ó ¡¼™\c@ s£dZddlmZmZddlmZddlmZmZm Z m Z ddl m Z ddl mZmZmZmZddlmZmZmZddlmZmZmZd!d „Zd „Zd!d „Zd!d „Zd „Zd„Z d„Z!d„Z"d„Z#d!d!d„Z$d„Z%ddd„Z&d!d„Z'd„Z(d„Z)e*d„Z+d!e,d„Z-d„Z.d„Z/d„Z0d„Z1d „Z2d!S("sø Module to implement integration of uni/bivariate polynomials over 2D Polytopes and uni/bi/trivariate polynomials over 3D Polytopes. Uses evaluation techniques as described in Chin et al. (2015) [1]. References =========== [1] : Chin, Eric B., Jean B. Lasserre, and N. Sukumar. "Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra." Computational Mechanics 56.6 (2015): 967-981 PDF link : http://dilbert.engr.ucdavis.edu/~suku/quadrature/cls-integration.pdf iÿÿÿÿ(tprint_functiontdivision(t cmp_to_key(tStdifftExprtSymbol(t nsimplify(t Segment2DtPolygontPointtPoint2D(txtytz(tLCtgcd_listt degree_listcK s4|jdtƒ}|jddƒ}|rft|tƒrWtdtt|jƒŒ}qftdƒ‚nt|tƒrt|ƒ}|j }n6t |dƒdkrft |ƒ}t |ddƒdkrWgt d|ƒD](}t ||d|||dƒ^qÙ} |}t | ƒ} gt d| ƒD]%}t | || |d| ƒ^q)}qÃtd ƒ‚n]|d} |d}t|| ƒ}|dkrÃ|dkr°td ƒ‚nt||| |ƒS|dk r i} t|tƒ r|dk rtd ƒ‚nt |ddƒd kr5td|| ||ƒ} ntd|||ƒ} |dkrZ| Sx¨|D] }|| kra|tjkr˜tj| tj|D]6}t|ƒ}t|ƒ\}}|| ||7}qºW|| |>> from sympy.abc import x, y >>> from sympy.geometry.polygon import Polygon >>> from sympy.geometry.point import Point >>> from sympy.integrals.intpoly import polytope_integrate >>> polygon = Polygon(Point(0, 0), Point(0, 1), Point(1, 1), Point(1, 0)) >>> polys = [1, x, y, x*y, x**2*y, x*y**2] >>> expr = x*y >>> polytope_integrate(polygon, expr) 1/4 >>> polytope_integrate(polygon, polys, max_degree=3) {1: 1, x: 1/2, y: 1/2, x*y: 1/4, x*y**2: 1/6, x**2*y: 1/6} t clockwiset max_degreetevaluates>clockwise=True works for only 2-PolytopeV-representation inputiiitplane2Ds<Integration for H-representation 3Dcase not implemented yet.s3Input expression be mustbe a valid SymPy expressions-Input polynomials must be list of expressionsitseparateN(tgettFalsetNonet isinstanceR t point_sorttverticest TypeErrorthyperplane_parameterstsidestlentranget intersectionRtNotImplementedErrortmain_integrate3dtlisttmain_integrateRtZerot decomposetTrueRtstrip(tpolytexprtkwargsRRt hp_paramstfacetstplentit intersectionstlintsRtresultt result_dicttintegral_valuetmonomstmonomtcoefftm((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pytpolytope_integratesh!   ; ;             cC sJ|tjkrdS|jr&|dfSt|ƒ}|t|ƒ|fSdS(Nii(ii(RR't is_numberR(R8R9((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyR*Šs    c#C si}tttf}t|ƒ}|r¢t|dƒ}g|D]&} | D]} | D] } | ^qQqGq=} x| D]} d|| d>> from sympy.abc import x, y >>> from sympy.integrals.intpoly import main_integrate3d, hyperplane_parameters >>> cube = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0), (5, 0, 5), (5, 5, 0), (5, 5, 5)], [2, 6, 7, 3], [3, 7, 5, 1], [7, 6, 4, 5], [1, 5, 4, 0], [3, 1, 0, 2], [0, 4, 6, 2]] >>> vertices = cube[0] >>> faces = cube[1:] >>> hp_params = hyperplane_parameters(faces, vertices) >>> main_integrate3d(1, faces, vertices, hp_params) -125 iiii( R R RR tgradient_termst enumerateRR'tintegration_reduction_dynamictnormR(tpolygon_integratettuple(#R,R/RR.RR4tdimst dim_lengtht grad_termstz_termstx_termttermt flat_listt facet_countthptatbtx0R1R8tx_dty_dtz_dtz_indexty_indextx_indext_tdegreetvalue_over_faceR6t polynomialstdegtpoly_contributetfacettpi((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyR$”sV!          ,      cC s>ttf}t|ƒ}i}tj}|r‘ddddggt|ƒ}x>t|ƒD]0\} } | d| d} } || jd} xût|ƒD]í\}}|\}}}}|j|dƒ}tj}| tjkrìtj}n:||}t || | | ||||||| ||ƒ }||d<|dk re||c|| t | ƒ||7>> from sympy.abc import x, y >>> from sympy.integrals.intpoly import main_integrate, hyperplane_parameters >>> from sympy.geometry.polygon import Polygon >>> from sympy.geometry.point import Point >>> triangle = Polygon(Point(0, 3), Point(5, 3), Point(1, 1)) >>> facets = triangle.sides >>> hp_params = hyperplane_parameters(triangle) >>> main_integrate(x**2 + y**2, facets, hp_params) 325/6 iiiN(R R R RR'R=R>tpointsRRR?R@R(tintegration_reduction(R,R/R.RRCRDR4R6RERJRKRLRMRNR1R8R:RORPRUtvalueRVtvalue_over_boundaryRXRYRZ((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyR&êsR            (      c C s5t|ƒ}|tjkr"tjStj}||d}xttt|ƒƒD]`} ||| ||| dt|ƒf} |t|| |dƒt|| | ||ƒ7}qLW|js#t|tƒ|dt|t ƒ|dt|t ƒ|d}|t |||||||dƒ7}n||d}|S(sÔHelper function to integrate the input uni/bi/trivariate polynomial over a certain face of the 3-Polytope. Parameters =========== facet : Particular face of the 3-Polytope over which `expr` is integrated index : The index of `facet` in `facets` facets : Faces of the 3-Polytope(expressed as indices of `vertices`) vertices : Vertices that constitute the facet expr : The input polynomial degree : Degree of `expr` Examples ======== >>> from sympy.abc import x, y >>> from sympy.integrals.intpoly import polygon_integrate >>> cube = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0), (5, 0, 5), (5, 5, 0), (5, 5, 5)], [2, 6, 7, 3], [3, 7, 5, 1], [7, 6, 4, 5], [1, 5, 4, 0], [3, 1, 0, 2], [0, 4, 6, 2]] >>> facet = cube[1] >>> facets = cube[1:] >>> vertices = cube[0] >>> polygon_integrate(facet, [(0, 1, 0), 5], 0, facets, vertices, 1, 0) -25 iii( RR'R!R tdistance_to_sidetlineseg_integrateR<RR R RRA( R[thp_paramtindexR/RR,RVR4RNR1tside((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyRA9s   * )c C s|\}}g|D] }dt|ƒt|ƒ^q}gtddƒD]}||||^qI}gtddƒD]}||t|ƒ^qw}td||ƒ}gtddƒD]}|d|||^q¹} tgtddƒD]}| |||^qîƒ} | S(syHelper function to compute the signed distance between given 3D point and a line segment. Parameters =========== point : 3D Point line_seg : Line Segment Examples ======== >>> from sympy.integrals.intpoly import distance_to_side >>> point = (0, 0, 0) >>> distance_to_side(point, [(0, 0, 1), (0, 1, 0)], (1, 0, 0)) -sqrt(2)/2 iÿÿÿÿii(iii(RR@R!t cross_producttsum( tpointtline_segtAtx1tx2R1t rev_normaltvectortn_sidetvectorx0t dot_product((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyRahs -.024c C sE|tjkrtjStj}|d}ttgtdƒD] }|d||d|^q<ƒƒ}t|tƒrÄi|ddt6|ddt6|ddt 6} |||j | ƒ7}n|||7}t |tƒ|dt |tƒ|dt |t ƒ|d}|t |||||dƒ7}||d}|S(sÓHelper function to compute the line integral of `expr` over `line_seg` Parameters =========== polygon : Face of a 3-Polytope index : index of line_seg in polygon line_seg : Line Segment Examples ======== >>> from sympy.integrals.intpoly import lineseg_integrate >>> polygon = [(0, 5, 0), (5, 5, 0), (5, 5, 5), (0, 5, 5)] >>> line_seg = [(0, 5, 0), (5, 5, 0)] >>> lineseg_integrate(polygon, 0, line_seg, 1, 0) 5 iiii( RR'R@RBR!RRR R RtsubsRRb( tpolygonRdRiR,RVR4RNR1tdistancet expr_dict((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyRb†s"   6) c C sã|tjkr|Stj}||jd}t|ƒ} ttf} t|| dƒ|dt|| dƒ|d} | dkr®|t||||| ||dƒ7}n|t| ||||| ƒ7}|t|ƒ|dS(sôHelper method for main_integrate. Returns the value of the input expression evaluated over the polytope facet referenced by a given index. Parameters =========== facets : List of facets of the polytope. index : Index referencing the facet to integrate the expression over. a : Hyperplane parameter denoting direction. b : Hyperplane parameter denoting distance. expr : The expression to integrate over the facet. dims : List of symbols denoting axes. degree : Degree of the homogeneous polynomial. Examples ======== >>> from sympy.abc import x, y >>> from sympy.integrals.intpoly import integration_reduction, hyperplane_parameters >>> from sympy.geometry.point import Point >>> from sympy.geometry.polygon import Polygon >>> triangle = Polygon(Point(0, 3), Point(5, 3), Point(1, 1)) >>> facets = triangle.sides >>> a, b = hyperplane_parameters(triangle)[0] >>> integration_reduction(facets, 0, a, b, 1, (x, y), 0) 5 ii( RR'R]R R R RR^tleft_integral2D( R/RdRLRMR,RCRVR_RNR:tgenst inner_product((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyR^¯s   4 c C sNtj}x>td|ƒD]-}d}||d|ksM||d|krjt||||dƒ}n|rtttd„||ƒƒƒ} t|ƒrFt|t ƒr2t |ƒdkrôi|d|d6|d|d6|d|d6} n$i|d|d6|d|d6} || |j | ƒ7}qC|| |7}qFqqW|S(sÀComputes the left integral of Eq 10 in Chin et al. For the 2D case, the integral is just an evaluation of the polynomial at the intersection of two facets which is multiplied by the distance between the first point of facet and that intersection. Parameters =========== m : No. of hyperplanes. index : Index of facet to find intersections with. facets : List of facets(Line Segments in 2D case). x0 : First point on facet referenced by index. expr : Input polynomial gens : Generators which generate the polynomial Examples ======== >>> from sympy.abc import x, y >>> from sympy.integrals.intpoly import left_integral2D >>> from sympy.geometry.point import Point >>> from sympy.geometry.polygon import Polygon >>> triangle = Polygon(Point(0, 3), Point(5, 3), Point(1, 1)) >>> facets = triangle.sides >>> left_integral2D(3, 0, facets, facets[0].points[0], 1, (x, y)) 5 iit segment2DcS s||S(N((R R ((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyttii(( RR'R!R"R@RBtmapt is_vertexRRR Rr( R:RdR/RNR,RwR_tjt intersecttdistance_originRu((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyRvßs& ( cC s&tj}t|ƒ}|tjkr(|St|ƒdkr|jsß| | \}}}}|dkrw| | |dnd}|dkr“| dnd}| |d| |d}}|||| d||| d7}n|t|||| ||ƒ7}n| }|jsñ|||||}}}|dkrV| |d|d|dnd}|dkr‚| |d||dnd}|dkr²| |d||ddnd}|||| d||| d||| d7}n|t|||| ||ƒ7}|t|ƒ|dS(sœThe same integration_reduction function which uses a dynamic programming approach to compute terms by using the values of the integral of previously computed terms. Parameters =========== facets : Facets of the Polytope index : Index of facet to find intersections with.