib/python3.11/site-packages/sympy/vector/implicitregion.pyr!zImplicitRegion.__new__9sY)U++ *y)I h # # 3|hl2HwwsIx888c|jdS)Nrargsselfs r&r#zImplicitRegion.variablesBy|r'c|jdS)Nr)r+s r&r$zImplicitRegion.equationFr-r'c*t|jSr)rr$r+s r&degreezImplicitRegion.degreeJsDM***r'c|j}t|jdkr;tt ||jdt jdfSt|jdkrW|jdkrLt|j|x}\}}}}}}|dzd|z|zkr|j |\} } n |j |\} } | | fSt|jdkr|j\} } } tddD]} tddD]} t | | | | | i|jdt jj s@| | tt | | | | | idfccSt|dkr%t|dSt) a1 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. Availaible: https://www3.risc.jku.at/publications/download/risc_1355/Rational%20Points%20on%20Conics.pdf r/r)domaini )r$lenr#listrrRealsr1 conic_coeff_regular_point_parabola_regular_point_ellipserangesubsis_emptysingular_pointsNotImplementedError)r,r$coeffsabcdefx_regy_regxyzs r& regular_pointzImplicitRegion.regular_pointNs6= t~  ! # $(DN1,=agNNNOOPQRT T  A % ${a $,7,Q,QQ)Aq!Qa41Q3q5=H#?4#?#HLE55#>4#>#GLE5e|# t~  ! # fnGAq!sB f f"3^^ffE#HMM1eQ2F$G$GXYIZcdcjkkktf %ud8HMM1eUVX]J^<_<_3`3`.a.abc.deeeeeeff t##%% & &! + 3,,../2 2!###r'c||fdko||fdko|dzd|z|zko||fdk}|std|dkr>d|z|zd|z|zz d|z|z|dzz } }|dkr| |z } ||| zz d|zz } nFd}nC|dkr=d|z|zd|z|zz d|z|z|dzz } }|dkr| |z } ||| zz d|zz } nd}|r| | fStd)N)rrr4r5*Rational Point on the conic does not existrF) ValueError) r,rDrErFrGrHrIokd_dashf_dashrKrJs r&r<z&ImplicitRegion._regular_point_parabolas]Q6!]q!f&6]1a41Q3q5=]aQRVW]M]B O !MNNNAv "#A#a%!A#a%-1QAQ;#GFNE!E'kNAaC0EEBBa "#A#a%!A#a%-1QAQ;#GFNE!E'kNAaC0EEB Oe|# !MNNNr'c f .d|z|z|dzz }|}|std|dkr|dkrd} d||z||zz z} n|dkrM|} d|dzz|dzzd|z|z|z|zz d|z|z|dzzzd|dzz|z|zzd|z|dzz|zz } n.|} d|dzz|dzzd|z|z|z|zz d|dzz|z|zz} | dko | dko| dk }|stdt| d} t| d} | j| j} } | j| j}} t | |}|| z|z }| |z|z }| | z |z }t |tt|dz}t||z }t |tt|dz}t||z }t |tt|dz}t||z }t t |||}||z }||z }||z }t ||}||z }||z }||z}t ||}||z }||z}||z }t ||}||z}||z }||z }td\}}}||dzz||dzzz||dzzz}t|} t| dkrtdd }!| D]}"t|"j}#d |#D.|"d}$|$dkrd }!.t|$t t"fs|$j}%t|%d kr|t%t'|%}&t)t*jt/t1|$d|&t*j}'t%t'|'.|&<t|%dkrt3t5|%\}&}(t*jD]})|$|&|)}*t)t*jt/t1|*d|(t*j}+|+js&|).|&<t%t'|+.|(<nt|#dkr t;.fd |"D\}}}n|"\}}}d }!n|!rtd||z|z }||z|z }||z|z }||z }||z }|dkr)|dkr#||zd|zz d|zz },||z d|zz d|zz }-nQ|dkr&|d|z|zz ||zz| z },|||,zz |z d|zz }-n%|d|z|zz ||zz| z }-|||-zz |z d|zz },|,|-fS)Nr5r4rQrlJ)zx y zFci|]}|dS)r6.0ss r& z9ImplicitRegion._regular_point_ellipse..***q!