o 8VaF@sdZddlZddlmZmZmZddlmZddlm Z ddl m Z m Z ddl mZddlmZdd lmZdd lmZdd lmZdd lmZdd lmZddlmZddlmZmZm Z ddl!m"Z"Gddde"Z#Gddde#Z$Gddde#Z%dS)aSGeometrical Points. Contains ======== Point Point2D Point3D When methods of Point require 1 or more points as arguments, they can be passed as a sequence of coordinates or Points: >>> from sympy.geometry.point import Point >>> Point(1, 1).is_collinear((2, 2), (3, 4)) False >>> Point(1, 1).is_collinear(Point(2, 2), Point(3, 4)) False N)SsympifyExpr) is_sequence)Tuple) nsimplifysimplify) GeometryError)sqrt)im)Matrix)Float)global_parameters)Add)uniq) filldedent func_name Undecidable)GeometryEntityc@sfeZdZdZdZddZddZddZd d Zd d Z d dZ ddZ ddZ ddZ ddZddZddZddZddZedd Zed!d"Zed#d$Zed%d&Zd'd(Zd)d*Zd+d,ZdLd.d/Zd0d1Zd2d3Zd4d5Z ed6d7Z!d8d9Z"ed:d;Z#edd?Z%ed@dAZ&edBdCZ'edDdEZ(dFdGZ)dHdIZ*edJdKZ+eZ,d-S)MPointaA 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) TcOs |dtj}|dd}t|dkr|dn|}t|tr.d}t||dt|kr.|St|s=ttd t |t|dkrR|ddrRt j f|d}t |}|dt|}t|d krjttd t||krd |t||}|dkr~n|d krt||d krt|nttdt||drtdtdd|Drtdtdd|Dstd|d|t j f|t|}|r|dd|tD}t|d krd|d<t|i|St|dkrd|d<t|i|Stj|g|RS)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 {}.errorwarnzf on_morph value should be 'error', 'warn' or 'ignore'.z&Nonzero coordinates cannot be removed.css|] }|jo t|VqdSN) is_numberr .0ar#6/usr/lib/python3/dist-packages/sympy/geometry/point.py z Point.__new__..z(Imaginary coordinates are not permitted.css|]}t|tVqdSr) isinstancerr r#r#r$r%z,Coordinates must be valid SymPy expressions.cSsi|] }|tt|ddqS)T)Zrational)rr)r!fr#r#r$ sz!Point.__new__..T_nocheck)getrrlenr'rr TypeErrorrformatrrZeror ValueErrorwarningsranyallZxreplaceZatomsr Point2DPoint3Dr__new__)clsargskwargsrrcoordsrmessager#r#r$r8js\            z Point.__new__cCstdgt|}t||S)z7Returns the distance between this point and the origin.r)rr.distance)selforiginr#r#r$__abs__s z Point.__abs__cCsZzt|t|dd\}}Wntytd|wddt||D}t|ddS)aGAdd 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 Frz+Don't know how to add {} and a Point objectcSsg|] \}}t||qSr#rr!r"br#r#r$ z!Point.__add__..)r_normalize_dimensionr/r r0zip)r?othersor<r#r#r$__add__s  z Point.__add__cCs ||jvSrr:r?itemr#r#r$ __contains__ zPoint.__contains__c(tfdd|jD}t|ddS)z'Divide point's coordinates by a factor.csg|]}t|qSr#rCr!xdivisorr#r$rFz%Point.__truediv__..FrBrr:r)r?rWr<r#rVr$ __truediv__s zPoint.__truediv__cCs.t|trt|jt|jkrdS|j|jkS)NF)r'rr.r:r?rJr#r#r$__eq__s z Point.__eq__cCs |j|SrrN)r?keyr#r#r$ __getitem__rRzPoint.__getitem__cC t|jSr)hashr:r?r#r#r$__hash__rRzPoint.__hash__cCs |jSr)r:__iter__rar#r#r$rcrRzPoint.__iter__cCr_r)r.r:rar#r#r$__len__rRz Point.__len__crS)a{Multiply point's coordinates by a factor. Notes ===== >>> from sympy.geometry.point import Point When multiplying a Point by a floating point number, the coordinates of the Point will be changed to Floats: >>> Point(1, 2)*0.1 Point2D(0.1, 0.2) If this is not desired, the `scale` method can be used or else only multiply or divide by integers: >>> Point(1, 2).scale(1.1, 1.1) Point2D(11/10, 11/5) >>> Point(1, 2)*11/10 Point2D(11/10, 11/5) See Also ======== sympy.geometry.point.Point.scale csg|]}t|qSr#rCrTfactorr#r$rFrXz!Point.__mul__..FrBrY)r?rfr<r#rer$__mul__s z Point.__mul__cCs ||S)z)Multiply a factor by point's coordinates.)rg)r?rfr#r#r$__rmul__s zPoint.__rmul__cCsdd|jD}t|ddS)zNegate the point.cSg|]}| qSr#r#rTr#r#r$rFz!Point.__neg__..FrBr:r)r?r<r#r#r$__neg__s z Point.__neg__cCs|dd|DS)zPSubtract two points, or subtract a factor from this point's coordinates.cSrir#r#rTr#r#r$rF#rjz!Point.__sub__..r#r[r#r#r$__sub__ sz Point.