(Used in left_integral()) a, b : Hyperplane parameters expr : Input monomial degree : Total degree of `expr` dims : Tuple denoting axes variables x_index : Exponent of 'x' in expr y_index : Exponent of 'y' in expr max_index : Maximum exponent of any monomial in monomial_values x0 : First point on facets[index] monomial_values : List of monomial values constituting the polynomial monom_index : Index of monomial whose integration is being found. Optional Parameters ------------------- vertices : Coordinates of vertices constituting the 3-Polytope hp_param : Hyperplane Parameter of the face of the facets[index] Examples ======== >>> from sympy.abc import x, y >>> from sympy.integrals.intpoly import integration_reduction_dynamic, hyperplane_parameters, gradient_terms >>> from sympy.geometry.point import Point >>> from sympy.geometry.polygon import Polygon >>> triangle = Polygon(Point(0, 3), Point(5, 3), Point(1, 1)) >>> facets = triangle.sides >>> a, b = hyperplane_parameters(triangle)[0] >>> x0 = facets[0].points[0] >>> monomial_values = [[0, 0, 0, 0], [1, 0, 0, 5], [y, 0, 1, 15], [x, 1, 0, None]] >>> integration_reduction_dynamic(facets, 0, a, b, x, 1, (x, y), 1, 0, 1, x0, monomial_values, 3) 25/2 iiiii(RR'R R<Rvtleft_integral3D(R/RdRLRMR,RVRCRTRSt max_indexRNtmonomial_valuest monom_indexRRcR_R:RUtx_degreety_degreetx_valuety_valueRRtz_degreetz_value((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyR?s0/   $)" 0,06c C sœtj}||}||d}xttt|ƒƒD]`} ||| ||| dt|ƒf} |t|| |dƒt|| | ||ƒ7}q4W|S(s0Computes the left integral of Eq 10 in Chin et al. For the 3D case, this is the sum of the integral values over constituting line segments of the face (which is accessed by facets[index]) multiplied by the distance between the first point of facet and that line segment. Parameters =========== facets : List of faces of the 3-Polytope. index : Index of face over which integral is to be calculated. expr : Input polynomial vertices : List of vertices that constitute the 3-Polytope hp_param : The hyperplane parameters of the face degree : Degree of the expr >>> from sympy.abc import x, y >>> from sympy.integrals.intpoly import left_integral3D >>> cube = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0), (5, 0, 5), (5, 5, 0), (5, 5, 5)], [2, 6, 7, 3], [3, 7, 5, 1], [7, 6, 4, 5], [1, 5, 4, 0], [3, 1, 0, 2], [0, 4, 6, 2]] >>> facets = cube[1:] >>> vertices = cube[0] >>> left_integral3D(facets, 3, 1, vertices, ([0, -1, 0], -5), 0) -50 ii(RR'R!R RaRb( R/RdR,RRcRVR_R[RNR1Re((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyRhs  *iicC sY|dkr¨d}dgt|dd|ddƒ}xtd|dƒD]T}xKtd||dƒD]2}t|t|||dg||<|d7}qkWqMWn­gtd|dƒD]“}gtd|dƒD]s}gt||ddƒD]P}t|t|t|||||||||||||dg^qó^qÖ^q¼}|S(s‚Returns a list of all the possible monomials between 0 and y**binomial_power for 2D case and z**binomial_power for 3D case. Parameters =========== binomial_power : Power upto which terms are generated. no_of_gens : Denotes whether terms are being generated for 2D or 3D case. Examples ======== >>> from sympy.abc import x, y >>> from sympy.integrals.intpoly import gradient_terms >>> gradient_terms(2) [[1, 0, 0, 0], [y, 0, 1, 0], [y**2, 0, 2, 0], [x, 1, 0, 0], [x*y, 