+ i50Rt^RIt^RIHtHtHt^RIHt^RIH t ^RI H t ^RI H t ^RIHtHt^RIHt^R IHt^R IHt^R IHtHt^R IHt^R IHt^RIHtH t ^RI!H"t"H#t#H$t$^RI%H&t&^RI'H(t(!RR]&4t)!RR])4t*!RR])4t+R#)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_dpscra]tRt^*toRtRtRtRtRtRt Rt Rt R t R t R tR tR tRtRtRt]R4t]R4t]R4t]R4tRtRtRtR)RltRtRt Rt!]R4t"Rt#]R4t$]R4t%R t&]R!4t']R"4t(]R#4t)R$t*R%t+]R&4t,R't-Vt.R(#)*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 VPR\P4pVPRR4p\V4^8Xd V^,MTp\ V\ 4'd.Rp\V4VPR\V448XdV#\ V4'g-\\RP\V4444h\V4^8Xd@VPRR4'd(\P3VPR4,p\V!pVPR\V44p\V4^8d\\R44h\V4V8wdkRPV\V4V4pVR8XdMFVR 8Xd \V4hVR 8Xd\P !V^R 7M\\R 44h\#WVR4'd \R 4h\";QJdRV4F 'gK RM RM !RV44'd \R4h\$;QJdRV4F 'dK RM RM !RV44'g \R4hVRV\P3V\V4, ,,pV'dJTP'VP)\*4Uu/uFpV\-\/VRR74bK up4p\V4^8XdRVR&\1V/VB#\V4^8XdRVR&\3V/VB#\4P6!V.VO5!#uupi)evaluateon_morphignoreFdimz< Expecting sequence of coordinates, not `{}`Nz[ 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.c3v"TF/qP;'d\V4PRJxK1 R#5i)FN) is_numberr is_zero.0as& t/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/geometry/point.py Point.__new__..s*Fv!{{55r!u}}55vs9 9Tz(Imaginary coordinates are not permitted.c3B"TFp\V\4xK R#5iN) isinstancerr&s& r)r*r+s71:a&&sz,Coordinates must be valid SymPy expressions.)rational_nocheck)getrrlenr.rr TypeErrorrformatrrZeror ValueErrorwarningsr!anyallxreplaceatomsrr r Point2DPoint3Dr__new__) clsargskwargsrrcoordsrmessagefs &*, r)r> 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 3FvF333FvF F FGH Hs77sss7777JK K 3V+< == __ ,,u-&/-Q8Ia$788-&/0F v;! !%F: F-f- - [A !%F: F-f- -%%c3F33&/s# M"c d\^.\V4,4p\PW4#)z7Returns the distance between this point and the origin.)rr2distance)selforigins& r)__abs__ Point.__abs__s%s3t9}%~~f++c \PV\VRR74wr#\ Y#4UUu.uFwrE\ YE,4NK ppp\TRR7# \d\RP T44hi;iuuppi)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_dimensionr3r r4zipr )rHothersor(brBs&& r)__add__ Point.__add__s> ]--dE%%4PQDA/2!i8ida(15/i8Ve,,  ] M T TUZ [\ \ ]9s#AB&BcWP9#r-r@rHitems&&r) __contains__Point.__contains__syy  rLc \V4pVPUu.uFp\W!, 4NK pp\VRR7#uupi)z'Divide point's coordinates by a factor.FrNrr@r r)rHdivisorxrBs&& r) __truediv__Point.__truediv__s='"/3yy9y!(19%y9Ve,,:Ac\V\4'd.\VP4\VP48wdR#VPVP8H#)F)r.rr2r@rHrQs&&r)__eq__ Point.__eq__s<%''3tyy>S_+LyyEJJ&&rLc(VPV,#r-rX)rHkeys&&r) __getitem__Point.__getitem__syy~rLc,\VP4#r-)hashr@rHs&r)__hash__Point.__hash__sDIIrLc6VPP4#r-)r@__iter__rns&r)rrPoint.__iter__syy!!##rLc,\VP4#r-)r2r@rns&r)__len__ Point.__len__s499~rLc \V4pVPUu.uFp\W!,4NK pp\VRR7#uupi)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 FrNr^)rHfactorr`rBs&& r)__mul__ Point.__mul__s>6.2ii8i(18$i8Ve,,9rcc $VPV4#)z)Multiply a factor by point's coordinates.)ry)rHrxs&&r)__rmul__Point.__rmul__s||F##rLc \VPUu.uFq)NK pp\VRR7#uupi)zNegate the point.FrN)r@r)rHr`rBs& r)__neg__ Point.