ó ~9­\c@ s¨dZddlmZmZddlZddlmZmZmZddl m Z ddl m Z m Z mZddlmZddlmZmZdd lmZdd lmZdd lmZdd lmZdd lmZddl mZddlm Z ddl!m"Z"ddl#m$Z$ddl%m&Z&ddl'm(Z(m)Z)m*Z*ddl+m,Z,de,fd„ƒYZ-de-fd„ƒYZ.de-fd„ƒYZ/dS(sSGeometrical Points. Contains ======== Point Point2D Point3D When methods of Point require 1 or more points as arguments, they can be passed as a sequence of coordinates or Points: >>> from sympy.geometry.point import Point >>> Point(1, 1).is_collinear((2, 2), (3, 4)) False >>> Point(1, 1).is_collinear(Point(2, 2), Point(3, 4)) False iÿÿÿÿ(tdivisiontprint_functionN(tStsympifytExpr(tNumber(titerablet is_sequencetas_int(tTuple(t nsimplifytsimplify(t GeometryError(tsqrt(tim(tMatrix(tEq(tFloat(tglobal_evaluate(tAdd(t FiniteSet(tuniq(t filldedentt func_namet Undecidablei(tGeometryEntitytPointcB s eZdZeZd„Zd„Zd„Zd„Zd„Z d„Z d„Z d„Z d „Z d „Zd „Zd „Zd „Zed„ƒZed„ƒZed„ƒZed„ƒZd„Zd„Zd„Zd$d„Zd„Zd„Zd„Z ed„ƒZ!d„Z"ed„ƒZ#ed„ƒZ$d„Z%ed„ƒZ&ed„ƒZ'ed „ƒZ(d!„Z)d"„Z*ed#„ƒZ+eZ,e Z-RS(%sâA point in a n-dimensional Euclidean space. Parameters ========== coords : sequence of n-coordinate values. In the special case where n=2 or 3, a Point2D or Point3D will be created as appropriate. evaluate : if `True` (default), all floats are turn into exact types. dim : number of coordinates the point should have. If coordinates are unspecified, they are padded with zeros. on_morph : indicates what should happen when the number of coordinates of a point need to be changed by adding or removing zeros. Possible values are `'warn'`, `'error'`, or `ignore` (default). No warning or error is given when `*args` is empty and `dim` is given. An error is always raised when trying to remove nonzero coordinates. Attributes ========== length origin: A `Point` representing the origin of the appropriately-dimensioned space. Raises ====== TypeError : When instantiating with anything but a Point or sequence ValueError : when instantiating with a sequence with length < 2 or when trying to reduce dimensions if keyword `on_morph='error'` is set. See Also ======== sympy.geometry.line.Segment : Connects two Points Examples ======== >>> from sympy.geometry import Point >>> from sympy.abc import x >>> Point(1, 2, 3) Point3D(1, 2, 3) >>> Point([1, 2]) Point2D(1, 2) >>> Point(0, x) Point2D(0, x) >>> Point(dim=4) Point(0, 0, 0, 0) Floats are automatically converted to Rational unless the evaluate flag is False: >>> Point(0.5, 0.25) Point2D(1/2, 1/4) >>> Point(0.5, 0.25, evaluate=False) Point2D(0.5, 0.25) c O s|jdtdƒ}|jddƒ}t|ƒdkrD|dn|}t|tƒrŠt}t|ƒ|jdt|ƒƒkrŠ|Snt|ƒsºttdj t |ƒƒƒƒ‚nt|ƒdkrú|jddƒrút j f|jdƒ}nt|Œ}|jdt|ƒƒ}t|ƒdkrEttd ƒƒ‚nt|ƒ|krÍd j |t|ƒ|ƒ}|dkrqÍ|d krœt|ƒ‚qÍ|d kr¸tj|ƒqÍttd ƒƒ‚ntd„||Dƒƒrötdƒ‚ntd„|Dƒƒrtdƒ‚ntd„|Dƒƒs@tdƒ‚n|| t j f|t|ƒ}|r´|jtg|jtƒD]$}|tt|dtƒƒf^qƒƒ}nt|ƒdkrÝt|d 1.s1Dimension of {} needs to be changedfrom {} to {}.terrortwarnsf on_morph value should be 'error', 'warn' or 'ignore'.cs s|] }|VqdS(N((t.0ti((s3lib/python2.7/site-packages/sympy/geometry/point.pys ›ss&Nonzero coordinates cannot be removed.cs s$|]}|jot|ƒVqdS(N(t is_numberR(R!ta((s3lib/python2.7/site-packages/sympy/geometry/point.pys