o 8Vad>@sddlmZmZmZmZmZmZmZddlm Z m Z m Z m Z m Z mZddlmZddlmZddlmZmZmZddlmZddlmZddlmZGd d d e Zd d Zd S))SRationalgcdsqrtsignsymbols 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() {(0, 0, 0)} >>> 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|ts t|}t|tr|j|j}t|||SN) isinstancer r ZlhsZrhssuper__new__)cls variablesequation __class__=/usr/lib/python3/dist-packages/sympy/vector/implicitregion.pyr3s   zImplicitRegion.__new__cC |jdS)Nrargsselfr"r"r#r< zImplicitRegion.variablescCr$)Nr%r'r"r"r#r@r)zImplicitRegion.equationcCs t|jSr)rrr'r"r"r#degreeDr)zImplicitRegion.degreec CsZ|j}t|jdkrtt||jdtjddfSt|jdkrT|jdkrTt|j|}\}}}}}}|dd||krI|j |\} } | | fS|j |\} } | | fSt|jdkr|j\} } } t ddD]3} t ddD]+} t| | | | | i|jdtjdj s| | tt| | | | | idfSqmqft|dkrt|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 )rlenrlistrrZRealsr+ conic_coeff_regular_point_parabola_regular_point_ellipserangesubsis_emptysingular_pointsNotImplementedError)r(rZcoeffsabcdefx_regy_regxyzr"r"r# regular_pointHs,   &,zImplicitRegion.regular_pointc Cs"||fdko||fdko|dd||ko||fdk}|s"td|dkrUd||d||d|||d}} |dkrR| |} |||  d|} n5d}n2|dkrd||d||d|||d}} |dkr| |} |||  d|} nd}|r| | fStd)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$8. . z&ImplicitRegion._regular_point_parabolac.s>d|||d}|}|std|dkr'|dkr'd} d||||} nY|dkr`|} d|d|dd||||d|||dd|d||d||d|} n |} d|d|dd||||d|d||} | dko| dko| dk }|stdt| d} t| d} | j| j} } | j| j} }t| |}|| |}| ||}| |  |}t|tt|d}t ||}t|tt|d}t ||}t|tt|d}t ||}tt|||}||}||}||}t||}||}||}||}t||}||}||}||}t||}||}||}||}t d\}}}||d||d||d}t |} t | dkritdd }!| D]}"t |"j}#d d |#D|"d}$|$dkrd }!qmt|$tst|$ts|$j}%t |%d krtt|%}&ttjtt|$d|&tj}'tt|'|&<t |%dkrtt|%\}&}(tjD])})|$|&|)}*ttjtt|*d|(tj}+|+js|)|&<tt|+|(<nqt |#dkr tfdd|"D\}}}n|"\}}}d }!nqm|!rtd|||}|||}|||}||}||}|dkr^|dkr^||d|d|},||d|d|}-|,|-fS|dkr|d||||| },|||,|d|}-|,|-fS|d||||| }-|||-|d|},|,|-fS)Nr.r-rGrlJ)zx y zFcSi|]}|dqS)r/r".0sr"r"r# z9ImplicitRegion._regular_point_ellipse..Tr*c3|]}|VqdSrr7rMZrepr"r# z8ImplicitRegion._regular_point_ellipse..)rHrZlimit_denominatorpqrrrabsrrrr1r free_symbolsrrintnextiterrrZIntegersrr r2rr7r8tuple).r(r;r<r=r>r?r@DrIKLZk1Zk2l1l2gZa1Zb1Zc1Za2Zr1Zb2Zr2Zc2Zr3Zg1Zg2Zg3rCrDrEeqZ solutionsflagZsolsymsZsol_zZsyms_zrWZp_valuesrXiZ subs_sol_zZq_valuesrArBr"rTr#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)} )rrr rr2)r(Zeq_listvarr"r"r#r9s zImplicitRegion.singular_pointscCst|tr|j}|j}t|jD]\}}|||||}qt|}t|jdkr8|j}t dd|D}|S|}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 rcSsg|]}t|qSr"r)rNtermr"r"r# Jsz/ImplicitRegion.multiplicity..) rrr&r enumeraterr7r r1minr)r(point modified_eqrhriZtermsmr"r"r# multiplicity,s zImplicitRegion.multiplicitytrONcs~|j}|j}|dkr0t|jdkr|fSt|jdkr-|j\}}tt||d}||fStd}|dkrR|dur=|}nt|dkrNt|d}n|}t|dkr|} | D],} t | j } dd| Dt| dkrt fdd | D} | | |dkr| }nq`t|dkrt|} t |jD]\} }| |||| } qt| } d}}| jD]}t||kr||7}q||7}qd |}t|t s|f}t|jdkr6|d}|d krtd d d}ntd d d}t|d d}||jd||jd|i}||jd||jd|i}||||d|d}||||d|d}||fSt|jdkr|\}}|dksL|dkrStdd d}ntdd d}t|d d}t|d d}||jd||jd||jd|i}||jd||jd||jd|i}||||d|d}||||d|d}||||d|d}|||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. Available: https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1827&context=cstech r*r-rr"NcSrL)r-r"rMr"r"r#rPrQz;ImplicitRegion.rational_parametrization..c3rRrrSrMrTr"r#rUrVz:ImplicitRegion.rational_parametrization..rJrOZs_T)realr/rZr_)rr+r1rr2rr:r9rFr rZr^rqrlr7r r&rrr)r(Z parametersZ reg_pointrr+rCrDZy_parrnr9ZspointrgrorhriZhnZhn_1rjZ parameter1rOrsZx_parZ parameter2ruZz_parr"rTr#rational_parametrizationQs/              (( z'ImplicitRegion.rational_parametrization)rrN)__name__ __module__ __qualname____doc__rpropertyrrr+rFr4r5r9rqrv __classcell__r"r"r r#r s '   7|%rc Cst|dkr t|d}|d}t|}||d}|||}||d}||d|d}||d|d}||d|d} |||||| fS)Nr-rr*)rrHr Zcoeff) rrrCrDr;r<r=r>r?r@r"r"r#r3s r3N)ZsympyrrrrrrrZ sympy.corer r r r r rZsympy.core.compatibilityrZsympy.core.symbolrZ sympy.solversrrrZ sympy.polysrZsympy.geometryrZsympy.ntheory.factor_rrr3r"r"r"r#s$       _