U Zʹj,@sdZddlZddlmZmZmZddlmZddlm Z ddl m Z ddl m Z ddlmZmZdd lmZdd lmZdd lmZdd lmZmZdd lmZddlmZddlmZm Z ddl!m"Z"m#Z#m$Z$ddl%m&Z&ddl'm(Z(Gddde&Z)Gddde)Z*Gddde)Z+dS)aDGeometrical Points. Contains ======== Point Point2D Point3D When methods of Point require 1 or more points as arguments, they can be passed as a sequence of coordinates or Points: >>> from sympy import Point >>> Point(1, 1).is_collinear((2, 2), (3, 4)) False >>> Point(1, 1).is_collinear(Point(2, 2), Point(3, 4)) False N)SsympifyExpr)Add)Tuple)Float)global_parameters) nsimplifysimplify) GeometryError)sqrt)im)cossin)Matrix) Transpose)uniq is_sequence) filldedent func_name Undecidable)GeometryEntity) prec_to_dpsc@sbeZdZdZdZddZddZddZd d Zd d Z d dZ ddZ ddZ ddZ ddZddZddZddZddZedd Zed!d"Zed#d$Zed%d&Zd'd(Zd)d*Zd+d,ZdMd.d/Zd0d1Zd2d3Zd4d5Z ed6d7Z!d8d9Z"ed:d;Z#edd?Z%ed@dAZ&edBdCZ'edDdEZ(dFdGZ)dHdIZ*edJdKZ+dLS)NPointaA 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) TcOs|dtj}|dd}t|dkr.|dn|}t|tr\d}t||dt|kr\|St|szttd t |t|dkr|ddrt j f|d}t |}|dt|}t|d krttd t||kr8d |t||}|dkrn:|d krt|n&|d kr,tj|d dn ttdt||drRtdtdd|Drntdtdd|Dstd|d|t j f|t|}|r|dd|tD}t|d krd|d<t||St|dkr d|d<t||Stj|f|S)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.css"|]}|jot|jdkVqdS)FN) is_numberr is_zero.0ar(S/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/geometry/point.py sz Point.__new__..z(Imaginary coordinates are not permitted.css|]}t|tVqdSN) isinstancerr%r(r(r)r*sz,Coordinates must be valid SymPy expressions.cSsi|]}|tt|ddqS)T)Zrational)r r )r&fr(r(r) sz!Point.__new__..T_nocheck)getrrlenr,rr TypeErrorrformatrrZeror ValueErrorwarningsr!anyallZxreplaceZatomsrPoint2DPoint3Dr__new__)clsargskwargsrrcoordsrmessager(r(r)r<ms`            z Point.__new__cCstdgt|}t||S)z7Returns the distance between this point and the origin.r)rr2distance)selforiginr(r(r)__abs__sz Point.__abs__cCs`zt|t|dd\}}Wn"tk r>td|YnXddt||D}t|ddS)a8Add other to self by incrementing self's coordinates by those of other. Notes ===== >>> from sympy import Point When sequences of coordinates are passed to Point methods, they are converted to a Point internally. This __add__ method does not do that so if floating point values are used, a floating point result (in terms of SymPy Floats) will be returned. >>> Point(1, 2) + (.1, .2) Point2D(1.1, 2.2) If this is not desired, the `translate` method can be used or another Point can be added: >>> Point(1, 2).translate(.1, .2) Point2D(11/10, 11/5) >>> Point(1, 2) + Point(.1, .2) Point2D(11/10, 11/5) See Also ======== sympy.geometry.point.Point.translate Frz+Don't know how to add {} and a Point objectcSsg|]\}}t||qSr(r r&r'br(r(r) sz!Point.__add__..)r_normalize_dimensionr3r r4zip)rCothersor@r(r(r)__add__s z Point.__add__cCs ||jkSr+r>rCitemr(r(r) __contains__szPoint.__contains__cs(tfdd|jD}t|ddS)z'Divide point's coordinates by a factor.csg|]}t|qSr(rGr&xdivisorr(r)rJsz%Point.__truediv__..FrFrr>r)rCrXr@r(rWr) __truediv__szPoint.__truediv__cCs.t|trt|jt|jkr"dS|j|jkS)NF)r,rr2r>rCrMr(r(r)__eq__sz Point.__eq__cCs |j|Sr+rQ)rCkeyr(r(r) __getitem__szPoint.__getitem__cCs t|jSr+)hashr>rCr(r(r)__hash__szPoint.__hash__cCs |jSr+)r>__iter__r`r(r(r)rbszPoint.__iter__cCs t|jSr+)r2r>r`r(r(r)__len__sz Point.__len__cs(tfdd|jD}t|ddS)alMultiply point's coordinates by a factor. Notes ===== >>> from sympy import Point When multiplying a Point by a floating point number, the coordinates of the Point will be changed to Floats: >>> Point(1, 2)*0.1 Point2D(0.1, 0.2) If this is not desired, the `scale` method can be used or else only multiply or divide by integers: >>> Point(1, 2).scale(1.1, 1.1) Point2D(11/10, 11/5) >>> Point(1, 2)*11/10 Point2D(11/10, 11/5) See Also ======== sympy.geometry.point.Point.scale csg|]}t|qSr(rGrUfactorr(r)rJsz!Point.__mul__..FrFrY)rCrer@r(rdr)__mul__sz Point.__mul__cCs ||S)z)Multiply a factor by point's coordinates.)rf)rCrer(r(r)__rmul__szPoint.