EdBdZddlZddlmZmZmZddlmZddlm Z ddl m Z ddl m Z ddlmZmZdd lmZdd lmZdd lmZdd lmZmZdd lmZddlmZddlmZm Z ddl!m"Z"m#Z#m$Z$ddl%m&Z&ddl'm(Z(Gdde&Z)Gdde)Z*Gdde)Z+dS)aDGeometrical 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 import Point >>> Point(1, 1).is_collinear((2, 2), (3, 4)) False >>> Point(1, 1).is_collinear(Point(2, 2), Point(3, 4)) False N)SsympifyExpr)Add)Tuple)Float)global_parameters) nsimplifysimplify) GeometryError)sqrt)im)cossin)Matrix) Transpose)uniq is_sequence) filldedent func_name Undecidable)GeometryEntity) prec_to_dpsceZdZdZdZdZdZdZdZdZ dZ d Z d Z d Z d Zd ZdZdZdZedZedZedZedZdZdZdZd)dZdZdZdZ edZ!dZ"edZ#ed Z$d!Z%ed"Z&ed#Z'ed$Z(d%Z)d&Z*ed'Z+d(S)*PointaA 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 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) Tc |dtj}|dd}t|dkr|dn|}t |t r8d}t||dt|kr|St |s 1.z2Dimension of {} needs to be changed from {} to {}.errorwarn) stacklevelzf on_morph value should be 'error', 'warn' or 'ignore'.z&Nonzero coordinates cannot be removed.c3PK|]!}|jot|jduV"dS)FN) is_numberris_zero.0as 4lib/python3.11/site-packages/sympy/geometry/point.py z Point.__new__..s6FF!q{5r!uu}5FFFFFFz(Imaginary coordinates are not permitted.c3@K|]}t|tVdSN) isinstancerr)s r,r-z Point.__new__..s,771:a&&777777r.z,Coordinates must be valid SymPy expressions.c Li|]!}|tt|d"S)T)rational)r r )r*fs r, z!Point.__new__..s?&/&/&/8Ia$77788&/&/&/r.T_nocheck)getr rlenr1rr TypeErrorrformatrrZeror ValueErrorwarningsr$anyallxreplaceatomsrPoint2DPoint3Dr__new__)clsargskwargsrrcoordsr!messages r,rEz Point.__new__ms::j*;*DEE::j(33 IIN4a fe $ $ H6{{fjjF <<<  6"" DJ(?(.y/@/@(A(ACCDD D v;;!  1 5$ 7 7 1fYvzz%000FjjF ,, v;;? (Z)&''(( ( v;;#  1()/F S)I)I 8# 1W$ 1 )))V# 1 g!44444 -/"0"0111 vcdd|   GEFF F FFvFFF F F IGHH H7777777 LJKK K 3V+< ==  0__&/&/ ,,u--&/&/&/00F v;;!  .!%F: F-f-- - [[A  .!%F: F-f-- -%c3F3333r.cxtdgt|z}t||S)z7Returns the distance between this point and the origin.r)rr9distance)selforigins r,__abs__z Point.__abs__s/s3t99}%%~~fd+++r.c t|t|d\}}n0#t$r#td|wxYwdt ||D}t|dS)a8Add other to self by incrementing self's coordinates by those of other. Notes ===== >>> from sympy 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 Frz+Don't know how to add {} and a Point objectc8g|]\}}t||zSr r*r+bs r, z!Point.__add__..s&888da(1q5//888r.)r_normalize_dimensionr:r r;zip)rMothersorIs r,__add__z Point.__add__s> ]--dE%%4P4P4PQQDAqq ] ] ] M T TUZ [ [\\ \ ]98c!Qii888Ve,,,,s -0-Ac||jvSr0rGrMitems r, __contains__zPoint.__contains__sty  r.cjtfd|jD}t|dS)z'Divide point's coordinates by a factor.c4g|]}t|z SrSrT)r*xdivisors r,rWz%Point.__truediv__..s%999!(1W9%%999r.FrQrrGr)rMrfrIs ` r, __truediv__zPoint.__truediv__s>'""9999ty999Ve,,,,r.ct|tr*t|jt|jkrdS|j|jkS)NF)r1rr9rGrMrZs r,__eq__z Point.__eq__sD%'' 3ty>>S__+L 5yEJ&&r.c|j|Sr0r_)rMkeys r, __getitem__zPoint.