U Zʹj?@sddlmZddlmZddlmZddlmZddlm Z ddl m Z ddl m Z ddlmZmZmZmZmZmZdd lmZdd lmZdd lmZmZmZdd lmZdd lmZddl m!Z!GdddeZ"ddZ#dS))Rational)S)symbols)sign)sqrt)gcd) Complement)BasicTuplediffexpandEqInteger)ordered)_symbol)solveset nonlinsolve diophantine total_degree)Point)corecsveZdZdZfddZeddZeddZedd Zd d Z d d Z ddZ ddZ ddZ dddZZS)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. cs8t|tst|}t|tr(|j|j}t|||SN) isinstancer r lhsrhssuper__new__)cls variablesequation __class__Z/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/vector/implicitregion.pyr9s    zImplicitRegion.__new__cCs |jdS)Nrargsselfr$r$r%r BszImplicitRegion.variablescCs |jdS)Nr&r(r$r$r%r!FszImplicitRegion.equationcCs t|jSr)rr!r(r$r$r%degreeJszImplicitRegion.degreec Cs\|j}t|jdkr4tt||jdtjddfSt|jdkr|jdkrt|j|}\}}}}}}|dd||kr|j |\} } n|j |\} } | | fSt|jdkr0|j\} } } t ddD]f} t ddD]V} t| | | | | i|jdtjdj s| | tt| | | | | idfSqqt|dkrRt|dStd S) a0 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 r*r)domaini N)r!lenr listrrZRealsr+ conic_coeff_regular_point_parabola_regular_point_ellipserangesubsis_emptysingular_pointsNotImplementedError)r)r!Zcoeffsabcdefx_regy_regxyzr$r$r% regular_pointNs&   &0zImplicitRegion.regular_pointc Cs,||fdko6||fdko6|dd||ko6||fdk}|sDtd|dkrd||d||d|||d}} |dkr| |} |||  d|} nd}nh|dkrd||d||d|||d}} |dkr| |} |||  d|} nd}|r | | fStddS)N)rrr-r.*Rational Point on the conic does not existrF) ValueError) r)r;r<r=r>r?r@okZd_dashZf_dashrBrAr$r$r%r4s$8.  .  z&ImplicitRegion._regular_point_parabolac.s4d|||d}|}|s$td|dkrN|dkrNd} d||||} n|dkr|} d|d|dd||||d|||dd|d||d||d|} n@|} d|d|dd||||d|d||} | dko| dko| dk }|s,tdt| d} t| d} | j| j} } | j| j} }t| |}|| |}| ||}| |  |}t|tt|d}t ||}t|tt|d}t ||}t|tt|d}t ||}tt|||}||}||}||}t||}||}||}||}t||}||}||}||}t||}||}||}||}t d\}}}||d||d||d}t |} t | dkrtdd }!| D]L}"t |"j}#t|#d |"d}$|$dkrd }!qt|$ttfs|$j}%t |%d krrtt|%}&ttjtt|$d|&tj}'tt|'|&<t |%dkrtt|%\}&}(tjD]T})|$|&|)}*ttjtt|*d|(tj}+|+js|)|&<tt|+|(<qqt |#dkrtfd d|"D\}}}n |"\}}}d }!q0q|!r>td|||}|||}|||}||}||}|dkr|dkr||d|d|},||d|d|}-nt|dkr|d||||| },|||,|d|}-n4|d||||| }-|||-|d|},|,|-fS)Nr.r-rGrlJ)zx y zFr/Tr*c3s|]}|VqdSrr7.0srepr$r% sz8ImplicitRegion._regular_point_ellipse..) rHrlimit_denominatorpqrrrabsrrrr1r free_symbolsdictfromkeysrintrnextiterrrZIntegersrr r2rr7r8tuple).r)r;r<r=r>r?r@DrIKLZk1Zk2l1l2gZa1b1c1Za2r1b2r2c2Zr3g1g2Zg3rCrDrEeqZ solutionsflagZsolsymsZsol_zZsyms_zrTZp_valuesrUiZ subs_sol_zZq_valuesrArBr$rPr%r5sf<         $              z%ImplicitRegion._regular_point_ellipsecCs6|jg}|jD]}|t|j|g7}qt|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 rr2)r)Zeq_listvarr$r$r%r9s zImplicitRegion.singular_pointscCs~t|tr|j}|j}t|jD]\}}|||||}q t|}t|jdkrn|j}t dd|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 rcss|]}t|VqdSrr)rNtermr$r$r%rRPsz.ImplicitRegion.multiplicity..) rrr'r! enumerater r7r r1minr)r)point modified_eqrorpZtermsmr$r$r% multiplicity2s zImplicitRegion.multiplicitytrONcs|j}|j}|dkr`t|jdkr(|fSt|jdkrZ|j\}}tt||d}||fStd}|dkr|dk rz|}n*t|dkrt|d}n|}t|dkr|} | D]Z} t | j } t | dt| dkrt fdd| D} || |dkr| }qqt|dkr2t|} t|jD]\} }| |||| } q@t| } d}}| jD]&}t||kr||7}n||7}qvd|}t|t s|f}t|jdkr||d}|d krtd d d }n td d d }t|d d }||jd||jd|i}||jd||jd|i}||||d|d}||||d|d}||fSt|jd kr~|\}}d|krtdd d }n tdd d }t|d d }t|d d }||jd||jd||jd|i}||jd||jd||jd|i}||||d|d}||||d|d}||||d|d}|||fStdS)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 r*r-rr$Nc3s|]}|VqdSrrLrMrPr$r%rRsz:ImplicitRegion.rational_parametrization..rJrOZs_T)realr/rZr_)r!r+r1r r2rr:r9rFr rWrXrYr]rwrrr7r r'rrr)r) parametersZ reg_pointr!r+rCrDZy_parrtr9ZspointrnrurorpZhnZhn_1rqZ parameter1rOryZx_parZ parameter2r{Zz_parr$rPr%rational_parametrizationWs/              (( z'ImplicitRegion.rational_parametrization)rxN)__name__ __module__ __qualname____doc__rpropertyr r!r+rFr4r5r9rwr} __classcell__r$r$r"r%rs'    7|%rc Cst|dkrt|d}|d}t|}||d}|||}||d}||d|d}||d|d}||d|d} |||||| fS)Nr-rr*)rrHr Zcoeff) r r!rCrDr;r<r=r>r?r@r$r$r%r3s r3N)$Zsympy.core.numbersrZsympy.core.singletonrZsympy.core.symbolrZ$sympy.functions.elementary.complexesrZ(sympy.functions.elementary.miscellaneousrZsympy.polys.polytoolsrZsympy.sets.setsrZ sympy.corer r r r r rZsympy.core.sortingrrZ sympy.solversrrrZ sympy.polysrZsympy.geometryrZsympy.ntheory.factor_rrr3r$r$r$r%s$             _