B [@s8dZddlmZmZddlZddlmZmZmZddl m Z ddl m Z m Z mZddlmZddlmZmZdd lmZdd lmZdd lmZdd lmZdd lmZddl mZddlm Z ddl!m"Z"ddl#m$Z$ddl%m&Z&ddl'm(Z(m)Z)m*Z*ddl+m,Z,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 )divisionprint_functionN)SsympifyExpr)Number)iterable is_sequenceas_int)Tuple) nsimplifysimplify) GeometryError)sqrt)im)Matrix)Eq)Float)global_evaluate)Add) FiniteSet)uniq) filldedent func_name Undecidable)GeometryEntityc@sbeZdZdZdZddZddZddZd d Zd d Z d dZ ddZ ddZ ddZ ddZddZddZddZeddZedd Zed!d"Zed#d$Zd%d&Zd'd(Zd)d*ZdJd,d-Zd.d/Zd0d1Zd2d3Zed4d5Z d6d7Z!ed8d9Z"ed:d;Z#dd?Z%ed@dAZ&edBdCZ'dDdEZ(dFdGZ)edHdIZ*eZ+e Z,d+S)KPointaA 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(|dtd}|dd}t|dkr0|dn|}t|tr^d}t||dt|kr^|St|s|ttdt |t|dkr|ddrt j f|d}t |}|dt|}t|d krt td t||kr8d |t||}|dkrn6|d krt |n"|d kr,t|n t tdtdd||dDr\t dtdd|Drxt dtdd|Dstd|d|t j f|t|}|r|tdd|tD}t|d krd|d<t||St|dkrd|d<t||Stj|f|S)Nevaluateron_morphignorerFdimz< Expecting sequence of coordinates, not `{}`z[ Point requires 2 or more coordinates or keyword `dim` > 1.z1Dimension of {} needs to be changedfrom {} to {}.errorwarnzf on_morph value should be 'error', 'warn' or 'ignore'.css|] }|VqdS)N).0ir%r%3lib/python3.7/site-packages/sympy/geometry/point.py sz Point.__new__..z&Nonzero coordinates cannot be removed.css|]}|jot|VqdS)N)Z is_numberr)r&ar%r%r(r)sz(Imaginary coordinates are not permitted.css|]}t|tVqdS)N) isinstancer)r&r*r%r%r(r)sz,Coordinates must be valid SymPy expressions.cSs g|]}|tt|ddfqS)T)Zrational)r r )r&fr%r%r( sz!Point.__new__..T_nocheck)getrlenr+rr TypeErrorrformatrrZeror ValueErrorwarningsr$anyallZxreplacedictZatomsrPoint2DPoint3Dr__new__)clsargskwargsrrcoordsr!messager%r%r(r<os^           z Point.__new__cCstdgt|}t||S)z7Returns the distance between this point and the origin.r)rr1distance)selforiginr%r%r(__abs__sz Point.__abs__cCs`yt|t|dd\}}Wn"tk r>td|YnXddt||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 F)rz+Don't know how to add {} and a Point objectcSsg|]\}}t||qSr%)r )r&r*br%r%r(r-sz!Point.__add__..)r_normalize_dimensionr2rr3zip)rCothersor@r%r%r(__add__s z Point.__add__cCs ||jkS)N)r>)rCitemr%r%r( __contains__szPoint.__contains__cs(tfdd|jD}t|ddS)z'Divide point's coordinates by a factor.csg|]}t|qSr%)r )r&x)divisorr%r(r-sz!Point.__div__..F)r)rr>r)rCrPr@r%)rPr(__div__sz Point.__div__cCs.t|trt|jt|jkr"dS|j|jkS)NF)r+rr1r>)rCrIr%r%r(__eq__sz Point.__eq__cCs |j|S)N)r>)rCkeyr%r%r( __getitem__szPoint.__getitem__cCs t|jS)N)hashr>)rCr%r%r(__hash__szPoint.__hash__cCs |jS)N)r>__iter__)rCr%r%r(rWszPoint.__iter__cCs t|jS)N)r1r>)rCr%r%r(__len__sz Point.__len__cs(tfdd|jD}t|ddS)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%)r )r&rO)factorr%r(r-sz!Point.__mul__..F)r)rr>r)rCrYr@r%)rYr(__mul__sz Point.__mul__cCsdd|jD}t|ddS)zNegate the point.cSsg|] }| qSr%r%)r&rOr%r%r(r-sz!Point.__neg__..F)r)r>r)rCr@r%r%r(__neg__sz Point.__neg__cCs|dd|DS)zPSubtract two points, or subtract a factor from this point's coordinates.cSsg|] }| qSr%r%)r&rOr%r%r(r-$sz!Point.__sub__..r%)rCrIr%r%r(__sub__!sz Point.