***r'Tr/c3BK|]}|VdSrr?r\r]reps r& z8ImplicitRegion._regular_point_ellipse..s-'A'As 'A'A'A'A'A'Ar')rRrlimit_denominatorpqrrrabsrrrr8r free_symbolsrintrnextiterr rIntegersrrr9rr?r@tuple)/r,rDrErFrGrHrIDrSKLk1k2l1l2ga1b1c1a2r1b2r2c2r3g1g2g3rLrMrNeq solutionsflagsolsymssol_zsyms_zrfp_valuesrgi subs_sol_zq_valuesrJrKrcs/ @r&r=z%ImplicitRegion._regular_point_ellipses!A1 AB O !MNNNAv 9!q& 9qsQqSyMa 9adF1a4K!A#a%'!)+ac!eAqDj81QT6!8A:E1QPQT RS SadF1a4K!A#a%'!)+a1fQhqj8a0A!a%0B O !MNNN --f55A --f55AS!#BS!#BB AR%BR%Bb5!Bb$s2ww***BbeBb$s2ww***BbeBb$s2ww***BbeBCBKK$$AABABABRBBBBBBBRBBBBBBBRBBBBBBBg&&GAq!AqD2ad7"R1W,B#BI9~~" O !MNNND " " c{/**T***AA:D!%#w88"/F6{{a'6 f..#-aj(2eQ<h>h'i'iH#+#4&)*A)-d8nn)=)=A %& 4yyA~("''A'A'A'AS'A'A'A"A"A1aa$'1a DE36 O !MNNN2r A2r A2r A!A!AAv 0!q& 0Q1qs+Q1qs+a 0QqSUQqS!+QuWq1Q3/QqSUQqS!+QuWq1Q3/%< r'c|jg}|jD]}|t|j|gz }t|t |jS)a Returns a set of singular points of the region. The singular points are those points on the region where all partial derivatives vanish. Examples ======== >>> from sympy.abc import x, y >>> from sympy.vector import ImplicitRegion >>> I = ImplicitRegion((x, y), (y-1)**2 -x**3 + 2*x**2 -x) >>> I.singular_points() {(1, 1)} )r$r#r rr9)r,eq_listvars r&rAzImplicitRegion.singular_pointssS"=/> 2 2C T]C001 1GG7D$8$8999r'cpt|tr|j}|j}t |jD]$\}}|||||z}%t|}t|jdkr!|j}td|D}n|}t|}|S)a Returns the multiplicity of a singular point on the region. A singular point (x,y) of region is said to be of multiplicity m if all the partial derivatives off to order m - 1 vanish there. Examples ======== >>> from sympy.abc import x, y, z >>> from sympy.vector import ImplicitRegion >>> I = ImplicitRegion((x, y, z), x**2 + y**3 - z**4) >>> I.singular_points() {(0, 0, 0)} >>> I.multiplicity((0, 0, 0)) 2 rc,g|]}t|SrZr)r\terms r& z/ImplicitRegion.multiplicity..Ps :::D\$'':::r') rrr*r$ enumerater#r?r r8minr)r,point modified_eqrrtermsms r& multiplicityzImplicitRegion.multiplicity2s& eU # # JEm // @ @FAs%**3eAh??KK[)) { A % $$E::E:::;;AAEU##Ar'tr]Ncn |j}|j}|dkrrt|jdkr|fSt|jdkr1|j\}}t t ||d}||fSt d}|dkrf||}nat|dkr(t |d}n|}t|dkr|} | D]j} t| j } d| Dt| dkrtfd| D} | | |dz kr| }nkt|dkrt |} t|jD]$\} }| |||| z} %t| } dx}}| jD] }t#||kr||z }||z }!d|z}t%|ts|f}t|jdkr|d}|d krt'd d }nt'd d }t'|d }||jd||jd|i}||jd||jd|i}|||z z|d|dz}|||z z|d|dz}||fSt|jd kr:|\}}d|vrt'dd }nt'dd }t'|d }t'|d }||jd||jd||jd|i}||jd||jd||jd|i}|||z z|d|dz}|||z z|d|dz}|||z z|d|dz}|||fSt )a Returns the rational parametrization of implict 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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