\h5.SrSSKrSSKJrJrJr SSKJr SSKJ r SSK J r SSK J r SSKJrJr SS KJr SS KJr SS KJr SS KJrJr SS KJr SSKJr SSKJrJ r SSK!J"r"J#r#J$r$ SSK%J&r& SSK'J(r( "SS\&5r)"SS\)5r*"SS\)5r+g)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_dpscf\rSrSrSrSrSrSrSrSr Sr S r S r S r S rS rSrSrSrSr\S5r\S5r\S5r\S5rSrSrSrS*SjrSrSr Sr!\S5r"Sr#\S5r$\S 5r%S!r&\S"5r'\S#5r(\S$5r)S%r*S&r+\S'5r,S(r-g))+Point*aA point in a n-dimensional Euclidean space. Parameters ========== coords : sequence of n-coordinate values. In the special case where n=2 or 3, a Point2D or Point3D will be created as appropriate. evaluate : if `True` (default), all floats are turn into exact types. dim : number of coordinates the point should have. If coordinates are unspecified, they are padded with zeros. on_morph : indicates what should happen when the number of coordinates of a point need to be changed by adding or removing zeros. Possible values are `'warn'`, `'error'`, or `ignore` (default). No warning or error is given when `*args` is empty and `dim` is given. An error is always raised when trying to remove nonzero coordinates. Attributes ========== length origin: A `Point` representing the origin of the appropriately-dimensioned space. Raises ====== TypeError : When instantiating with anything but a Point or sequence ValueError : when instantiating with a sequence with length < 2 or when trying to reduce dimensions if keyword `on_morph='error'` is set. See Also ======== sympy.geometry.line.Segment : Connects two Points Examples ======== >>> from sympy import Point >>> from sympy.abc import x >>> Point(1, 2, 3) Point3D(1, 2, 3) >>> Point([1, 2]) Point2D(1, 2) >>> Point(0, x) Point2D(0, x) >>> Point(dim=4) Point(0, 0, 0, 0) Floats are automatically converted to Rational unless the evaluate flag is False: >>> Point(0.5, 0.25) Point2D(1/2, 1/4) >>> Point(0.5, 0.25, evaluate=False) Point2D(0.5, 0.25) Tc URS[R5nURSS5n[U5S:XaUSOUn[ U[ 5(a,Sn[U5URS[U55:XaU$[ U5(d,[[SR[U5555e[U5S:Xa:URSS5(a#[R4URS5-n[U6nURS[U55n[U5S :a[[S 55e[U5U:wadS RU[U5U5nUS:XaOAUS :Xa [U5eUS :Xa[R "US S9 O[[S55e[#XVS5(a [S5e[#SU55(a [S5e[%SU55(d [S5eUSU[R4U[U5- --nU(aGUR'UR)[*5Vs0sHnU[-[/USS95_M sn5n[U5S :XaSUS'[1U0UD6$[U5S:XaSUS'[3U0UD6$[4R6"U/UQ76$s snf)Nevaluateon_morphignorerrFdimz< Expecting sequence of coordinates, not `{}`z[ Point requires 2 or more coordinates or keyword `dim` > 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.c3t# UH.oR=(a [U5RSLv M0 g7f)FN) is_numberris_zero.0as L/var/www/auris/envauris/lib/python3.13/site-packages/sympy/geometry/point.py Point.__new__..s'Fv!{{5r!u}}55vs68z(Imaginary coordinates are not permitted.c3B# UHn[U[5v M g7fN) isinstancerr*s r-r.r/s71:a&&sz,Coordinates must be valid SymPy expressions.T)rational_nocheck)getr rlenr2rr TypeErrorrformatrrZeror ValueErrorwarningsr%anyallxreplaceatomsrr r Point2DPoint3Dr__new__) clsargskwargsrr coordsr"messagefs r-rC Point.__new__ms::j*;*D*DE::j(3 INa fe $ $H6{fjjF << 6""J(?(.y/@(ACD D v;!  5$ 7 7ffYvzz%00FjjF , v;?Z)&'( ( v;# ()/F S)I 8#W$ ))V# g!4 -/"011 vd|  EF F FvF F FGH H7777JK K 3V+< == __ ,,u-&/-Q8Ia$788-&/0F v;! !%F: F-f- - [A !%F: F-f- -%%c3F33&/s8K2c\[S/[U5-5n[RX5$)z7Returns the distance between this point and the origin.r)rr7distance)selforigins r-__abs__ Point.