__neg__s,"ii(i"i(Ve,,)s )c <YUu.uFq")NK up,#uupi)zHSubtract two points, or subtract a factor from this point's coordinates.)rHrQr`s&& r)__sub__ Point.__sub__#s!5)5ar5))))s c za\VRR4oVPRS4oSf\RV44o\;QJdV3RlV4F 'dK RM RM!V3RlV44'd \ V4#SVR&VPRR 4VR&VUu.uFp\ V3/VBNK up#uupi) znEnsure that points have the same dimension. By default `on_morph='warn'` is passed to the `Point` constructor._ambient_dimensionNrc38"TFqPxK R#5ir-ambient_dimensionr'is& r)r*-Point._normalize_dimension..3s:6a))6sc3@<"TFqPS8HxK R#5ir-r)r'rrs& r)r*r4s:6a""c)6sFTrr!)getattrr1maxr9listr)r?pointsrArrs&*, @r)rOPoint._normalize_dimension(s c/6jj$ ;:6::C 3:6:333:6: : :< u #ZZ F;z,23Fqa"6"F333s B8c Z\V4^8XdR#\P!VUu.uFp\V4NK up!pV^,pVR,Uu.uF qV, NK pp\VUu.uFqPNK up4pVP RR7#uupiuupiuupi)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.:NNcxVP'd\VP^44R8#VP#)g-q=)r$absnr%)r`s&r)#Point.affine_rank..Ms*#$;;;CAK%  =AII =rL) iszerofunc)r2rrOrr@rank)r@rrrIms* r) affine_rankPoint.affine_rank:s t9>I++-E1eAh-EF&,Rj1jf**j1 F+FqFFF+ ,vv$>v? ? .F1+sBB#0B(c .\VR\V44#)z$Number of components this point has.r)rr2rns&r)rPoint.ambient_dimensionPst13t9==rLc \V4^8:dR#VP!VUu.uFp\V4NK up!pV^,P^8XdR#\ \ V44p\P !V!^8*#uupi)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 T)r2rOrrrrr)r?rrs&* r) are_coplanarPoint.are_coplanarUsvH v;! ))f+EfE!Hf+EF !9 & &! +d6l#  &)Q.. ,FsBc  \V\4'g\WPR7p\V\4'dD\P V\V44wr#\\R\W#44!4#\VRR4pVf\ R\ V4,4hV!V4# \d\ R\ T4,4hi;i)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) rz'not recognized as a GeometricEntity: %sc3D"TFwrW, ^,xK R#5irNrr'r(rTs& r)r*!Point.distance..s?YTQquqjjYs rGNz,distance between Point and %s is not defined) r.rrrr3typerOr rrPr)rHrQrRprGs&& r)rGPoint.distancesN%00 Ye)?)?@ eU # #--dE%LADA?SY?@A A5*d3  JTRW[XY Y~ Y IDQVK WXX Ys B;;'C"c l\V4'g \V4p\R\W44!#)z.Return dot product of self with another Point.c36"TFwrW,xK R#5ir-rrs& r)r*Point.dot..s2\TQQSS\s)rrrrP)rHrs&&r)dot Point.dots)1~~aA2S\233rLc \V\4'd\V4\V48wdR#\;QJd%R\ W44F 'dK R# R#!R\ W444#)z8Returns whether the coordinates of self and other agree.Fc3H"TFwrVPV4xK R#5ir-)equalsrs& r)r*Point.equals..s<+;41188A;;+; "T)r.rr2r9rPres&&r)r Point.equalssV%''3t9E +Bs<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)rr@evalfr)rHprecoptionsdpsr`rBs&&, r) _eval_evalfPoint._eval_evalfsK8$59YY?Y''+C+7+Y?f-u--@sAc \V\4'g \V4p\V\4'd2W8XdV.#\PW4wr#W 8Xd W#8XdV.#.#VP V4#)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)] )r.rrrO intersection)rHrQp1p2s&& 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 )rrOrrr)rHr@rrs&* r) is_collinearPoint.is_collinear s]B4++-G1eAh-GHd6l#  &)Q...HsA#c  V3V,p\P!VUu.uFp\V4NK up!p\\V44p\P!V!^8:gR#V^,pVUu.uF qUV, NK pp\ VUu.uF%p\V4VP V4.,NK' up4pVP4wrx\V4V9dR#R#uupiuupiuupi)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 FT) rrOrrrrrrrefr2) rHr@rrrIrmatrpivotss &* r) is_concyclicPoint.is_concyclic3sL4++-G1eAh-GHd6l#  &)Q.