__getitem__sy~r.c*t|jSr0)hashrGrMs r,__hash__zPoint.__hash__sDIr.c4|jSr0)rG__iter__rqs r,rtzPoint.__iter__sy!!###r.c*t|jSr0)r9rGrqs r,__len__z Point.__len__s49~~r.cjtfd|jD}t|dS)alMultiply point's coordinates by a factor. Notes ===== >>> from sympy 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 c4g|]}t|zSrSrT)r*refactors r,rWz!Point.__mul__..s%888(1V8$$888r.FrQrg)rMryrIs ` r,__mul__z Point.__mul__s>68888di888Ve,,,,r.c,||S)z)Multiply a factor by point's coordinates.)rz)rMrys r,__rmul__zPoint.__rmul__s||F###r.cFd|jD}t|dS)zNegate the point.cg|]}| SrSrSr*res r,rWz!Point.__neg__.. s(((1"(((r.FrQ)rGr)rMrIs r,__neg__z Point.__neg__s*((di(((Ve,,,,r.c |d|DzS)zPSubtract two points, or subtract a factor from this point's coordinates.cg|]}| SrSrSrs r,rWz!Point.__sub__..&s)))ar)))r.rSrjs r,__sub__z Point.__sub__#s))5)))))r.c6t|dddtd|Dtfd|Drt |Sd<ddd<fd|DS) z~Ensure that points have the same dimension. By default `on_morph='warn'` is passed to the `Point` constructor._ambient_dimensionNr!c3$K|] }|jV dSr0ambient_dimensionr*is r,r-z-Point._normalize_dimension..3s%::aa)::::::r.c3.K|]}|jkVdSr0r)r*rr!s r,r-z-Point._normalize_dimension..4s+::aq"c)::::::r.rr$c*g|]}t|fiSrSr)r*rrHs r,rWz.Point._normalize_dimension..8s)333qa""6""333r.)getattrr8maxr@list)rFpointsrHr!s `@r,rXzPoint._normalize_dimension(s c/66jj$$  ;::6:::::C ::::6::: : : << u #ZZ F;;z3333F3333r.ct|dkrdStjd|D}|dfd|ddD}td|D}|d S) agThe 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.rc,g|]}t|SrSrrs r,rWz%Point.affine_rank..Gs-E-E-E1eAhh-E-E-Er.cg|]}|z SrSrS)r*rrNs r,rWz%Point.affine_rank..Is111!f*111r.rNcg|] }|j SrSr_rs r,rWz%Point.affine_rank..Ks+++qAF+++r.cj|jr&t|ddkn|jS)Nr"g-q=)r'absnr()res r,z#Point.affine_rank..Ms-#$; =CAKK%  AIr.) iszerofunc)r9rrXrrank)rGrmrNs @r, affine_rankzPoint.affine_rank:s t99> 2+-E-E-E-E-EF1111fQRRj111 ++F+++ , ,vv$>$>v?? ?r.c>t|dt|S)z$Number of components this point has.r)rr9rqs r,rzPoint.ambient_dimensionPst13t99===r.ct|dkrdS|jd|D}|djdkrdStt |}t j|dkS)aReturn 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 rTc,g|]}t|SrSrrs r,rWz&Point.are_coplanar..|s+E+E+EE!HH+E+E+Er.rr")r9rXrrrrr)rFrs r, are_coplanarzPoint.are_coplanarUszH v;;!  4))+E+Ef+E+E+EF !9 &! + 4d6ll## &)Q..r.c  t|tsE t||j}n-#t$r t dt |zwxYwt|trYt|t|\}}ttdt||DSt|dd}|t dt |z||S)azThe Euclidean distance between self and another GeometricEntity. Returns ======= distance : number or symbolic expression. Raises ====== TypeError : if other is not recognized as a GeometricEntity or is a GeometricEntity for which distance is not defined. See Also ======== sympy.geometry.line.Segment.length sympy.geometry.point.Point.taxicab_distance Examples ======== >>> from sympy import Point, Line >>> p1, p2 = Point(1, 1), Point(4, 5) >>> l = Line((3, 1), (2, 2)) >>> p1.distance(p2) 5 >>> p1.distance(l) sqrt(2) The computed distance may be symbolic, too: >>> from sympy.abc import x, y >>> p3 = Point(x, y) >>> p3.distance((0, 0)) sqrt(x**2 + y**2) r!z'not recognized as a GeometricEntity: %sc3,K|]\}}||z dzVdSr"NrSrUs r,r-z!Point.distance..s.??TQq1uqj??????r.rLNz,distance between Point and %s is not defined) r1rrrr:typerXr rrYr)rMrZr[prLs r,rLzPoint.distancesN%00 Y Ye)?