ss(Imaginary coordinates are not permitted.cs s|]}t|tƒVqdS(N(t isinstanceR(R!R$((s3lib/python2.7/site-packages/sympy/geometry/point.pys Ÿss,Coordinates must be valid SymPy expressions.trationalt_nochecki(tgetRtlenR%RtFalseRt TypeErrorRtformatRtNoneRtZeroR t ValueErrortwarningsR tanytalltxreplacetdicttatomsRR R tTruetPoint2DtPoint3DRt__new__( tclstargstkwargsRRtcoordsRtmessagetf((s3lib/python2.7/site-packages/sympy/geometry/point.pyR9os^"$ $    " @    cC s)tdgt|ƒƒ}tj||ƒS(s7Returns the distance between this point and the origin.i(RR)tdistance(tselftorigin((s3lib/python2.7/site-packages/sympy/geometry/point.pyt__abs__·scC s“y(tj|t|dtƒƒ\}}Wn&tk rPtdj|ƒƒ‚nXgt||ƒD]\}}t||ƒ^qa}t|dtƒS(sGAdd other to self by incrementing self's coordinates by those of other. Notes ===== >>> from sympy.geometry.point import Point When sequences of coordinates are passed to Point methods, they are converted to a Point internally. This __add__ method does not do that so if floating point values are used, a floating point result (in terms of SymPy Floats) will be returned. >>> Point(1, 2) + (.1, .2) Point2D(1.1, 2.2) If this is not desired, the `translate` method can be used or another Point can be added: >>> Point(1, 2).translate(.1, .2) Point2D(11/10, 11/5) >>> Point(1, 2) + Point(.1, .2) Point2D(11/10, 11/5) See Also ======== sympy.geometry.point.Point.translate Rs+Don't know how to add {} and a Point object(Rt_normalize_dimensionR*R+R R,tzipR (RAtothertstoR$tbR=((s3lib/python2.7/site-packages/sympy/geometry/point.pyt__add__¼s ( 2cC s ||jkS(N(R;(RAtitem((s3lib/python2.7/site-packages/sympy/geometry/point.pyt __contains__ãscC sBt|ƒ}g|jD]}t||ƒ^q}t|dtƒS(s'Divide point's coordinates by a factor.R(RR;R RR*(RAtdivisortxR=((s3lib/python2.7/site-packages/sympy/geometry/point.pyt__div__æs &cC sBt|tƒ s.t|jƒt|jƒkr2tS|j|jkS(N(R%RR)R;R*(RARF((s3lib/python2.7/site-packages/sympy/geometry/point.pyt__eq__ìs.cC s |j|S(N(R;(RAtkey((s3lib/python2.7/site-packages/sympy/geometry/point.pyt __getitem__ñscC s t|jƒS(N(thashR;(RA((s3lib/python2.7/site-packages/sympy/geometry/point.pyt__hash__ôscC s |jjƒS(N(R;t__iter__(RA((s3lib/python2.7/site-packages/sympy/geometry/point.pyRU÷scC s t|jƒS(N(R)R;(RA((s3lib/python2.7/site-packages/sympy/geometry/point.pyt__len__úscC sBt|ƒ}g|jD]}t||ƒ^q}t|dtƒS(s{Multiply point's coordinates by a factor. Notes ===== >>> from sympy.geometry.point import Point When multiplying a Point by a floating point number, the coordinates of the Point will be changed to Floats: >>> Point(1, 2)*0.1 Point2D(0.1, 0.2) If this is not desired, the `scale` method can be used or else only multiply or divide by integers: >>> Point(1, 2).scale(1.1, 1.1) Point2D(11/10, 11/5) >>> Point(1, 2)*11/10 Point2D(11/10, 11/5) See Also ======== sympy.geometry.point.Point.scale R(RR;R RR*(RAtfactorRNR=((s3lib/python2.7/site-packages/sympy/geometry/point.pyt__mul__ýs &cC s-g|jD] }| ^q }t|dtƒS(sNegate the point.R(R;RR*(RARNR=((s3lib/python2.7/site-packages/sympy/geometry/point.pyt__neg__scC s|g|D] }| ^q S(sPSubtract two points, or subtract a factor from this point's coordinates.