&,-ff**f- F;Fqd1gq **F;<xxz  v;f $.H . .s*'QDy'sTN)r@rr8)rHr`nonzeros& r)r% Point.is_zerosV*.3A<<3 w<< 3*'*333*'* * * 4sA6c "\P#)z Treating a Point as a Line, this returns 0 for the length of a Point. Examples ======== >>> from sympy import Point >>> p = Point(0, 1) >>> p.length 0 )rr5rns&r)length Point.lengths vv rLc  \PV\V44wr!\\W!4UUu.uF+wr4\W4,\P ,4NK- upp4#uuppi)aThe midpoint between self and point p. Parameters ========== p : Point Returns ======= midpoint : Point See Also ======== sympy.geometry.line.Segment.midpoint Examples ======== >>> from sympy import Point >>> p1, p2 = Point(1, 1), Point(13, 5) >>> p1.midpoint(p2) Point2D(7, 3) )rrOrPr rHalf)rHrrRr(rTs&& r)midpointPoint.midpointsN6))$a93q9E941hqvv~.9EFFEs1A. c >\^.\V4,RR7#)zGA point of all zeros of the same ambient dimension as the current pointFrN)rr2rns&r)rI Point.originsaST]U33rLc lVPpV^,P'd#\^.V^, ^.,,4#V^,P'd$\^^.V^, ^.,,4#\V^,)V^,.V^, ^.,,4#)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 )rr%r)rHrs& r)orthogonal_directionPoint.orthogonal_directions $$ 7???!a!},- - 7???!A#'A3./ /tAwhQ(C!GaS=899rLc \P\V4\V44wrVP'd \R4hWP V4VP V4, ,#)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.)rrOr%r6r)r(rTs&&r)project Point.projectsTD))%(E!H= 999AB B%%(QUU1X%&&rLc v\PV\V44wr!\R\W!44!#)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 c3H"TFwr\W, 4xK R#5ir-rrs& r)r*)Point.taxicab_distance..%s6IDASZZIr)rrOrrPrHrrRs&& r)taxicab_distancePoint.taxicab_distances1<))$a96CI677rLc \PV\V44wr!VP'dVP'd \R4h\ R\ W!44!#)amThe Canberra Distance from self to point p. Returns the weighted sum of horizontal and vertical distances to point p. Parameters ========== p : Point Returns ======= canberra_distance : The weighted sum of horizontal and vertical distances to point p. The weight used is the sum of absolute values of the coordinates. Examples ======== >>> from sympy import Point >>> p1, p2 = Point(1, 1), Point(3, 3) >>> p1.canberra_distance(p2) 1 >>> p1, p2 = Point(0, 0), Point(3, 3) >>> p1.canberra_distance(p2) 2 Raises ====== ValueError when both vectors are zero. See Also ======== sympy.geometry.point.Point.distance rc3"TF8wr\W, 4\V4\V4,, xK: R#5ir-rrs& r)r**Point.canberra_distance..Ss,J c!%j#a&3q6/22 sAA)rrOr%r6rrPrs&& r)canberra_distancePoint.canberra_distance'sOR))$a9 <<rJrUr[rarfrjrorrruryr|rr classmethodrO staticmethodrpropertyrrrGrrrrrrrrr%rrrIrrrrr__static_attributes____classdictcell__ __classdict__s@r)rr*sy>@HF4P, %-N!- ' $->$- * 44"??*>>+/+/Z2h4 =.