@@@ Y Y Y IDQVKK WXXX Y eU # # B--dE%LLAADAq??SAYY???@AA A5*d33  ZJTRW[[XYY Yx~~s .*Act|st|}tdt||DS)z.Return dot product of self with another Point.c3&K|] \}}||zV dSr0rSrUs r,r-zPoint.dot..s*22TQQqS222222r.)rrrrY)rMrs r,dotz Point.dots=1~~ aA22Sq\\22233r.ct|tr t|t|krdStdt ||DS)z8Returns whether the coordinates of self and other agree.Fc3FK|]\}}||VdSr0)equalsrUs r,r-zPoint.equals..s0<<41a188A;;<<<<<>> from sympy import Point, Rational >>> p1 = Point(Rational(1, 2), Rational(3, 2)) >>> p1 Point2D(1/2, 3/2) >>> p1.evalf() Point2D(0.5, 1.5) c.g|]}|jddiS)rrS)evalf)r*redpsoptionss r,rWz%Point._eval_evalf..s0???'!'++C+7++???r.rF)rrGr)rMprecrrIrs ` @r, _eval_evalfzPoint._eval_evalfsD8$?????TY???f-u---r.ct|tst|}t|tr8||kr|gSt||\}}||kr ||kr|gSgS||S)a|The intersection between this point and another GeometryEntity. Parameters ========== other : GeometryEntity or sequence of coordinates 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)] )r1rrrX intersection)rMrZp1p2s r,rzPoint.intersections<%00 !%LLE eU # # u} v //e<>> 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 c,g|]}t|SrSrrs r,rWz&Point.is_collinear../-G-G-G1eAhh-G-G-Gr.r)rrXrrr)rMrGrs r, is_collinearzPoint.is_collinear sRB4+-G-G-G-G-GHd6ll## &)Q..r.cV|f|z}tjd|D}tt|}tj|dksdS|dfd|D}t d|D}|\}}t|vrdSdS)aDo `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 c,g|]}t|SrSrrs r,rWz&Point.is_concyclic..Zrr.r"Frcg|]}|z SrSrS)r*rrNs r,rWz&Point.is_concyclic.._s---!f*---r.cZg|](}t|||gz)SrS)rrrs r,rWz&Point.is_concyclic..es/;;;qd1ggq *;;;r.T)rrXrrrrrrefr9)rMrGrmatrpivotsrNs @r, is_concycliczPoint.is_concyclic3sL4+-G-G-G-G-GHd6ll## &)Q. 5----f--- ;;F;;;<<xxzz f v;;f $ 4ur.c|j}|dS| S)zrTrue if any coordinate is nonzero, False if every coordinate is zero, and None if it cannot be determined.N)r()rMr(s r, is_nonzerozPoint.is_nonzeroks ,  4{r.ct|t|\}}|jdkrW|j|jc\}}\}}||z||zz d}|"t t d|d|t|j|jg} | dkS)z{Returns whether each coordinate of `self` is a scalar multiple of the corresponding coordinate in point p. r"rNzCannot determine if z- is a scalar multiple of ) rrXrrGrrrrr) rMrr[r\x1y1x2y2rvrs r,is_scalar_multiplezPoint.is_scalar_multiplets))$a991 ! # %!" HRhr2R%"R%-''**B %!**QQ##$#$%%% AFAF# $ $vvxx!