((RARFRN((s3lib/python2.7/site-packages/sympy/geometry/point.pyt__sub__!sc s¯t|ddƒ‰|jdˆƒ‰ˆdkrItd„|Dƒƒ‰nt‡fd†|Dƒƒrot|ƒSˆ|d<|jddƒ|d1sc3 s|]}|jˆkVqdS(N(R\(R!R"(R(s3lib/python2.7/site-packages/sympy/geometry/point.pys 2sRR N(tgetattrR-R(tmaxR2tlistR(R:tpointsR<R"((Rs3lib/python2.7/site-packages/sympy/geometry/point.pyRD&s   cG s•t|ƒdkrdStjg|D]}t|ƒ^q#Œ}|d}g|dD]}||^qS}tg|D]}|j^qsƒ}|jƒS(sgThe affine rank of a set of points is the dimension of the smallest affine space containing all the points. For example, if the points lie on a line (and are not all the same) their affine rank is 1. If the points lie on a plane but not a line, their affine rank is 2. By convention, the empty set has affine rank -1.iiÿÿÿÿi(R)RRDRR;trank(R;R"R`RBtm((s3lib/python2.7/site-packages/sympy/geometry/point.pyt affine_rank8s ( !"cC st|dt|ƒƒS(s$Number of components this point has.R[(R]R)(RA((s3lib/python2.7/site-packages/sympy/geometry/point.pyR\LscG szt|ƒdkrtS|jg|D]}t|ƒ^q#Œ}|djdkrUtStt|ƒƒ}tj|ŒdkS(sñReturn True if there exists a plane in which all the points lie. A trivial True value is returned if `len(points) < 3` or all Points are 2-dimensional. Parameters ========== A set of points Raises ====== ValueError : if less than 3 unique points are given Returns ======= boolean Examples ======== >>> from sympy import Point3D >>> p1 = Point3D(1, 2, 2) >>> p2 = Point3D(2, 7, 2) >>> p3 = Point3D(0, 0, 2) >>> p4 = Point3D(1, 1, 2) >>> Point3D.are_coplanar(p1, p2, p3, p4) True >>> p5 = Point3D(0, 1, 3) >>> Point3D.are_coplanar(p1, p2, p3, p5) False iii(R)R6RDRR\R_RRc(R:R`R"((s3lib/python2.7/site-packages/sympy/geometry/point.pyt are_coplanarQs$(cC sæt|tƒsUyt|d|jƒ}WqUtk rQtdt|ƒƒ‚qUXnt|tƒr¥tj|t|ƒƒ\}}ttd„t ||ƒDƒŒƒSt |ddƒ}|dkrÜtdt|ƒƒ‚n||ƒS(sƒThe Euclidean distance between self and another GeometricEntity. Returns ======= distance : number or symbolic expression. Raises ====== TypeError : if other is not recognized as a GeometricEntity or is a GeometricEntity for which distance is not defined. See Also ======== sympy.geometry.line.Segment.length sympy.geometry.point.Point.taxicab_distance Examples ======== >>> from sympy.geometry import Point, Line >>> p1, p2 = Point(1, 1), Point(4, 5) >>> l = Line((3, 1), (2, 2)) >>> p1.distance(p2) 5 >>> p1.distance(l) sqrt(2) The computed distance may be symbolic, too: >>> from sympy.abc import x, y >>> p3 = Point(x, y) >>> p3.distance((0, 0)) sqrt(x**2 + y**2) Rs'not recognized as a GeometricEntity: %scs s#|]\}}||dVqdS(iN((R!R$RI((s3lib/python2.7/site-packages/sympy/geometry/point.pys ­sR@s,distance between Point and %s is not definedN( R%RRR\R+ttypeRDR RRER]R-(RARFRGtpR@((s3lib/python2.7/site-packages/sympy/geometry/point.pyR@s' # cC s8t|ƒst|ƒ}ntd„t||ƒDƒŒS(s.Return dot product of self with another Point.cs s|]\}}||VqdS(N((R!R$RI((s3lib/python2.7/site-packages/sympy/geometry/point.pys ·s(RRRRE(RARf((s3lib/python2.7/site-packages/sympy/geometry/point.pytdot³s cC sIt|tƒ s(t|ƒt|ƒkr,tStd„t||ƒDƒƒS(s8Returns whether the coordinates of self and other