__sub__cszt|ddddkr2tdd|Dtfdd|DrPt|Sd<ddd<fd d |DS) z~Ensure that points have the same dimension. By default `on_morph='warn'` is passed to the `Point` constructor._ambient_dimensionNr!css|] }|jVqdS)N)ambient_dimension)r&r'r%r%r(r)1sz-Point._normalize_dimension..c3s|]}|jkVqdS)N)r^)r&r')r!r%r(r)2srr$csg|]}t|fqSr%)r)r&r')r?r%r(r-6sz.Point._normalize_dimension..)getattrr0maxr8list)r=pointsr?r%)r!r?r(rG&s  zPoint._normalize_dimensioncs`t|dkrdStjdd|D}|dfdd|ddD}tdd|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.rcSsg|] }t|qSr%)r)r&r'r%r%r(r-Esz%Point.affine_rank..csg|] }|qSr%r%)r&r')rDr%r(r-GsrNcSsg|] }|jqSr%)r>)r&r'r%r%r(r-Is)r1rrGrrank)r>rbmr%)rDr( affine_rank8s zPoint.affine_rankcCst|dt|S)z$Number of components this point has.r])r_r1)rCr%r%r(r^LszPoint.ambient_dimensioncGsPt|dkrdS|jdd|D}|djdkr6dStt|}tj|dkS)aReturn True if there exists a plane in which all the points lie. A trivial True value is returned if `len(points) < 3` or all Points are 2-dimensional. Parameters ========== A set of points Raises ====== ValueError : if less than 3 unique points are given Returns ======= boolean Examples ======== >>> from sympy import Point3D >>> p1 = Point3D(1, 2, 2) >>> p2 = Point3D(2, 7, 2) >>> p3 = Point3D(0, 0, 2) >>> p4 = Point3D(1, 1, 2) >>> Point3D.are_coplanar(p1, p2, p3, p4) True >>> p5 = Point3D(0, 1, 3) >>> Point3D.are_coplanar(p1, p2, p3, p5) False rTcSsg|] }t|qSr%)r)r&r'r%r%r(r-xsz&Point.are_coplanar..rr")r1rGr^rarrrf)r=rbr%r%r( are_coplanarQs$  zPoint.are_coplanarcCs0t|t|\}}ttddt||DS)apThe Euclidean distance from self to point p. Parameters ========== p : Point Returns ======= distance : number or symbolic expression. See Also ======== sympy.geometry.line.Segment.length sympy.geometry.point.Point.taxicab_distance Examples ======== >>> from sympy.geometry import Point >>> p1, p2 = Point(1, 1), Point(4, 5) >>> p1.distance(p2) 5 >>> from sympy.abc import x, y >>> p3 = Point(x, y) >>> p3.distance(Point(0, 0)) sqrt(x**2 + y**2) css|]\}}||dVqdS)r"Nr%)r&r*rFr%r%r(r)sz!Point.distance..)rrGrrrH)rCprJr%r%r(rBs!zPoint.distancecCs(t|st|}tddt||DS)z.Return dot product of self with another Point.css|]\}}||VqdS)Nr%)r&r*rFr%r%r(r)szPoint.dot..)r rrrH)rCrhr%r%r(dotsz Point.dotcCs6t|trt|t|krdStddt||DS)z8Returns whether the coordinates of self and other agree.Fcss|]\}}||VqdS)N)equals)r&r*rFr%r%r(r)szPoint.equals..)r+rr1r8rH)rCrIr%r%r(rjsz 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|]}|jfqSr%)evalf)r&rO)optionsprecr%r(r-szPoint.evalf..rF)r>r)rCrmrlr@r%)rlrmr(rksz Point.evalfcCs^t|tst|}t|trT||kr*|gSt||\}}||krP||krP|gSgS||S)aXThe intersection between this point and another GeometryEntity. Parameters ========== other : Point Returns ======= intersection : list of Points Notes ===== The return value will either be an empty list if there is no intersection, otherwise it will contain this point. Examples ======== >>> from sympy import Point >>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 0) >>> p1.intersection(p2) [] >>> p1.intersection(p3) [Point2D(0, 0)] )r+rrrG intersection)rCrIZp1Zp2r%r%r(rns  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 cSsg|] }t|qSr%)r)r&r'r%r%r(r-sz&Point.is_collinear..r)rrGrarrf)rCr>rbr%r%r( is_collinears!  zPoint.is_collinearcs|f|}tjdd|D}tt|}tj|dks>> 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 cSsg|] }t|qSr%)r)r&r'r%r%r(r-Esz&Point.is_concyclic..r"Frcsg|] }|qSr%r%)r&rh)rDr%r(r-JscSs g|]}t|||gqSr%)rari)r&r'r%r%r(r-PsT)rrGrarrfrrrefr1)rCr>rbZmatrpZpivotsr%)rDr( is_concyclics&    zPoint.is_concycliccCs|j}|dkrdS| S)zrTrue if any coordinate is nonzero, False if every coordinate is zero, and None if it cannot be determined.N)is_zero)rCrrr%r%r( is_nonzeroVszPoint.is_nonzeroc Cst|t|\}}|jdkrf|j|j\}}\}}||||d}|dkrfttd||ft|j|jg} | dkS)z{Returns whether each coordinate of `self` is a scalar multiple of the corresponding coordinate in point p. r"rNzDcan't determine if %s is a scalar multiple of %s) rrGr^r>rjrrrrd) rCrhrJrKZx1Zy1Zx2Zy2rvrer%r%r(is_scalar_multiple_s zPoint.is_scalar_multiplecCs6dd|jD}t|rdStdd|Dr2dSdS)zsTrue