@'(R$/L6p&  G<44 ::2$'$'L8B,L\  rLrca]tRtRtoRt^tRR/RltRt]R4t RR lt RR lt R t RR lt ]R 4t]R4t]R4tRtVtR#)r<i\a%A point in a 2-dimensional Euclidean space. Parameters ========== coords A sequence of 2 coordinate values. Attributes ========== x y length Raises ====== TypeError When trying to add or subtract points with different dimensions. When trying to create a point with more than two dimensions. When `intersection` is called with object other than a Point. See Also ======== sympy.geometry.line.Segment : Connects two Points Examples ======== >>> from sympy import Point2D >>> from sympy.abc import x >>> Point2D(1, 2) Point2D(1, 2) >>> Point2D([1, 2]) Point2D(1, 2) >>> Point2D(0, x) Point2D(0, x) Floats are automatically converted to Rational unless the evaluate flag is False: >>> Point2D(0.5, 0.25) Point2D(1/2, 1/4) >>> Point2D(0.5, 0.25, evaluate=False) Point2D(0.5, 0.25) r0FcbV'g^VR&\V/VBp\P!V.VO5!#)rrrrr>r?r0r@rAs&$*,r)r>Point2D.__new__3F5M$)&)D%%c1D11rLc W8H#r-rrYs&&r)r[Point2D.__contains__ |rLc ^VPVPVPVP3#)zgReturn a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure. )r`yrns&r)boundsPoint2D.boundss#//rLNc \V4p\V4pTpVe\V^R7pWR,pVPwrg\W6,WG,, WF,W7,,4pVe WR, pV#)zRotate ``angle`` radians counterclockwise about Point ``pt``. See Also ======== translate, scale Examples ======== >>> from sympy import Point2D, pi >>> t = Point2D(1, 0) >>> t.rotate(pi/2) Point2D(0, 1) >>> t.rotate(pi/2, (2, 0)) Point2D(2, -1) r)rrrr@)rHangleptcrRrr`rs&&& r)rotatePoint2D.rotatesk& J J  >rq!B HBww 139acACi ( > HB rLc V'dP\V^R7pVP!V)P!PW4P!VP!#\VPV,VP V,4#)a}Scale the coordinates of the Point by multiplying by ``x`` and ``y`` after subtracting ``pt`` -- default is (0, 0) -- and then adding ``pt`` back again (i.e. ``pt`` is the point of reference for the scaling). See Also ======== rotate, translate Examples ======== >>> from sympy import Point2D >>> t = Point2D(1, 1) >>> t.scale(2) Point2D(2, 1) >>> t.scale(2, 2) Point2D(2, 2) r)r translater@scaler`r)rHr`rrs&&&&r)r! Point2D.scales_, rq!B>>RC::.44Q:DDbggN NTVVAXtvvax((rLc  VP'dVPR8Xg \R4hVPwr#\ \ ^^W#^.4V,P 4^,R,!#)zReturn the point after applying the transformation described by the 3x3 Matrix, ``matrix``. See Also ======== sympy.geometry.point.Point2D.rotate sympy.geometry.point.Point2D.scale sympy.geometry.point.Point2D.translate zmatrix must be a 3x3 matrix:NrN)r$) is_Matrixshaper6r@rrtolist)rHmatrixr`rs&& r) transformPoint2D.transformsb   V\\V%;:; ;yyvaQ1I.v5==?B2FGGrLc ^\VPV,VPV,4#)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) )rr`r)rHr`rs&&&r)r Point2D.translates!*TVVaZ!,,rLc VP#)z Returns the two coordinates of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.coordinates (0, 1) rXrns&r) coordinatesPoint2D.coordinatesyyrLc (VP^,#)zz Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.x 0 rXrns&r)r` Point2D.xyy|rLc (VP^,#)zz Returns the Y coordinate of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.y 1 rXrns&r)r Point2D.y"r3rLrr-)rrN)r6)rrrrrrr>r[rrrr!r)r r.r`rrr r s@r)r<r<\s0d2U2 00@)6 H-.      rLr<ca]tRtRtoRt^tRR/RltRt]R4t Rt R t R t RR lt R tRRlt]R4t]R4t]R4t]R4tRtVtR #)r=i1aA 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) r0FcbV'g^VR&\V/VBp\P!V.VO5!