__abs__s%s3t9}%~~f++c[RU[USS95up#[ X#5VVs/sHupE[ XE-5PM nnn[USS9$![a [SR U55ef=fs snnf)aAdd 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 object)r_normalize_dimensionr8r r9zipr )rMothersor,brGs r-__add__ Point.__add__s> ]--dE%%4PQDA/2!i8ida(15/i8Ve,,  ] M T TUZ [\ \ ]9s AA>%A;cXR;$r1rErMitems r- __contains__Point.__contains__syy  rQc[U5nURVs/sHn[X!- 5PM nn[USS9$s snf)z'Divide point's coordinates by a factor.FrSrrEr r)rMdivisorxrGs r- __truediv__Point.__truediv__s='"/3yy9y!(19%y9Ve,,:=c[U[5(a,[UR5[UR5:wagURUR:H$)NF)r2rr7rErMrVs r-__eq__ Point.__eq__s<%''3tyy>S_+LyyEJJ&&rQc URU$r1r])rMkeys r- __getitem__Point.__getitem__syy~rQc,[UR5$r1)hashrErMs r-__hash__Point.__hash__sDIIrQc6URR5$r1)rE__iter__rss r-rwPoint.__iter__syy!!##rQc,[UR5$r1)r7rErss r-__len__ Point.__len__s499~rQc[U5nURVs/sHn[X!-5PM nn[USS9$s snf)aMultiply 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 FrSrc)rMfactorrerGs r-__mul__ Point.__mul__s>6.2ii8i(18$i8Ve,,9rhc$URU5$)z)Multiply a factor by point's coordinates.)r~)rMr}s r-__rmul__Point.__rmul__s||F##rQcXURVs/sHo*PM nn[USS9$s snf)zNegate the point.FrS)rEr)rMrerGs r-__neg__ Point.__neg__s,"ii(i"i(Ve,,)s 'c6XVs/sHo"*PM sn-$s snf)zHSubtract two points, or subtract a factor from this point's coordinates.)rMrVres r-__sub__ Point.__sub__#s!5)5ar5))))s c.^[USS5mURST5mTc[SU55m[U4SjU55(a [ U5$TUS'URSS5US'UVs/sHn[ U40UD6PM sn$s snf)znEnsure that points have the same dimension. By default `on_morph='warn'` is passed to the `Point` constructor._ambient_dimensionNr"c38# UHoRv M g7fr1ambient_dimensionr+is r-r.-Point._normalize_dimension..3s:6a))6sc3@># UHoRT:Hv M g7fr1r)r+rr"s r-r.r4s:6a""c)6sr r%)getattrr6maxr>listr)rDpointsrFrr"s @r-rTPoint._normalize_dimension(s c/6jj$ ;:6::C :6: : :< u #ZZ F;z,23Fqa"6"F333s9Bc<[U5S:Xag[R"UVs/sHn[U5PM sn6nUSnUSSVs/sHoU- PM nn[UVs/sHoRPM sn5nUR SS9$s snfs snfs snf)a?The affine rank of a set of points is the dimension of the smallest affine space containing all the points. For example, if the points lie on a line (and are not all the same) their affine rank is 1. If the points lie on a plane but not a line, their affine rank is 2. By convention, the empty set has affine rank -1.rrNcvUR(a[URS55S:$UR$)Nr#g-q=)r(absnr))res r-#Point.affine_rank..Ms(#$;;CAK%  =AII =rQ) iszerofunc)r7rrTrrErank)rErrrNms r- affine_rankPoint.affine_rank:s t9>++-E1eAh-EF&,QRj1jf*j1 F+FqFFF+ ,vv$>v? ? .F1+sB B#Bc.[US[U55$)z$Number of components this point has.r)rr7rss r-rPoint.ambient_dimensionPst13t9==rQc[U5S::agUR"UVs/sHn[U5PM sn6nUSRS:Xag[ [ U55n[R "U6S:*$s snf)a1Return 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 rTrr#)r7rTrrrrr)rDrrs r- are_coplanarPoint.are_coplanarUsvH v;! ))f+EfE!Hf+EF !9 & &! +d6l#  &)Q.. ,FsA8c [U[5(d[XRS9n[U[5(aB[R U[U55up#[[S[X#5565$[USS5nUc[ S[ U5-5eU"U5$![a [ S[ U5-5ef=f)aThe 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: %sc34# UHupX- S-v M g7fr#Nrr+r,rYs r-r.!Point.distance..s?YTQquqjYsrLNz,distance between Point and %s is not defined) r2rrrr8typerTr rrUr)rMrVrWprLs r-rLPoint.distancesN%00 Ye)?)?@ eU # #--dE%LADA?SY?@A A5*d3  JTRW[XY Y~ Y IDQVK WXX Ys B00"Cch[U5(d [U5n[S[X556$)z.Return dot product of self with another Point.c3.