agree.cs s$|]\}}|j|ƒVqdS(N(tequals(R!R$RI((s3lib/python2.7/site-packages/sympy/geometry/point.pys ¾s(R%RR)R*R2RE(RARF((s3lib/python2.7/site-packages/sympy/geometry/point.pyRh¹s(cK s8g|jD]}|j||^q }tdt|ŒS(sFEvaluate the coordinates of the point. This method will, where possible, create and return a new Point where the coordinates are evaluated as floating point numbers to the precision indicated (default=15). Parameters ========== prec : int Returns ======= point : Point Examples ======== >>> from sympy import Point, Rational >>> p1 = Point(Rational(1, 2), Rational(3, 2)) >>> p1 Point2D(1/2, 3/2) >>> p1.evalf() Point2D(0.5, 1.5) R(R;tevalfRR*(RAtprectoptionsRNR=((s3lib/python2.7/site-packages/sympy/geometry/point.pyRiÀs(cC sˆt|tƒst|ƒ}nt|tƒr{||kr@|gStj||ƒ\}}||krw||krw|gSgS|j|ƒS(sXThe intersection between this point and another GeometryEntity. Parameters ========== other : Point Returns ======= intersection : list of Points Notes ===== The return value will either be an empty list if there is no intersection, otherwise it will contain this point. Examples ======== >>> from sympy import Point >>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 0) >>> p1.intersection(p2) [] >>> p1.intersection(p3) [Point2D(0, 0)] (R%RRRDt intersection(RARFtp1tp2((s3lib/python2.7/site-packages/sympy/geometry/point.pyRlßs cG sZ|f|}tjg|D]}t|ƒ^qŒ}tt|ƒƒ}tj|ŒdkS(sÛReturns `True` if there exists a line that contains `self` and `points`. Returns `False` otherwise. A trivially True value is returned if no points are given. Parameters ========== args : sequence of Points Returns ======= is_collinear : boolean See Also ======== sympy.geometry.line.Line Examples ======== >>> from sympy import Point >>> from sympy.abc import x >>> p1, p2 = Point(0, 0), Point(1, 1) >>> p3, p4, p5 = Point(2, 2), Point(x, x), Point(1, 2) >>> Point.is_collinear(p1, p2, p3, p4) True >>> Point.is_collinear(p1, p2, p3, p5) False i(RRDR_RRc(RAR;R`R"((s3lib/python2.7/site-packages/sympy/geometry/point.pyt is_collinears! (c G sè|f|}tjg|D]}t|ƒ^qŒ}tt|ƒƒ}tj|Œdks`tS|d}g|D]}||^qq}tg|D]"}t|ƒ|j|ƒg^q‘ƒ}|jƒ\}}t |ƒ|krät StS(sDo `self` and the given sequence of points lie in a circle? Returns True if the set of points are concyclic and False otherwise. A trivial value of True is returned if there are fewer than 2 other points. Parameters ========== args : sequence of Points Returns ======= is_concyclic : boolean Examples ======== >>> from sympy import Point Define 4 points that are on the unit circle: >>> p1, p2, p3, p4 = Point(1, 0), (0, 1), (-1, 0), (0, -1) >>> p1.is_concyclic() == p1.is_concyclic(p2, p3, p4) == True True Define a point not on that circle: >>> p = Point(1, 1) >>> p.is_concyclic(p1, p2, p3) False ii( RRDR_RRcR*RRgtrrefR)R6( RAR;R`R"RBRftmatRptpivots((s3lib/python2.7/site-packages/sympy/geometry/point.pyt is_concyclic.s& ( 5cC s|j}|dkrdS| S(srTrue if any coordinate is nonzero, False if every coordinate is zero, and None if it cannot be determined.N(tis_zeroR-(RARt((s3lib/python2.7/site-packages/sympy/geometry/point.pyt is_nonzerofs  c C