|r.cd|jD}t|rdStd|DrdSdS)zsTrue if every coordinate is zero, False if any coordinate is not zero, and None if it cannot be determined.cg|] }|j SrS)rrs r,rWz!Point.is_zero..s333A1<333r.Fc3K|]}|duV dSr0rSrs r,r-z Point.is_zero..s&**QqDy******r.NT)rGr?)rMnonzeros r,r(z Point.is_zerosU43333 w<< 5 **'*** * * 4tr.ctjS)z Treating a Point as a Line, this returns 0 for the length of a Point. Examples ======== >>> from sympy import Point >>> p = Point(0, 1) >>> p.length 0 )rr<rqs r,lengthz Point.lengths v r.ct|t|\}}tdt||DS)aThe midpoint between self and point p. Parameters ========== p : Point Returns ======= midpoint : Point See Also ======== sympy.geometry.line.Segment.midpoint Examples ======== >>> from sympy import Point >>> p1, p2 = Point(1, 1), Point(13, 5) >>> p1.midpoint(p2) Point2D(7, 3) cRg|]$\}}t||ztjz%SrS)r rHalfrUs r,rWz"Point.midpoint..s.EEE41ahAqv~..EEEr.)rrXrYrMrr[s r,midpointzPoint.midpointsH6))$a991EE3q!99EEEFFFr.cFtdgt|zdS)zOA point of all zeros of the same ambient dimension as the current pointrFrQ)rr9rqs r,rNz Point.origins#aST]U3333r.c|j}|djrtdg|dz dgzzS|djrtddg|dz dgzzSt|d |dg|dz dgzzS)auReturns a non-zero point that is orthogonal to the line containing `self` and the origin. Examples ======== >>> from sympy 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 rrr")rr(r)rMr!s r,orthogonal_directionzPoint.orthogonal_directions $ 7? .!a!},-- - 7? 0!A#'A3.// /tAwhQ(C!GaS=8999r.ctt|t|\}}|jrtd|||||z zS)aProject 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 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 "Cannot project to the zero vector.)rrXr(r=r)r+rVs r,projectz Point.projectshD))%((E!HH==1 9 CABB B!%%((QUU1XX%&&r.ct|t|\}}tdt||DS)a2The 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 import Point >>> p1, p2 = Point(1, 1), Point(4, 5) >>> p1.taxicab_distance(p2) 7 c3@K|]\}}t||z VdSr0rrUs r,r-z)Point.taxicab_distance..%s066DAqSQZZ666666r.)rrXrrYrs r,taxicab_distancezPoint.taxicab_distancesE<))$a99166C1II66677r.ct|t|\}}|jr|jrtdt dt ||DS)a=The Canberra Distance from self to point p. Returns the weighted sum of horizontal and vertical distances to point p. Parameters ========== p : Point Returns ======= canberra_distance : The weighted sum of horizontal and vertical distances to point p. The weight used is the sum of absolute values of the coordinates. Examples ======== >>> from sympy import Point >>> p1, p2 = Point(1, 1), Point(3, 3) >>> p1.canberra_distance(p2) 1 >>> p1, p2 = Point(0, 0), Point(3, 3) >>> p1.canberra_distance(p2) 2 Raises ====== ValueError when both vectors are zero. See Also ======== sympy.geometry.point.Point.distance rc3K|]9\}}t||z t|t|zz V:dSr0rrUs r,r-z*Point.canberra_distance..SsEJJ1c!a%jj#a&&3q66/2JJJJJJr.)rrXr(r=rrYrs r,canberra_distancezPoint.canberra_distance'sjR))$a991 < CAI CABB BJJAq JJJKKr.c&|t|z S)zdReturn the Point that is in the same direction as `self` and a distance of 1 from the originrrqs r,unitz Point.unitUsc$iir.N)r),__name__ __module__ __qualname____doc__is_PointrErOr]rbrhrkrnrrrtrvrzr|rr classmethodrX staticmethodrpropertyrrrLrrrrrrrrr(rrrNrrrrrrSr.r,rr*s>>@HF4F4F4P,,, %-%-%-N!!!