# UH upX-v M g7fr1rrs r-r.Point.dot..s2\TQQS\s)rrrrU)rMrs r-dot Point.dots)1~~aA2S\233rQc[U[5(a[U5[U5:wag[S[ X555$)z8Returns whether the coordinates of self and other agree.Fc3H# UHupURU5v M g7fr1)equalsrs r-r.Point.equals..s<+;41188A;;+;s ")r2rr7r>rUrjs r-r Point.equalss9%''3t9E +B<3t+;<<>> 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) rrFr)rrEevalfr)rMprecoptionsdpsrerGs r- _eval_evalfPoint._eval_evalfsK8$59YY?Y''+C+7+Y?f-u--@sAc[U[5(d [U5n[U[5(a.X:XaU/$[RX5up#X :XaX#:XaU/$/$UR U5$)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 Point >>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 0) >>> p1.intersection(p2) [] >>> p1.intersection(p3) [Point2D(0, 0)] )r2rrrT intersection)rMrVp1p2s r-rPoint.intersectionsk<%00%LE eU # #}v //>> 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 r)rrTrrr)rMrErrs r- is_collinearPoint.is_collinear s]B4++-G1eAh-GHd6l#  &)Q...HsAc U4U-n[R"UVs/sHn[U5PM sn6n[[U55n[R"U6S::dgUSnUVs/sHoUU- PM nn[ UVs/sH!n[U5UR U5/-PM# sn5nUR5upx[U5U;aggs snfs snfs snf)agDo `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 r#FrT) rrTrrrrrrrefr7) rMrErrrNrmatrpivotss r- is_concyclicPoint.is_concyclic3sL4++-G1eAh-GHd6l#  &)Q.&,-ff*f- F;Fqd1gq *F;<xxz  v;f $.H . .s*'QDy'sNT)rErr=)rMrenonzeros r-r) Point.is_zerosF*.3A<<3 w<< *'* * * 4sAc"[R$)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:rss r-length Point.lengths vv rQc [RU[U55up![[X!5VVs/sH#up4[X4-[R -5PM% snn5$s snnf)aThe midpoint between self and point p. Parameters ========== p : Point Returns ======= midpoint : Point See Also ======== sympy.geometry.line.Segment.midpoint Examples ======== >>> from sympy import Point >>> p1, p2 = Point(1, 1), Point(13, 5) >>> p1.midpoint(p2) Point2D(7, 3) )rrTrUr rHalf)rMrrWr,rYs r-midpointPoint.midpointsN6))$a93q9E941hqvv~.9EFFEs*A' c0[S/[U5-SS9$)zGA point of all zeros of the same ambient dimension as the current pointrFrS)rr7rss r-rN Point.originsaST]U33rQcURnUSR(a[S/US- S/--5$USR(a[SS/US- S/--5$[US*US/US- S/--5$)aReturns 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_directionPoint.orthogonal_directions $$ 7??!a!},- - 7??!A#'A3./ /tAwhQ(C!GaS=899rQc[R[U5[U55upUR(a [S5eXR U5UR U5- -$)aProject the point `a` onto the line between the origin and point `b` along the normal direction. Parameters ========== a : Point b : Point Returns ======= p : Point See Also ======== sympy.geometry.line.LinearEntity.projection Examples ======== >>> from sympy import Line, Point >>> a = Point(1, 2) >>> b = Point(2, 5) >>> z = a.origin >>> p = Point.project(a, b) >>> Line(p, a).is_perpendicular(Line(p, b)) True >>> Point.is_collinear(z, p, b) True "Cannot project to the zero vector.)rrTr)r;r)r,rYs r-project Point.projectsRD))%(E!H= 99AB B%%(QUU1X%&&rQct[RU[U55up![S[X!556$)aThe 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@# UHup[X- 5v M g7fr1rrs r-r.)Point.taxicab_distance..%s6IDASZZIs)rrTrrUrMrrWs r-taxicab_distancePoint.taxicab_distances1<))$a96CI677rQc[RU[U55up!UR(aUR(a [S5e[ S[ X!556$)amThe 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 rc3p# UH,up[X- 5[U5[U5-- v M. g7fr1rrs r-r.*Point.canberra_distance..Ss)J c!%j#a&3q6/2 s46)rrTr)r;rrUrs r-canberra_distancePoint.canberra_distance'sKR))$a9 <@HF4P, %-N!