s½tj|t|ƒƒ\}}|jdkr•|j|j\}}\}}||||jdƒ}|dkr•ttd||fƒƒ‚q•nt|j|jgƒ} | j ƒdkS(s{Returns whether each coordinate of `self` is a scalar multiple of the corresponding coordinate in point p. iisDcan't determine if %s is a scalar multiple of %sN( RRDR\R;RhR-RRRRa( RARfRGRHtx1ty1tx2ty2trvRb((s3lib/python2.7/site-packages/sympy/geometry/point.pytis_scalar_multipleos cC sMg|jD]}|j^q }t|ƒr/tStd„|DƒƒrIdStS(ssTrue if every coordinate is zero, False if any coordinate is not zero, and None if it cannot be determined.cs s|]}|dkVqdS(N(R-(R!RN((s3lib/python2.7/site-packages/sympy/geometry/point.pys ‰sN(R;RuR1R*R-R6(RARNtnonzero((s3lib/python2.7/site-packages/sympy/geometry/point.pyRt‚s  cC stjS(sÚ Treating a Point as a Line, this returns 0 for the length of a Point. Examples ======== >>> from sympy import Point >>> p = Point(0, 1) >>> p.length 0 (RR.(RA((s3lib/python2.7/site-packages/sympy/geometry/point.pytlengths cC s[tj|t|ƒƒ\}}tgt||ƒD]#\}}t||tjƒ^q1ƒS(s¨The midpoint between self and point p. Parameters ========== p : Point Returns ======= midpoint : Point See Also ======== sympy.geometry.line.Segment.midpoint Examples ======== >>> from sympy.geometry import Point >>> p1, p2 = Point(1, 1), Point(13, 5) >>> p1.midpoint(p2) Point2D(7, 3) (RRDRER RtHalf(RARfRGR$RI((s3lib/python2.7/site-packages/sympy/geometry/point.pytmidpointœscC stdgt|ƒdtƒS(sOA point of all zeros of the same ambient dimension as the current pointiR(RR)R*(RA((s3lib/python2.7/site-packages/sympy/geometry/point.pyRBºscC s’|j}|dtjkr8tdg|ddgƒS|dtjkrjtddg|ddgƒSt|d |dg|ddgƒS(s~Returns a non-zero point that is orthogonal to the line containing `self` and the origin. Examples ======== >>> from sympy.geometry import Line, Point >>> a = Point(1, 2, 3) >>> a.orthogonal_direction Point3D(-2, 1, 0) >>> b = _ >>> Line(b, b.origin).is_perpendicular(Line(a, a.origin)) True iii(R\RR.R(RAR((s3lib/python2.7/site-packages/sympy/geometry/point.pytorthogonal_directionÀs  cC sZtjt|ƒt|ƒƒ\}}|jr<tdƒ‚n||j|ƒ|j|ƒS(s‹Project the point `a` onto the line between the origin and point `b` along the normal direction. Parameters ========== a : Point b : Point Returns ======= p : Point See Also ======== sympy.geometry.line.LinearEntity.projection Examples ======== >>> from sympy.geometry import Line, Point >>> a = Point(1, 2) >>> b = Point(2, 5) >>> z = a.origin >>> p = Point.project(a, b) >>> Line(p, a).is_perpendicular(Line(p, b)) True >>> Point.is_collinear(z, p, b) True s"Cannot project to the zero vector.(RRDRtR/Rg(R$RI((s3lib/python2.7/site-packages/sympy/geometry/point.pytprojectÚs"$ cC s;tj|t|ƒƒ\}}td„t||ƒDƒŒS(s;The Taxicab Distance from self to point p. Returns the sum of the horizontal and vertical distances to point p. Parameters ========== p : Point Returns ======= taxicab_distance : The sum of the horizontal and vertical distances to point p. See Also ======== sympy.geometry.point.Point.distance Examples ======== >>> from sympy.geometry import Point >>> p1, p2 = Point(1, 1), Point(4, 5) >>> p1.taxicab_distance(p2) 7 cs s%|]\}}t||ƒVqdS(N(tabs(R!R$RI((s3lib/python2.7/site-packages/sympy/geometry/point.pys s(RRDRRE(RARfRG((s3lib/python2.7/site-packages/sympy/geometry/point.pyttaxicab_distancescC s\tj|t|ƒƒ\}}|jr?