__sub__cszt|ddddurtdd|Dtfdd|Dr(t|Sd<ddd<fd d |DS) z~Ensure that points have the same dimension. By default `on_morph='warn'` is passed to the `Point` constructor._ambient_dimensionNrcss|]}|jVqdSrambient_dimensionr!ir#r#r$r%0sz-Point._normalize_dimension..c3s|]}|jkVqdSrrorqrr#r$r%1r(rrcsg|] }t|fiqSr#rrq)r;r#r$rF5rGz.Point._normalize_dimension..)getattrr-maxr5list)r9pointsr;r#)rr;r$rH%s  zPoint._normalize_dimensioncsht|dkrdStjdd|D}|dfdd|ddD}tdd|D}|jd 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.rcSg|]}t|qSr#rtrqr#r#r$rFDz%Point.affine_rank..cg|]}|qSr#r#rqr@r#r$rFFr{rNcSg|]}|jqSr#rNrqr#r#r$rFHrjcSs|jr t|ddkS|jS)Nrg-q=)rabsnis_zero)rUr#r#r$Jsz#Point.affine_rank..)Z iszerofunc)r.rrHr rank)r:rxmr#r}r$ affine_rank7s zPoint.affine_rankcCst|dt|S)z$Number of components this point has.rn)rur.rar#r#r$rpMszPoint.ambient_dimensioncGsPt|dkrdS|jdd|D}|djdkrdStt|}tj|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 rTcSrzr#rtrqr#r#r$rFyr{z&Point.are_coplanar..rr)r.rHrprwrrr)r9rxr#r#r$ are_coplanarRs $ zPoint.are_coplanarcCst|tsz t||jd}Wntytdt|wt|tr;t|t|\}}ttddt ||DSt |dd}|durMtdt|||S)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.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) rsz'not recognized as a GeometricEntity: %scss |] \}}||dVqdSrNr#rDr#r#r$r%z!Point.distance..r>Nz,distance between Point and %s is not defined) r'rrrpr/typerHr rrIru)r?rJrKpr>r#r#r$r>s '   zPoint.distancecCs(t|st|}tddt||DS)z.Return dot product of self with another Point.css|] \}}||VqdSrr#rDr#r#r$r%szPoint.dot..)rrrrI)r?rr#r#r$dotsz Point.dotcCs6t|tr t|t|krdStddt||DS)z8Returns whether the coordinates of self and other agree.Fcss|] \}}||VqdSr)equalsrDr#r#r$r%r&zPoint.equals..)r'rr.r5rIr[r#r#r$rsz Point.equalsNc s$fdd|jD}t|ddiS)aFEvaluate the coordinates of the point. This method will, where possible, create and return a new Point where the coordinates are evaluated as floating point numbers to the precision indicated (default=15). Parameters ========== prec : int Returns ======= point : Point Examples ======== >>> from sympy import Point, Rational >>> p1 = Point(Rational(1, 2), Rational(3, 2)) >>> p1 Point2D(1/2, 3/2) >>> p1.evalf() Point2D(0.5, 1.5) csg|] }|jfiqSr#)evalfrToptionsprecr#r$rFszPoint.evalf..rFrk)r?rrr<r#rr$rsz Point.evalfcCs^t|ts t|}t|tr*||kr|gSt||\}}||kr(||kr(|gSgS||S)a|The intersection between this point and another GeometryEntity. Parameters ========== other : GeometryEntity or sequence of coordinates Returns ======= intersection : list of Points Notes ===== The return value will either be an empty list if there is no intersection, otherwise it will contain this point. Examples ======== >>> from sympy import Point >>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 0) >>> p1.intersection(p2) [] >>> p1.intersection(p3) [Point2D(0, 0)] )r'rrrH intersection)r?rJZp1Zp2r#r#r$rs   zPoint.intersectioncGs8|f|}tjdd|D}tt|}tj|dkS)aReturns `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 cSrzr#rtrqr#r#r$rF+r{z&Point.is_collinear..r)rrHrwrr)r?r:rxr#r#r$ is_collinear s ! zPoint.is_collinearcs|f|}tjdd|D}tt|}tj|dksdS|dfdd|D}tdd|D}|\}}t|vrBdSdS) 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 cSrzr#rtrqr#r#r$rFVr{z&Point.is_concyclic..rFrcr|r#r#)r!rr}r#r$rF[r{cSs g|] }t|||gqSr#)rwrrqr#r#r$rFas T)rrHrwrrr rrefr.)r?r:rxZmatrZpivotsr#r}r$ is_concyclic/s &   zPoint.is_concycliccCs|j}|dur dS| S)zrTrue if any coordinate is nonzero, False if every coordinate is zero, and None if it cannot be determined.N)r)r?rr#r#r$ is_nonzerogszPoint.is_nonzeroc Cst|t|\}}|jdkr3|j|j\}}\}}||||d}|dur3ttd||ft|j|jg} | dkS)z{Returns whether