--- ''' $$$--->$$$--- *** 44[4"??\?*>>X>+/+/[+/Z222h444 ===....@'('('(R$/$/$/L666pX&X  X GGG<44X4 ::X:2$'$'\$'L888B,L,L,L\  X   r.rceZdZdZdZdddZdZedZdd Z dd Z d Z ddZ edZ edZedZdS)rCaA point in a 2-dimensional Euclidean space. Parameters ========== coords : sequence of 2 coordinate values. Attributes ========== x y length Raises ====== TypeError When trying to add or subtract points with different dimensions. When trying to create a point with more than two dimensions. When `intersection` is called with object other than a Point. See Also ======== sympy.geometry.line.Segment : Connects two Points Examples ======== >>> from sympy import Point2D >>> from sympy.abc import x >>> Point2D(1, 2) Point2D(1, 2) >>> Point2D([1, 2]) Point2D(1, 2) >>> Point2D(0, x) Point2D(0, x) Floats are automatically converted to Rational unless the evaluate flag is False: >>> Point2D(0.5, 0.25) Point2D(1/2, 1/4) >>> Point2D(0.5, 0.25, evaluate=False) Point2D(0.5, 0.25) r"Fr6cL|sd|d<t|i|}tj|g|RS)Nr"r!rrrErFr6rGrHs r,rEzPoint2D.__new__> *F5M$)&))D%c1D1111r.c||kSr0rSr`s r,rbzPoint2D.__contains__ t|r.c6|j|j|j|jfS)zwReturn a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure. )reyrqs r,boundszPoint2D.boundss//r.Nct|}t|}|}|t|d}||z}|j\}}t||z||zz ||z||zz}|||z }|S)a[Rotate ``angle`` radians counterclockwise about Point ``pt``. See Also ======== translate, scale Examples ======== >>> 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) Nr"r)rrrrG)rMangleptcr[rrers r,rotatezPoint2D.rotates& JJ JJ   rq!!!B "HBw1 1Q319acAaCi ( (   "HB r.rc|rBt|d}|j| j||j|jSt|j|z|j|zS)aScale 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) r"r)r translaterGscalerer)rMrerrs r,rz Point2D.scalesj,  Orq!!!BD>4>RC:.44Q::DbgN NTVAXtvax(((r.c |jr |jdkstd|j\}}t t dd||dg|zdddS)aReturn the point after applying the transformation described by the 3x3 Matrix, ``matrix``. See Also ======== sympy.geometry.point.Point2D.rotate sympy.geometry.point.Point2D.scale sympy.geometry.point.Point2D.translate )r7r7zmatrix must be a 3x3 matrixrr7rNr") is_Matrixshaper=rGrrtolist)rMmatrixrers r, transformzPoint2D.transformss  >> 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) )rrer)rMrers r,rzPoint2D.translates *TVaZ!,,,r.c|jS)z Returns the two coordinates of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.coordinates (0, 1) r_rqs r, coordinateszPoint2D.coordinates yr.c|jdS)z Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.x 0 rr_rqs r,rez Point2D.xy|r.c|jdS)z Returns the Y coordinate of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.y 1 rr_rqs r,rz Point2D.y!rr.r0)rrN)rr)rrrrrrErbrrrrr rr rerrSr.r,rCrC\s //b%*22222 00X0@))))6 H H H----.  X   X   X   r.rCceZdZdZdZdddZdZedZdZ d Z d Z dd Z dZ ddZedZedZedZedZd S)rDa6A 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) r7FrcL|sd|d<t|i|}tj|g|RS)Nr7r!rrs r,rEzPoint3D.