|jr?tdƒ‚ntd„t||ƒDƒŒS(s–The Canberra Distance from self to point p. Returns the weighted sum of horizontal and vertical distances to point p. Parameters ========== p : Point Returns ======= canberra_distance : The weighted sum of horizontal and vertical distances to point p. The weight used is the sum of absolute values of the coordinates. See Also ======== sympy.geometry.point.Point.distance Examples ======== >>> from sympy.geometry import Point >>> p1, p2 = Point(1, 1), Point(3, 3) >>> p1.canberra_distance(p2) 1 >>> p1, p2 = Point(0, 0), Point(3, 3) >>> p1.canberra_distance(p2) 2 Raises ====== ValueError when both vectors are zero. See Also ======== sympy.geometry.point.Point.distance s"Cannot project to the zero vector.cs s9|]/\}}t||ƒt|ƒt|ƒVqdS(N(R‚(R!R$RI((s3lib/python2.7/site-packages/sympy/geometry/point.pys Ss(RRDRtR/RRE(RARfRG((s3lib/python2.7/site-packages/sympy/geometry/point.pytcanberra_distance"s.cC s|t|ƒS(sdReturn the Point that is in the same direction as `self` and a distance of 1 from the origin(R‚(RA((s3lib/python2.7/site-packages/sympy/geometry/point.pytunitUsN(.t__name__t __module__t__doc__R6tis_PointR9RCRJRLRORPRRRTRURVRXRYRZt classmethodRDt staticmethodRctpropertyR\RdR@RgRhR-RiRlRoRsRuR{RtR}RRBR€RRƒR„R…tnt __truediv__(((s3lib/python2.7/site-packages/sympy/geometry/point.pyR,sN? H  '          . 4    ) & 8   ' ! 3R7cB s‰eZdZdZd„Zd„Zed„ƒZd d„Z ddd d„Z d„Z d d d „Z ed „ƒZ ed „ƒZRS(s²A point in a 2-dimensional Euclidean space. Parameters ========== coords : sequence of 2 coordinate values. Attributes ========== x y length Raises ====== TypeError When trying to add or subtract points with different dimensions. When trying to create a point with more than two dimensions. When `intersection` is called with object other than a Point. See Also ======== sympy.geometry.line.Segment : Connects two Points Examples ======== >>> from sympy.geometry import Point2D >>> from sympy.abc import x >>> Point2D(1, 2) Point2D(1, 2) >>> Point2D([1, 2]) Point2D(1, 2) >>> Point2D(0, x) Point2D(0, x) Floats are automatically converted to Rational unless the evaluate flag is False: >>> Point2D(0.5, 0.25) Point2D(1/2, 1/4) >>> Point2D(0.5, 0.25, evaluate=False) Point2D(0.5, 0.25) icO s>|jdtƒs.d|d>> from sympy import Point2D, pi >>> t = Point2D(1, 0) >>> t.rotate(pi/2) Point2D(0, 1) >>> t.rotate(pi/2, (2, 0)) Point2D(2, -1) iÿÿÿÿ(tcostsinRRiN(tsympyR’R“RR-R;( RAtangletptR’R“RtcRGRzRNR((s3lib/python2.7/site-packages/sympy/geometry/point.pytrotate¥s    '  icC s\|rAt|ddƒ}|j| jŒj||ƒj|jŒSt|j||j|ƒS(sõScale the coordinates of the Point by multiplying by ``x`` and ``y`` after subtracting ``pt`` -- default is (0, 0) -- and then adding ``pt`` back again (i.e. ``pt`` is the point of reference for the scaling). See Also ======== rotate, translate Examples ======== >>> from sympy import Point2D >>> t = Point2D(1, 1) >>> t.scale(2) Point2D(2, 1) >>> t.scale(2, 2) Point2D(2, 2) Ri(Rt