- ' $->$- * 44"??*>>+/+/Z2h4 =.@'(R$/L6p&  G<44 ::2$'$'L8B,L\  rQrc\rSrSrSrSrSS.SjrSr\S5r SS jr SS jr S r SS jr \S5r\S5r\S5rSrg )rAi\a%A point in a 2-dimensional Euclidean space. Parameters ========== coords A sequence of 2 coordinate values. Attributes ========== x y length Raises ====== TypeError When trying to add or subtract points with different dimensions. When trying to create a point with more than two dimensions. When `intersection` is called with object other than a Point. See Also ======== sympy.geometry.line.Segment : Connects two Points Examples ======== >>> from sympy import Point2D >>> from sympy.abc import x >>> Point2D(1, 2) Point2D(1, 2) >>> Point2D([1, 2]) Point2D(1, 2) >>> Point2D(0, x) Point2D(0, x) Floats are automatically converted to Rational unless the evaluate flag is False: >>> Point2D(0.5, 0.25) Point2D(1/2, 1/4) >>> Point2D(0.5, 0.25, evaluate=False) Point2D(0.5, 0.25) r#Fr4c^U(dSUS'[U0UD6n[R"U/UQ76$)Nr#r"rrrCrDr4rErFs r-rCPoint2D.__new__3F5M$)&)D%%c1D11rQc X:H$r1rr^s r-r`Point2D.__contains__ |rQc^URURURUR4$)zgReturn a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure. )reyrss r-boundsPoint2D.boundss#//rQNc[U5n[U5nUnUb[USS9nXR-nURupg[X6-XG-- XF-X7--5nUbXR- nU$)zRotate ``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) r#r)rrrrE)rMangleptcrWrrers r-rotatePoint2D.rotatesk& J J  >rq!B HBww 139acACi ( > HB rQcU(aJ[USS9nUR"U*R6RX5R"UR6$[URU-UR U-5$)a}Scale 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 translaterEscalerer)rMrerrs r-r! Point2D.scales_, rq!B>>RC::.44Q:DDbggN NTVVAXtvvax((rQc UR(aURS:Xd [S5eURup#[ [ SSX#S/5U-R 5SSS6$)zReturn the point after applying the transformation described by the 3x3 Matrix, ``matrix``. See Also ======== sympy.geometry.point.Point2D.rotate sympy.geometry.point.Point2D.scale sympy.geometry.point.Point2D.translate )r5r5zmatrix must be a 3x3 matrixrr5rNr#) is_Matrixshaper;rErrtolist)rMmatrixrers r- transformPoint2D.transformsa  V\\V%;:; ;yyvaQ1I.v5==?B2AFGGrQcN[URU-URU-5$)a4Shift the Point by adding x and y to the coordinates of the Point. See Also ======== sympy.geometry.point.Point2D.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) )rrer)rMrers r-r Point2D.translates!*TVVaZ!,,rQcUR$)z Returns the two coordinates of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.coordinates (0, 1) r]rss r- coordinatesPoint2D.coordinatesyyrQc URS$)zz Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.x 0 rr]rss r-re Point2D.xyy|rQc URS$)zz Returns the Y coordinate of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.y 1 rr]rss r-r Point2D.y"r2rQrr1)rrN)rr)rrrrrrrCr`r rrr!r(r r-rerr rrQr-rArA\s0d%*2 00@)6 H-.      rQrAc\rSrSrSrSrSS.SjrSr\S5r S r S r S r SS jr SrSSjr\S5r\S5r\S5r\S5rSrg )rBi1aA point in a 3-dimensional Euclidean space. Parameters ========== coords A sequence of 3 coordinate values. Attributes ========== x y z length Raises ====== TypeError When trying to add or subtract points with different dimensions. When `intersection` is called with object other than a Point. Examples ======== >>> from sympy import Point3D >>> from sympy.abc import x >>> Point3D(1, 2, 3) Point3D(1, 2, 3) >>> Point3D([1, 2, 3]) Point3D(1, 2, 3) >>> Point3D(0, x, 3) Point3D(0, x, 3) Floats are automatically converted to Rational unless the evaluate flag is False: >>> Point3D(0.5, 0.25, 2) Point3D(1/2, 1/4, 2) >>> Point3D(0.5, 0.25, 3, evaluate=False) Point3D(0.5, 0.25, 3) r5Fr c^U(dSUS'[U0UD6n[R"U/UQ76$)Nr5r"rrs r-rCPoint3D.