if every coordinate is zero, False if any coordinate is not zero, and None if it cannot be determined.cSsg|] }|jqSr%)rs)r&rOr%r%r(r-vsz!Point.is_zero..Fcss|]}|dkVqdS)Nr%)r&rOr%r%r(r)ysz Point.is_zero..NT)r>r7)rCZnonzeror%r%r(rrrs 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 )rr4)rCr%r%r(length}s 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%)r rZHalf)r&r*rFr%r%r(r-sz"Point.midpoint..)rrGrH)rCrhrJr%r%r(midpointszPoint.midpointcCstdgt|ddS)zOA point of all zeros of the same ambient dimension as the current pointrF)r)rr1)rCr%r%r(rDsz Point.origincCsx|j}|dtjkr,tdg|ddgS|dtjkrTtddg|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")r^rr4r)rCr!r%r%r(orthogonal_directions zPoint.orthogonal_directioncCs>tt|t|\}}|jr&td|||||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 z"Cannot project to the zero vector.)rrGrrr5ri)r*rFr%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||VqdS)N)abs)r&r*rFr%r%r(r)sz)Point.taxicab_distance..)rrGrrH)rCrhrJr%r%r(taxicab_distanceszPoint.taxicab_distancecCs@t|t|\}}|jr(|jr(tdtddt||DS)aThe Canberra Distance from self to point p. Returns the weighted sum of horizontal and vertical distances to point p. Parameters ========== p : Point Returns ======= canberra_distance : The weighted sum of horizontal and vertical distances to point p. The weight used is the sum of absolute values of the coordinates. See Also ======== sympy.geometry.point.Point.distance Examples ======== >>> from sympy.geometry import Point >>> p1, p2 = Point(1, 1), Point(3, 3) >>> p1.canberra_distance(p2) 1 >>> p1, p2 = Point(0, 0), Point(3, 3) >>> p1.canberra_distance(p2) 2 Raises ====== ValueError when both vectors are zero. See Also ======== sympy.geometry.point.Point.distance z"Cannot project to the zero vector.css.|]&\}}t||t|t|VqdS)N)rz)r&r*rFr%r%r(r)Csz*Point.canberra_distance..)rrGrrr5rrH)rCrhrJr%r%r(canberra_distances. zPoint.canberra_distancecCs |t|S)zdReturn the Point that is in the same direction as `self` and a distance of 1 from the origin)rz)rCr%r%r(unitEsz Point.unit)N)-__name__ __module__ __qualname____doc__Zis_Pointr<rErLrNrQrRrTrVrWrXrZr[r\ classmethodrG staticmethodrfpropertyr^rgrBrirjrkrnrorqrsrurrrvrwrDrxryr{r|r}n __truediv__r%r%r%r(r,sN?H'    .$ )&8     '!3 rc@sneZdZdZdZddZddZeddZdd d Z dd dZ ddZ dddZ eddZ eddZd S)r:aA 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) r"cOs.|ddsd|d<t||}tj|f|S)Nr.Fr"r!)poprrr<)r=r>r?r%r%r(r<s  zPoint2D.__new__cCs||kS)Nr%)rCrMr%r%r(rNszPoint2D.__contains__cCs|j|j|j|jfS)zwReturn a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure. )rOy)rCr%r%r(boundsszPoint2D.boundsNc Csddlm}m}m}||}||}|}|dk rD||dd}||8}|j\} } ||| || || || }|dk r||7}|S)aXRotate ``angle`` radians counterclockwise about Point ``pt``. See Also ======== rotate, 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)cossinrNr")r!)Zsympyrrrr>) rCZangleptrrrcrJrtrOrr%r%r(rotates  "zPoint2D.rotatercCsD|r.t|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) r")r!)r translater>scalerOr)rCrOrrr%r%r(rs z Point2D.scalecCsvy|j\}}|jo|dk}Wntk r4d}YnX|sBtd|j\}}ttdd||dg|dddS)zReturn the point after applying the transformation described by the 3x3 Matrix, ``matrix``. See Also ======== geometry.entity.rotate geometry.entity.scale geometry.entity.translate r/Fz;The argument to the transform function must be a 3x3 matrixrrNr")shape is_squareAttributeErrorr5r>rrtolist)rCmatrixcolrow valid_matrixrOrr%r%r( transforms    zPoint2D.transformrcCst|j||j|S)aShift the Point by adding x and y to the coordinates of the