translateR;tscaleRNR(RARNRR–((s3lib/python2.7/site-packages/sympy/geometry/point.pyRšÇs)cC s‹|jo|jdks'tdƒ‚n|j\}}|joH|dk}|j\}}ttdd||dgƒ|jƒdd ŒS(sîReturn the point after applying the transformation described by the 3x3 Matrix, ``matrix``. See Also ======== geometry.entity.rotate geometry.entity.scale geometry.entity.translate ismatrix must be a 3x3 matrixiii(ii(t is_MatrixtshapeR/t is_squareR;RRttolist(RAtmatrixtcoltrowt valid_matrixRNR((s3lib/python2.7/site-packages/sympy/geometry/point.pyt transformâs icC st|j||j|ƒS(s‡Shift the Point by adding x and y to the coordinates of the Point. See Also ======== rotate, scale Examples ======== >>> from sympy import Point2D >>> t = Point2D(0, 1) >>> t.translate(2) Point2D(2, 1) >>> t.translate(2, 2) Point2D(2, 3) >>> t + Point2D(2, 2) Point2D(2, 3) (RRNR(RARNR((s3lib/python2.7/site-packages/sympy/geometry/point.pyR™ôscC s |jdS(sº Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.x 0 i(R;(RA((s3lib/python2.7/site-packages/sympy/geometry/point.pyRN s cC s |jdS(sº Returns the Y coordinate of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.y 1 i(R;(RA((s3lib/python2.7/site-packages/sympy/geometry/point.pyRs N(R†R‡RˆR[R9RLRŒR‘R-R˜RšR£R™RNR(((s3lib/python2.7/site-packages/sympy/geometry/point.pyR7_s0   " R8cB s­eZdZdZd„Zd„Zed„ƒZd„Zd„Z d„Z ddddd „Z d „Z d d d d „Zed „ƒZed„ƒZed„ƒZRS(s6A point in a 3-dimensional Euclidean space. Parameters ========== coords : sequence of 3 coordinate values. Attributes ========== x y z length Raises ====== TypeError When trying to add or subtract points with different dimensions. When `intersection` is called with object other than a Point. Examples ======== >>> from sympy import Point3D >>> from sympy.abc import x >>> Point3D(1, 2, 3) Point3D(1, 2, 3) >>> Point3D([1, 2, 3]) Point3D(1, 2, 3) >>> Point3D(0, x, 3) Point3D(0, x, 3) Floats are automatically converted to Rational unless the evaluate flag is False: >>> Point3D(0.5, 0.25, 2) Point3D(1/2, 1/4, 2) >>> Point3D(0.5, 0.25, 3, evaluate=False) Point3D(0.5, 0.25, 3) icO s>|jdtƒs.d|d>> from sympy import Point3D, Matrix >>> from sympy.abc import x >>> p1, p2 = Point3D(0, 0, 0), Point3D(1, 1, 1) >>> p3, p4, p5 = Point3D(2, 2, 2), Point3D(x, x, x), Point3D(1, 2, 6) >>> Point3D.are_collinear(p1, p2, p3, p4) True >>> Point3D.are_collinear(p1, p2, p3, p5) False (RRo(R`((s3lib/python2.7/site-packages/sympy/geometry/point.pyt are_collinearas"cC sb|j|ƒ}ttd„|DƒŒƒ}|j|j||j|j||j|j|gS(sp Gives the direction cosine between 2 points Parameters ========== p : Point3D Returns ======= list Examples ======== >>> from sympy import Point3D >>> p1 = Point3D(1, 2, 3) >>> p1.direction_cosine(Point3D(2, 3, 5)) [sqrt(6)/6, sqrt(6)/6, sqrt(6)/3] cs s|]}|dVqdS(iN((R!R"((s3lib/python2.7/site-packages/sympy/geometry/point.pys œs(tdirection_ratioR RRNRtz(RAtpointR$RI((s3lib/python2.7/site-packages/sympy/geometry/point.pytdirection_cosine…s"cC s+|j|j|j|j|j|jgS(sV Gives the direction ratio between 2 points Parameters ========== p : Point3D Returns ======= list Examples ======== >>> from sympy import Point3D >>> p1 = Point3D(1, 