1, 1, 0], [x**2, 2, 0, 0]] >>> gradient_terms(2, 3) [[[[1, 0, 0, 0, 0, 0, 0, 0]]], [[[y, 0, 1, 0, 1, 0, 0, 0], [z, 0, 0, 1, 1, 0, 1, 0]], [[x, 1, 0, 0, 1, 1, 0, 0]]], [[[y**2, 0, 2, 0, 2, 0, 0, 0], [y*z, 0, 1, 1, 2, 0, 1, 0], [z**2, 0, 0, 2, 2, 0, 2, 0]], [[x*y, 1, 1, 0, 2, 1, 0, 0], [x*z, 1, 0, 1, 2, 1, 1, 0]], [[x**2, 2, 0, 0, 2, 2, 0, 0]]]] iiiiiÿÿÿÿN(RtintR!R R R(tbinomial_powert no_of_genstcountttermstx_countty_counttz_count((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyR=Œs 'ªcC st|tƒrt|jƒ|jdg}dgt|ƒd}xÏtt|ƒdƒD]¹}||}||d}|d|d}|d|d}|d|d|d|d}t|||gƒ} t|ƒ| }t|ƒ| t|ƒ| f} | |f||>> from sympy.geometry.point import Point >>> from sympy.geometry.polygon import Polygon >>> from sympy.integrals.intpoly import hyperplane_parameters >>> hyperplane_parameters(Polygon(Point(0, 3), Point(5, 3), Point(1, 1))) [((0, 1), 3), ((1, -2), -1), ((-2, -1), -3)] >>> cube = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0), (5, 0, 5), (5, 5, 0), (5, 5, 5)], [2, 6, 7, 3], [3, 7, 5, 1], [7, 6, 4, 5], [1, 5, 4, 0], [3, 1, 0, 2], [0, 4, 6, 2]] >>> hyperplane_parameters(cube[1:], cube[0]) [([0, -1, 0], -5), ([0, 0, -1], -5), ([-1, 0, 0], -5), ([0, 1, 0], 0), ([1, 0, 0], 0), ([0, 0, 1], 0)] iiiN( RR R%RRR R!RRR>RfRgR'(R+RtparamsR1tv1tv2ta1ta2RMtfactorRLRstvertextv3tnormalR~tfac((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyR¹s2 " *4   cC s½gtddƒD]}||||^q}gtddƒD]}||||^q>}|d|d|d|d|d|d|d|d|d|d|d|dgS(seReturns the cross-product of vectors (v2 - v1) and (v3 - v1) That is : (v2 - v1) X (v3 - v1) iiii(R!(R”R•RšR~((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyRfòs ..cC sá|jd\}}d„}d„}ttf}i} x|D]} tj| | Wt|ƒdkrÏt|ƒdkrþ|dtjkrº||ƒr­tj||dfS||fSqþ|dtjkrþ||ƒrî||dtjfS||fSqþnt|tƒrØ|jrûx¼|j D]Ô} | j rf| j d|krô| | j dc| j d7 Firstly, check for edge cases. Here that would refer to vertical or horizontal lines. 2 > If input expression is a polynomial containing more than one generator then find out the total power of each of the generators. x**2 + 3 + x*y + x**4*y**5 ---> {x: 7, y: 6} If expression is a constant value then pick the first boundary point of the line segment. 3 > First check if a point exists on the line segment where the value of the highest power generator becomes 0. If not check if the value of the next highest becomes 0. If none becomes 0 within line segment constraints then pick the first boundary point of the line segment. Actually, any point lying on the segment can be picked as best origin in the last case. Examples ======== >>> from sympy.integrals.intpoly import best_origin >>> from sympy.abc import x, y >>> from sympy.geometry.line import Segment2D >>> from sympy.geometry.point import Point >>> l = Segment2D(Point(0, 3), Point(1, 1)) >>> expr = x**3*y**7 >>> best_origin((2, 1), 3, l, expr) (0, 3.0) icS s›|j\}}|jtjkr+t|ƒS|jtjkrGt|ƒS|j|jtjkr“|j|j|j|j|j|jtjfSdSdS(s–Returns the point where the input line segment intersects the x-axis. Parameters ========== ls : Line segment N((R]R RR'RBR (tlstptq((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyt x_axis_cut/s  3cS s›|j\}}|jtjkr+t|ƒS|jtjkrGt|ƒS|j|jtjkr“tj|j|j|j|j|j|jfSdSdS(s–Returns the point where the