__new___rr.c||kSr0rSr`s r,rbzPoint3D.__contains__err.ctj|S)aIs a sequence of points collinear? Test whether or not a set of points are collinear. Returns True if the set of points are collinear, or False otherwise. Parameters ========== points : sequence of Point Returns ======= are_collinear : boolean See Also ======== sympy.geometry.line.Line3D Examples ======== >>> from sympy import Point3D >>> 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 )rr)rs r, are_collinearzPoint3D.are_collinearhsD!6**r.c||}ttd|D}|j|jz |z |j|jz |z |j|jz |z gS)ap 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] c3 K|] }|dzV dSrrSrs r,r-z+Point3D.direction_cosine..s&''q!t''''''r.)direction_ratior rrerz)rMpointr+rVs r,direction_cosinezPoint3D.direction_cosinest,   ' ' ''Q'''( ) )46!Q&$&(8A'=46!Q&( (r.cZ|j|jz |j|jz |j|jz gS)aV 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] )rerr)rMrs r,rzPoint3D.direction_ratios,,46!EGdf$4uw7GIIr.ct|tst|d}t|tr ||kr|gSgS||S)aThe intersection between this point and another GeometryEntity. Parameters ========== other : GeometryEntity or sequence of coordinates Returns ======= intersection : list of Points Notes ===== The return value will either be an empty list if there is no intersection, otherwise it will contain this point. Examples ======== >>> from sympy import Point3D >>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, 0, 0) >>> p1.intersection(p2) [] >>> p1.intersection(p3) [Point3D(0, 0, 0)] r7r)r1rrrDrrjs r,rzPoint3D.intersectionsf<%00 (%Q'''E eW % % u} v I!!$'''r.rNc|rAt|}|j| j|||j|jSt|j|z|j|z|j|zS)aScale 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) )rDrrGrrerr)rMrerrrs r,rz Point3D.scalesm,  RBG>4>RC:.44Q1==GQ Qtvax46!8444r.c |jr |jdkstd|j\}}}t |}t t dd|||dg|zdddS)zReturn the point after applying the transformation described by the 4x4 Matrix, ``matrix``. See Also ======== sympy.geometry.point.Point3D.scale sympy.geometry.point.Point3D.translate )r!zmatrix must be a 4x4 matrixrr!rNr7)rrr=rGrrDrr)rMr rerrrs r,r zPoint3D.transforms  >> 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) )rDrerr)rMrerrs r,rzPoint3D.translates(*tvz46A:tvz:::r.c|jS)z Returns the three coordinates of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 2) >>> p.coordinates (0, 1, 2) r_rqs r,r zPoint3D.coordinates&rr.c|jdS)z Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 3) >>> p.x 0 rr_rqs r,rez Point3D.x5rr.c|jdS)z Returns the Y coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 2) >>> p.y 1 rr_rqs r,rz Point3D.yDrr.c|jdS)z Returns the Z coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 1) >>> p.z 1 r"r_rqs r,rz Point3D.zSrr.)rrrN)rrr)rrrrrrErbrrrrrrr rrr rerrrSr.r,rDrD0sE**X%*22222 !+!+\!+F(((6JJJ0$($($(L55556 H H H;;;;.  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