2, 3) >>> p1.direction_ratio(Point3D(2, 3, 5)) [1, 1, 2] (RNRR¦(RAR§((s3lib/python2.7/site-packages/sympy/geometry/point.pyR¥ scC sWt|tƒs$t|ddƒ}nt|tƒrJ||krF|gSgS|j|ƒS(scThe intersection between this point and another point. Parameters ========== other : Point Returns ======= intersection : list of Points Notes ===== The return value will either be an empty list if there is no intersection, otherwise it will contain this point. Examples ======== >>> from sympy import Point3D >>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, 0, 0) >>> p1.intersection(p2) [] >>> p1.intersection(p3) [Point3D(0, 0, 0)] Ri(R%RRR8Rl(RARF((s3lib/python2.7/site-packages/sympy/geometry/point.pyRl¸s icC sc|r>t|ƒ}|j| jŒj|||ƒj|jŒSt|j||j||j|ƒS(söScale the coordinates of the Point by multiplying by ``x`` and ``y`` after subtracting ``pt`` -- default is (0, 0) -- and then adding ``pt`` back again (i.e. ``pt`` is the point of reference for the scaling). See Also ======== translate Examples ======== >>> from sympy import Point3D >>> t = Point3D(1, 1, 1) >>> t.scale(2) Point3D(2, 1, 1) >>> t.scale(2, 2) Point3D(2, 2, 1) (R8R™R;RšRNRR¦(RARNRR¦R–((s3lib/python2.7/site-packages/sympy/geometry/point.pyRšÞs ,c C s­|jo|jdks'tdƒ‚n|j\}}|joH|dk}ddlm}|j\}}}||ƒ} ttdd|||dgƒ| j ƒdd ŒS( sîReturn the point after applying the transformation described by the 4x4 Matrix, ``matrix``. See Also ======== geometry.entity.rotate geometry.entity.scale geometry.entity.translate ismatrix must be a 4x4 matrixiÿÿÿÿ(t Transposeiii(ii( R›RœR/Rtsympy.matrices.expressionsR©R;R8RRž( RARŸR R¡R¢R©RNRR¦Rb((s3lib/python2.7/site-packages/sympy/geometry/point.pyR£ùs  icC s%t|j||j||j|ƒS(s–Shift the Point by adding x and y to the coordinates of the Point. See Also ======== rotate, scale Examples ======== >>> from sympy import Point3D >>> t = Point3D(0, 1, 1) >>> t.translate(2) Point3D(2, 1, 1) >>> t.translate(2, 2) Point3D(2, 3, 1) >>> t + Point3D(2, 2, 2) Point3D(2, 3, 3) (R8RNRR¦(RARNRR¦((s3lib/python2.7/site-packages/sympy/geometry/point.pyR™ scC s |jdS(s½ Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 3) >>> p.x 0 i(R;(RA((s3lib/python2.7/site-packages/sympy/geometry/point.pyRN$s cC s |jdS(s½ Returns the Y coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 2) >>> p.y 1 i(R;(RA((s3lib/python2.7/site-packages/sympy/geometry/point.pyR3s cC s |jdS(s½ Returns the Z coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 1) >>> p.z 1 i(R;(RA((s3lib/python2.7/site-packages/sympy/geometry/point.pyR¦Bs N(R†R‡RˆR[R9RLR‹R¤R¨R¥RlR-RšR£R™RŒRNRR¦(((s3lib/python2.7/site-packages/sympy/geometry/point.pyR8)s+  $   & (0Rˆt __future__RRR0t sympy.coreRRRtsympy.core.numbersRtsympy.core.compatibilityRRRtsympy.core.containersR tsympy.simplifyR R tsympy.geometry.exceptionsR t(sympy.functions.elementary.miscellaneousR t$sympy.functions.elementary.complexesRtsympy.matricesRtsympy.core.relationalRRtsympy.core.evaluateRtsympy.core.addRt sympy.setsRtsympy.utilities.iterablesRtsympy.utilities.miscRRRtentityRRR7R8(((s3lib/python2.7/site-packages/sympy/geometry/point.pyts2 ÿÿÿ6Ê