#)r$rrrs&$*,r)r>Point3D.__new__arrLc W8H#r-rrYs&&r)r[Point3D.__contains__grrLc *\P!V!#)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**rLc ,VPV4p\\RV4!4pVPVP, V, VPVP, V, VP VP , V, .#)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] c32"TF q^,xK R#5irrrs& r)r*+Point3D.direction_cosine..s'Q!ttQs)direction_ratior rr`rz)rHpointr(rTs&& r)direction_cosinePoint3D.direction_cosinesq,   ' 'Q'( )466!Q&$&&(8A'=466!Q&( (rLc VPVP, VPVP, VPVP, .#)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] )r`rrC)rHrDs&&r)rBPoint3D.direction_ratios8,466!EGGdff$4uww7GIIrLc \V\4'g\V^R7p\V\4'd W8XdV.#.#VP V4#)aThe intersection between this point and another GeometryEntity. Parameters ========== other : GeometryEntity or sequence of coordinates Returns ======= intersection : list of Points Notes ===== The return value will either be an empty list if there is no intersection, otherwise it will contain this point. Examples ======== >>> from sympy import Point3D >>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, 0, 0) >>> p1.intersection(p2) [] >>> p1.intersection(p3) [Point3D(0, 0, 0)] r)r.rrr=rres&&r)rPoint3D.intersectionsL<%00%Q'E eW % %}v I!!$''rLNc .V'dO\V4pVP!V)P!PWV4P!VP!#\VPV,VP V,VP V,4#)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) )r=r r@r!r`rrC)rHr`rrCrs&&&&&r)r! Point3D.scalesh, B>>RC::.44Q1=GGQ Qtvvax466!844rLc  VP'dVPR8Xg \R4hVPwr#p\ V4p\ \ ^^W#V^.4V,P4^,R,!#)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 zmatrix must be a 4x4 matrix:Nr$N)rN)r%r&r6r@rr=rr')rHr(r`rrCrs&& r)r)Point3D.transformsp   V\\V%;:; ;))a f 1qQl3A5==?B2FGGrLc \VPV,VPV,VPV,4#)aShift the Point by adding x and y to the coordinates of the Point. See Also ======== scale Examples ======== >>> from sympy import Point3D >>> t = Point3D(0, 1, 1) >>> t.translate(2) Point3D(2, 1, 1) >>> t.translate(2, 2) Point3D(2, 3, 1) >>> t + Point3D(2, 2, 2) Point3D(2, 3, 3) )r=r`rrC)rHr`rrCs&&&&r)r Point3D.translates+*tvvz466A:tvvz::rLc VP#)z Returns the three coordinates of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 2) >>> p.coordinates (0, 1, 2) rXrns&r)r.Point3D.coordinates(r0rLc (VP^,#)z} Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 3) >>> p.x 0 rXrns&r)r` Point3D.x7r3rLc (VP^,#)z} Returns the Y coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 2) >>> p.y 1 rXrns&r)r Point3D.yFr3rLc (VP^,#)z} Returns the Z coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 1) >>> p.z 1 rXrns&r)rC Point3D.zUr3rLr)rrrN)r6r6r6)rrrrrrr>r[rr=rErBrr!r)r rr.r`rrCrr r s@r)r=r=1s+Z2U2 !+!+F(6J0$(L56 H;.        rLr=),rr7 sympy.corerrrsympy.core.addrsympy.core.containersrsympy.core.numbersrsympy.core.parametersrsympy.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.libmpfrrr<r=rrLr)rjsq&'''$37393=!07CC"+o  No  dSeSjqeqrL