input line segment intersects the y-axis. Parameters ========== ls : Line segment N((R]R RR'RBR (RRžRŸ((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyt y_axis_cut@s  3iitkeycS st|dƒS(Ni(tstr(tk((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyRzƒR{(R]R R RR'R RRtis_Addtargstis_Powtis_Mult is_Symboltsortedtitems(RLRMtlinesegR,R–tb1R R¡Rwt power_gensR1tmonomialt univariatet term_typeRHRN((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyt best_originýsp0         $ (       c C sŸi}t|tƒr{|j r{|jr5||d}|\}}|j |ƒrÍ||c|7>> from sympy.abc import x, y >>> from sympy.integrals.intpoly import decompose >>> decompose(x**2 + x*y + x + y + x**3*y**2 + y**5) {1: x + y, 2: x**2 + x*y, 5: x**3*y**2 + y**5} >>> decompose(x**2 + x*y + x + y + x**3*y**2 + y**5, True) {x, x**2, y, y**5, x*y, x**3*y**2} icS sh|]}|d’qS(i((t.0R8((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pys »s ii(RRR<R©R¥tatomsRR¦RgRRR§R tsettvalues( R,Rt poly_dicttsymbolsR8tdegreesRVRHRUR±((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyR(—s<   .        c s[t|tƒr|jn|}t|ƒ}|dkr@t|ƒS|rRtdƒn tdƒ‰t|dƒ}|dkr¸tttd„|ƒƒ|ttd„|ƒƒ|ƒ‰nTtttd„|ƒƒ|ttd„|ƒƒ|ttd „|ƒƒ|ƒ‰‡‡fd †}‡‡‡fd †}t |d t |dkrQ|n|ƒƒS( sUReturns the same polygon with points sorted in clockwise or anti-clockwise order. Note that it's necessary for input points to be sorted in some order (clockwise or anti-clockwise) for the integration algorithm to work. As a convention algorithm has been implemented keeping clockwise orientation in mind. Parameters ========== poly: 2D or 3D Polygon Optional Parameters: --------------------- normal : The normal of the plane which the 3-Polytope is a part of. clockwise : Returns points sorted in clockwise order if True and anti-clockwise if False. Examples ======== >>> from sympy.integrals.intpoly import point_sort >>> from sympy.geometry.point import Point >>> point_sort([Point(0, 0), Point(1, 0), Point(1, 1)]) [Point2D(1, 1), Point2D(1, 0), Point2D(0, 0)] iiiÿÿÿÿicS s|jS(N(R (R™((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyRzüR{cS s|jS(N(R (R™((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyRzýR{cS s|jS(N(R (R™((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyRzÿR{cS s|jS(N(R (R™((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyRzR{cS s|jS(N(R(R™((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyRzR{c sñ|jˆjtjkr7|jˆjtjkr7ˆ S|jˆjtjkrm|jˆjtjkrmˆS|jˆjtjkr|jˆjtjkr|jˆjtjksÑ|jˆjtjkrì|j|jkrèˆ SˆS|j|jkrˆ SˆS|jˆj|jˆj|jˆj|jˆj}|tjkrUˆ S|tjkrhˆS|jˆj|jˆj|jˆj|jˆj}|jˆj|jˆj|jˆj|jˆj}||kríˆ SˆS(N(R RR'R (RLRMtdettfirsttsecond(tcentertorder(s6lib/python2.7/site-packages/sympy/integrals/intpoly.pytcompares&2222c sqtˆ||ƒ}tgtddƒD]}||ˆ|^q%ƒ}|tjkrZˆ S|tjkrmˆSdS(Nii(RfRgR!RR'(RLRMRºR1Rq(R½R›R¾(s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyt compare3ds 4R¢( RR RR R%RR RgR|RªR(R+R›RtptstntdimR¿RÀ((R½R›R¾s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyR×s    "cC s×tdƒd}t|ttfƒrJtg|D]}|d^q/ƒ|St|tƒr¦t|tƒr‚|jd|jd|S|jd|jd|j |Sn-t|t ƒrÓtd„|j ƒDƒƒ|SdS(s6Returns the Euclidean norm of a point from origin. Parameters ========== point: This denotes a point in the dimension_al spac_e. Examples ======== >>> from sympy.integrals.intpoly import norm >>> from sympy.geometry.point import Point >>> norm(Point(2, 7)) sqrt(53) iics s|]}|dVqdS(iN((R³R1((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pys >sN( RRR%RBRgR R R R RtdictR¶(Rhthalftcoord((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyR@%s%$cC s|d dkrg|dkrk|jd\}}|jd\}}|jd\}}|jd\} } ng|dkrÒ|jd\}}} |jd\}}} |jd\}}} |jd\} } }n|||| |||| }|rg||||}|| | |}t||| |||ƒ|t||| |||ƒ|fSn|d dkr |dkr |d\}}|d\}}|d|d}}||||}|r t||||ƒ|t||||ƒ|fSq nd S( sxReturns intersection between geometric objects. Note that this function is meant for use in integration_reduction and at that point in the calling function the lines denoted by the segments surely intersect within segment boundaries. Coincident lines are taken to be non-intersecting. Also, the hyperplane intersection for 2D case is also implemented. Parameters ========== geom_1, geom_2: The input line segments Examples ======== >>> from sympy.integrals.intpoly import intersection >>> from sympy.geometry.point import Point >>> from sympy.geometry.line import Segment2D >>> l1 = Segment2D(Point(1, 1), Point(3, 5)) >>> l2 = Segment2D(Point(2, 0), Point(2, 5)) >>> intersection(l1, l2, "segment2D") (2, 3) >>> p1 = ((-1, 0), 0) >>> p2 = ((0, 1), 1) >>> intersection(p1, p2, "plane2D") (0, 1) iþÿÿÿtsegmentRyiit segment3DtplaneRN(R]R(tgeom_1tgeom_2tintersection_typeRkty1Rlty2tx3ty3tx4ty4tz1tz2tz3tz4tdenomtt1tt2ta1xta1yta2xta2yR­tb2((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyR"As4  "!( cC s?t|tƒr(t|ƒdkr;tSnt|tƒr;tStS(syIf the input entity is a vertex return True Parameter ========= ent : Denotes a geometric entity representing a point Examples ======== >>> from sympy.geometry.point import Point >>> from sympy.integrals.intpoly import is_vertex >>> is_vertex((2, 3)) True >>> is_vertex((2, 3, 6)) True >>> is_vertex(Point(2, 3)) True ii(ii(RRBR R)R R(tent((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyR}|s cC s©ddlm}m}ttd„|jƒƒ}ttd„|jƒƒ}|j|jdjƒ|j|jdjƒ|||ƒ}||ddƒ}|j ƒdS( s±Plots the 2D polytope using the functions written in plotting module which in turn uses matplotlib backend. Parameter ========= poly: Denotes a 2-Polytope iÿÿÿÿ(tPlott List2DSeriescS s|jS(N(R (R™((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyRz¢R{cS s|jS(N(R (R™((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyRz£R{itaxesslabel_axes=TrueN( tsympy.plotting.plotRàRáR%R|RtappendR R tshow(R+RàRátxltyltl2dsRž((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyt plot_polytope—s cC sLddlm}m}|j}t|ƒdkr>||ƒn ||ƒdS(sÂPlots the polynomial using the functions written in plotting module which in turn uses matplotlib backend. Parameter ========= expr: Denotes a polynomial(SymPy expression) iÿÿÿÿ(tplot3dtplotiN(RãRêRët free_symbolsR (R,RêRëRw((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pytplot_polynomial­s   N(3t__doc__t __future__RRt functoolsRt sympy.coreRRRRtsympy.simplify.simplifyRtsympy.geometryRR R R t sympy.abcR R Rtsympy.polys.polytoolsRRRRR;R*R$R&RARaRbR^RvR?RR=RRfR²RR(R)RR@R"R}RéRí(((s6lib/python2.7/site-packages/sympy/integrals/intpoly.pyts<"" m V O /  ) 0 5 S $- 9 š @N  ;