each coordinate of `self` is a scalar multiple of the corresponding coordinate in point p. rrNzDcan't determine if %s is a scalar multiple of %s) rrHrpr:rrrr r) r?rrKrLZx1Zy1Zx2Zy2rvrr#r#r$is_scalar_multipleps  zPoint.is_scalar_multiplecCs6dd|jD}t|rdStdd|DrdSdS)zsTrue if every coordinate is zero, False if any coordinate is not zero, and None if it cannot be determined.cSr~r#)rrTr#r#r$rFrjz!Point.is_zero..Fcss|]}|duVqdSrr#rTr#r#r$r%z Point.is_zero..NT)r:r4)r?Znonzeror#r#r$rs z Point.is_zerocCstjS)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 )rr1rar#r#r$length z Point.lengthcCs,t|t|\}}tddt||DS)aThe midpoint between self and point p. Parameters ========== p : Point Returns ======= midpoint : Point See Also ======== sympy.geometry.line.Segment.midpoint Examples ======== >>> from sympy.geometry import Point >>> p1, p2 = Point(1, 1), Point(13, 5) >>> p1.midpoint(p2) Point2D(7, 3) cSs"g|] \}}t||tjqSr#)rrZHalfrDr#r#r$rFs"z"Point.midpoint..)rrHrIr?rrKr#r#r$midpointszPoint.midpointcCstdgt|ddS)zOA point of all zeros of the same ambient dimension as the current pointrFrB)rr.rar#r#r$r@sz Point.origincCsp|j}|djrtdg|ddgS|djr&tddg|ddgSt|d |dg|ddgS)a~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 rrr)rprr)r?rr#r#r$orthogonal_directions   $zPoint.orthogonal_directioncCs>tt|t|\}}|jrtd|||||S)aProject the point `a` onto the line between the origin and point `b` along the normal direction. Parameters ========== a : Point b : Point Returns ======= p : Point See Also ======== sympy.geometry.line.LinearEntity.projection Examples ======== >>> from sympy.geometry import Line, Point >>> a = Point(1, 2) >>> b = Point(2, 5) >>> z = a.origin >>> p = Point.project(a, b) >>> Line(p, a).is_perpendicular(Line(p, b)) True >>> Point.is_collinear(z, p, b) True "Cannot project to the zero vector.)rrHrr2r)r"rEr#r#r$projects"z Point.projectcCs,t|t|\}}tddt||DS)a;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 css |] \}}t||VqdSrrrDr#r#r$r%!rz)Point.taxicab_distance..)rrHrrIrr#r#r$taxicab_distanceszPoint.taxicab_distancecCs@t|t|\}}|jr|jrtdtddt||DS)aFThe 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.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 rcss0|]\}}t||t|t|VqdSrrrDr#r#r$r%Os.z*Point.canberra_distance..)rrHrr2rrIrr#r#r$canberra_distance#s) zPoint.canberra_distancecCs |t|S)zdReturn the Point that is in the same direction as `self` and a distance of 1 from the originrrar#r#r$unitQs z Point.unitr)-__name__ __module__ __qualname____doc__Zis_Pointr8rArMrQrZr\r^rbrcrdrgrhrlrm classmethodrH staticmethodrpropertyrprr>rrrrrrrrrrrr@rrrrrrr#r#r#r$r'sf@H'    -4 )&8      &!. rc@seZdZdZdZddddZddZed d Zdd d Z dddZ ddZ dddZ eddZ eddZeddZd S)r6aA 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) rFr+cO,|s d|d<t|i|}tj|g|RS)Nrrrrr8r9r+r:r;r#r#r$r8zPoint2D.__new__cC||kSrr#rOr#r#r$rQzPoint2D.__contains__cCs|j|j|j|jfS)zwReturn a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure. )rUyrar#r#r$boundsszPoint2D.boundsNc Csddlm}m}m}||}||}|}|dur"||dd}||8}|j\} } ||| || || || }|dur@||7}|S)a[Rotate ``angle`` radians counterclockwise about Point ``pt``. See Also ======== translate, scale Examples ======== >>> from sympy import Point2D, pi >>> t = Point2D(1, 0) >>> t.rotate(pi/2) Point2D(0, 1) >>> t.rotate(pi/2, (2, 0)) Point2D(2, -1) r)cossinrNrrs)Zsympyrrrr:) r?ZangleptrrrcrKrrUrr#r#r$rotates  "zPoint2D.rotatercCsD|rt|dd}|j| j||j|jSt|j||j|S)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) rrs)r translater:scalerUr)r?rUrrr#r#r$rs z Point2D.scalecCsL|jr|jdks td|j\}}ttdd||dg|dddS)aReturn the point after applying the