__new__arrQc X:H$r1rr^s r-r`Point3D.__contains__grrQc([R"U6$)aKIs 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_collinearPoint3D.are_collinearjsD!!6**rQcURU5n[[SU565nURUR- U- URUR- U- UR UR - U- /$)a Gives the direction cosine between 2 points Parameters ========== p : Point3D Returns ======= list Examples ======== >>> from sympy import Point3D >>> p1 = Point3D(1, 2, 3) >>> p1.direction_cosine(Point3D(2, 3, 5)) [sqrt(6)/6, sqrt(6)/6, sqrt(6)/3] c3*# UH oS-v M g7frrrs r-r.+Point3D.direction_cosine..s'Q!tQs)direction_ratior rrerz)rMpointr,rYs r-direction_cosinePoint3D.direction_cosinesq,   ' 'Q'( )466!Q&$&&(8A'=466!Q&( (rQcURUR- URUR- URUR- /$)z 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] )rerrA)rMrBs r-r@Point3D.direction_ratios8,466!EGGdff$4uww7GIIrQc[U[5(d [USS9n[U[5(a X:XaU/$/$UR U5$)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)] r5r)r2rrrBrrjs r-rPoint3D.intersectionsL<%00%Q'E eW % %}v I!!$''rQNcU(aL[U5nUR"U*R6RXU5R"UR6$[URU-UR U-UR U-5$)a~Scale the coordinates of the Point by multiplying by ``x`` and ``y`` after subtracting ``pt`` -- default is (0, 0) -- and then adding ``pt`` back again (i.e. ``pt`` is the point of reference for the scaling). See Also ======== translate Examples ======== >>> from sympy import Point3D >>> t = Point3D(1, 1, 1) >>> t.scale(2) Point3D(2, 1, 1) >>> t.scale(2, 2) Point3D(2, 2, 1) )rBr rEr!rerrA)rMrerrArs r-r! Point3D.scalesh, B>>RC::.44Q1=GGQ Qtvvax466!844rQc UR(aURS:Xd [S5eURup#n[ U5n[ [ SSX#US/5U-R5SSS6$)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 )rLzmatrix must be a 4x4 matrixrrLrNr5)r$r%r;rErrBrr&)rMr'rerrArs r-r(Point3D.transformso  V\\V%;:; ;))a f 1qQl3A5==?B2AFGGrQcj[URU-URU-URU-5$)aShift the Point by adding x and y to the coordinates of the Point. See Also ======== 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) )rBrerrA)rMrerrAs r-r Point3D.translates+*tvvz466A:tvvz::rQcUR$)z Returns the three coordinates of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 2) >>> p.coordinates (0, 1, 2) r]rss r-r-Point3D.coordinates(r/rQc URS$)z} Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 3) >>> p.x 0 rr]rss r-re Point3D.x7r2rQc URS$)z} Returns the Y coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 2) >>> p.y 1 rr]rss r-r Point3D.yFr2rQc URS$)z} Returns the Z coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 1) >>> p.z 1 r#r]rss r-rA Point3D.zUr2rQr)rrrN)rrr)rrrrrrrCr`rr;rCr@rr!r(r r r-rerrAr rrQr-rBrB1s+Z%*2 !+!+F(6J0$(L56 H;.        rQrB),rr< sympy.corerrrsympy.core.addrsympy.core.containersrsympy.core.numbersrsympy.core.parametersr sympy.simplify.simplifyr r sympy.geometry.exceptionsr (sympy.functions.elementary.miscellaneousr $sympy.functions.elementary.complexesr(sympy.functions.elementary.trigonometricrrsympy.matricesrsympy.matrices.expressionsrsympy.utilities.iterablesrrsympy.utilities.miscrrrentityrmpmath.libmp.libmpfrrrArBrrQr-rhsq&'''$37393=!07CC"+o  No  dSeSjqeqrQ