Point. See Also ======== rotate, scale Examples ======== >>> from sympy import Point2D >>> t = Point2D(0, 1) >>> t.translate(2) Point2D(2, 1) >>> t.translate(2, 2) Point2D(2, 3) >>> t + Point2D(2, 2) Point2D(2, 3) )rrOr)rCrOrr%r%r(rszPoint2D.translatecCs |jdS)z Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.x 0 r)r>)rCr%r%r(rOs z Point2D.xcCs |jdS)z Returns the Y coordinate of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.y 1 r)r>)rCr%r%r(rs z Point2D.y)N)rrN)rr)r~rrrr]r<rNrrrrrrrOrr%r%r%r(r:Os0 "   r:c@seZdZdZdZddZddZeddZd d Z d d Z d dZ dddZ ddZ dddZeddZeddZeddZdS) r;a6A 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/cOs.|ddsd|d<t||}tj|f|S)Nr.Fr/r!)rrrr<)r=r>r?r%r%r(r<Ls  zPoint3D.__new__cCs||kS)Nr%)rCrMr%r%r(rNRszPoint3D.__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, Matrix >>> from sympy.abc import x >>> p1, p2 = Point3D(0, 0, 0), Point3D(1, 1, 1) >>> p3, p4, p5 = Point3D(2, 2, 2), Point3D(x, x, x), Point3D(1, 2, 6) >>> Point3D.are_collinear(p1, p2, p3, p4) True >>> Point3D.are_collinear(p1, p2, p3, p5) False )rro)rbr%r%r( are_collinearUs"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|]}|dVqdS)r"Nr%)r&r'r%r%r(r)sz+Point3D.direction_cosine..)direction_ratiorrrOrz)rCpointr*rFr%r%r(direction_cosineys 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] )rOrr)rCrr%r%r(rszPoint3D.direction_ratiocCs<t|tst|dd}t|tr2||kr.|gSgS||S)acThe intersection between this point and another point. Parameters ========== other : Point Returns ======= intersection : list of Points Notes ===== The return value will either be an empty list if there is no intersection, otherwise it will contain this point. Examples ======== >>> from sympy import Point3D >>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, 0, 0) >>> p1.intersection(p2) [] >>> p1.intersection(p3) [Point3D(0, 0, 0)] r/)r!)r+rrr;rn)rCrIr%r%r(rns   zPoint3D.intersectionrNcCsJ|r,t|}|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) )r;rr>rrOrr)rCrOrrrr%r%r(rs z Point3D.scalec Csy|j\}}|jo|dk}Wntk r4d}YnX|sBtdddlm}|j\}}}||} ttdd|||dg|  dddS) zReturn the point after applying the transformation described by the 4x4 Matrix, ``matrix``. See Also ======== geometry.entity.rotate geometry.entity.scale geometry.entity.translate Fz;The argument to the transform function must be a 4x4 matrixr) TransposerNr/) rrrr5Zsympy.matrices.expressionsrr>r;rr) rCrrrrrrOrrrer%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 ======== rotate, scale Examples ======== >>> from sympy import Point3D >>> t = Point3D(0, 1, 1) >>> t.translate(2) Point3D(2, 1, 1) >>> t.translate(2, 2) Point3D(2, 3, 1) >>> t + Point3D(2, 2, 2) Point3D(2, 3, 3) )r;rOrr)rCrOrrr%r%r(rszPoint3D.translatecCs |jdS)z Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 3) >>> p.x 0 r)r>)rCr%r%r(rOs z Point3D.xcCs |jdS)z Returns the Y coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 2) >>> p.y 1 r)r>)rCr%r%r(r+s z Point3D.ycCs |jdS)z Returns the Z coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 1) >>> p.z 1 r")r>)rCr%r%r(r:s z Point3D.z)rrrN)rrr)r~rrrr]r<rNrrrrrnrrrrrOrrr%r%r%r(r;s+ $&    r;)0rZ __future__rrr6Z sympy.corerrrZsympy.core.numbersrZsympy.core.compatibilityrr r Zsympy.core.containersr Zsympy.simplifyr r Zsympy.geometry.exceptionsrZ(sympy.functions.elementary.miscellaneousrZ$sympy.functions.elementary.complexesrZsympy.matricesrZsympy.core.relationalrrZsympy.core.evaluaterZsympy.core.addrZ sympy.setsrZsympy.utilities.iterablesrZsympy.utilities.miscrrrZentityrrr:r;r%r%r%r(s:             )O