transformation described by the 3x3 Matrix, ``matrix``. See Also ======== sympy.geometry.point.Point2D.rotate sympy.geometry.point.Point2D.scale sympy.geometry.point.Point2D.translate )r,r,zmatrix must be a 3x3 matrixrr,rNr) is_Matrixshaper2r:rr tolist)r?matrixrUrr#r#r$ transforms  *zPoint2D.transformrcCst|j||j|S)aShift 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) )rrUr)r?rUrr#r#r$rszPoint2D.translatecC|jS)z Returns the two coordinates of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.coordinates (0, 1) rNrar#r#r$ coordinatesrzPoint2D.coordinatescC |jdS)z Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.x 0 rrNrar#r#r$rU z Point2D.xcCr)z Returns the Y coordinate of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.y 1 rrNrar#r#r$r rz Point2D.yr)rrN)rr)rrrrrnr8rQrrrrrrrrUrr#r#r#r$r6Ys"1   "   r6c@seZdZdZdZddddZddZed d Zd d Z d dZ ddZ d"ddZ ddZ d#ddZeddZeddZeddZed d!ZdS)$r7a6A 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) r,FrcOr)Nr,rrrr#r#r$r8^rzPoint3D.__new__cCrrr#rOr#r#r$rQdrzPoint3D.__contains__cGs tj|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)rxr#r#r$ are_collineargs "zPoint3D.are_collinearcCsN||}ttdd|D}|j|j||j|j||j|j|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] css|]}|dVqdSrr#rqr#r#r$r%rz+Point3D.direction_cosine..)direction_ratior rrUrz)r?pointr"rEr#r#r$direction_cosines zPoint3D.direction_cosinecCs"|j|j|j|j|j|jgS)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] )rUrr)r?rr#r#r$rs"zPoint3D.direction_ratiocCs<t|ts t|dd}t|tr||kr|gSgS||S)aThe intersection between this point and another GeometryEntity. Parameters ========== other : GeometryEntity or sequence of coordinates Returns ======= intersection : list of Points Notes ===== The return value will either be an empty list if there is no intersection, otherwise it will contain this point. Examples ======== >>> from sympy import Point3D >>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, 0, 0) >>> p1.intersection(p2) [] >>> p1.intersection(p3) [Point3D(0, 0, 0)] r,rs)r'rrr7rr[r#r#r$rs    zPoint3D.intersectionrNcCsJ|rt|}|j| j|||j|jSt|j||j||j|S)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) )r7rr:rrUrr)r?rUrrrr#r#r$rs z Point3D.scalecCsd|jr|jdks tdddlm}|j\}}}||}ttdd|||dg|dddS) zReturn the point after applying the transformation described by the 4x4 Matrix, ``matrix``. See Also ======== sympy.geometry.point.Point3D.scale sympy.geometry.point.Point3D.translate )rzmatrix must be a 4x4 matrixr) TransposerrNr,) rrr2Zsympy.matrices.expressionsrr:r7r r)r?rrrUrrrr#r#r$rs    ,zPoint3D.transformrcCst|j||j||j|S)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) )r7rUrr)r?rUrrr#r#r$rszPoint3D.translatecCr)z Returns the three coordinates of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 2) >>> p.coordinates (0, 1, 2) rNrar#r#r$r&rzPoint3D.coordinatescCr)z Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 3) >>> p.x 0 rrNrar#r#r$rU5rz Point3D.xcCr)z Returns the Y coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 2) >>> p.y 1 rrNrar#r#r$rDrz Point3D.ycCr)z Returns the Z coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 1) >>> p.z 1 rrNrar#r#r$rSrz Point3D.z)rrrN)rrr)rrrrrnr8rQrrrrrrrrrrrUrrr#r#r#r$r7/s*, # &    r7)&rr3Z sympy.corerrrZsympy.core.compatibilityrZsympy.core.containersrZsympy.simplifyrrZsympy.geometry.exceptionsr Z(sympy.functions.elementary.miscellaneousr Z$sympy.functions.elementary.complexesr Zsympy.matricesr Zsympy.core.numbersr Zsympy.core.parametersrZsympy.core.addrZsympy.utilities.iterablesrZsympy.utilities.miscrrrZentityrrr6r7r#r#r#r$s4           8W