__rmul__cCsdd|jD}t|ddS)zNegate the point.cSsg|] }| qSr(r(rUr(r(r)rJ sz!Point.__neg__..FrF)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(rUr(r(r)rJ&sz!Point.__sub__..r(r[r(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_dimensionNrcss|] }|jVqdSr+ambient_dimensionr&ir(r(r)r*3sz-Point._normalize_dimension..c3s|]}|jkVqdSr+rkrmrr(r)r*4srr!csg|]}t|fqSr(rrm)r?r(r)rJ8sz.Point._normalize_dimension..)getattrr1maxr9list)r=pointsr?r()rr?r)rK(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.rcSsg|] }t|qSr(rprmr(r(r)rJGsz%Point.affine_rank..csg|] }|qSr(r(rmrDr(r)rJIsrNcSsg|] }|jqSr(rQrmr(r(r)rJKscSs|jrt|ddkS|jS)Nrg-q=)r#absnr$)rVr(r(r)Msz#Point.affine_rank..)Z iszerofunc)r2rrKrrank)r>rtmr(rvr) affine_rank:s zPoint.affine_rankcCst|dt|S)z$Number of components this point has.rj)rqr2r`r(r(r)rlPszPoint.ambient_dimensioncGsPt|dkrdS|jdd|D}|djdkr6dStt|}tj|dkS)aReturn True if there exists a plane in which all the points lie. A trivial True value is returned if `len(points) < 3` or all Points are 2-dimensional. Parameters ========== A set of points Raises ====== ValueError : if less than 3 unique points are given Returns ======= boolean Examples ======== >>> from sympy import Point3D >>> p1 = Point3D(1, 2, 2) >>> p2 = Point3D(2, 7, 2) >>> p3 = Point3D(0, 0, 2) >>> p4 = Point3D(1, 1, 2) >>> Point3D.are_coplanar(p1, p2, p3, p4) True >>> p5 = Point3D(0, 1, 3) >>> Point3D.are_coplanar(p1, p2, p3, p5) False rTcSsg|] }t|qSr(rprmr(r(r)rJ|sz&Point.are_coplanar..rr)r2rKrlrsrrr|)r=rtr(r(r) are_coplanarUs$  zPoint.are_coplanarcCst|tsBzt||jd}Wn$tk r@tdt|YnXt|tr|t|t|\}}ttddt ||DSt |dd}|dkrtdt|||S)azThe Euclidean distance between self and another GeometricEntity. Returns ======= distance : number or symbolic expression. Raises ====== TypeError : if other is not recognized as a GeometricEntity or is a GeometricEntity for which distance is not defined. See Also ======== sympy.geometry.line.Segment.length sympy.geometry.point.Point.taxicab_distance Examples ======== >>> from sympy import Point, Line >>> p1, p2 = Point(1, 1), Point(4, 5) >>> l = Line((3, 1), (2, 2)) >>> p1.distance(p2) 5 >>> p1.distance(l) sqrt(2) The computed distance may be symbolic, too: >>> from sympy.abc import x, y >>> p3 = Point(x, y) >>> p3.distance((0, 0)) sqrt(x**2 + y**2) roz'not recognized as a GeometricEntity: %scss|]\}}||dVqdSrNr(rHr(r(r)r*sz!Point.distance..rBNz,distance between Point and %s is not defined) r,rrrlr3typerKr rrLrq)rCrMrNprBr(r(r)rBs'   zPoint.distancecCs(t|st|}tddt||DS)z.Return dot product of self with another Point.css|]\}}||VqdSr+r(rHr(r(r)r*szPoint.dot..)rrrrL)rCrr(r(r)dotsz Point.dotcCs6t|trt|t|krdStddt||DS)z8Returns whether the coordinates of self and other agree.Fcss|]\}}||VqdSr+)equalsrHr(r(r)r*szPoint.equals..)r,rr2r9rLr[r(r(r)rsz Point.equalsc s,t|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) cs g|]}|jfdiqS)rx)ZevalfrUZdpsoptionsr(r)rJsz%Point._eval_evalf..rF)rr>r)rCprecrr@r(rr) _eval_evalfszPoint._eval_evalfcCs^t|tst|}t|trT||kr*|gSt||\}}||krP||krP|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,rrrK intersection)rCrMp1Zp2r(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 cSsg|] }t|qSr(rprmr(r(r)rJ/sz&Point.is_collinear..r)rrKrsrr|)rCr>rtr(r(r) is_collinear s!  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(rprmr(r(r)rJZsz&Point.is_concyclic..rFrcsg|] }|qSr(r()r&rrvr(r)rJ_scSs g|]}t|||gqSr()rsrrmr(r(r)rJesT)rrKrsrr|rrrefr2)rCr>rtZmatrZpivotsr(rvr) is_concyclic3s&    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)r$)rCr$r(r(r) is_nonzerokszPoint.is_nonzeroc Cst|t|\}}|jdkrf|j|j\}}\}}||||d}|dkrfttd||ft|j|jg} | dkS)z{Returns whether each coordinate of `self` is a scalar multiple of the corresponding coordinate in point p. rrNzECannot determine if %s is a scalar multiple of %s) rrKrlr>rrrrrz) rCrrNrOx1y1Zx2y2rvr{r(r(r)is_scalar_multiplets zPoint.is_scalar_multiplecCs6dd|jD}t|rdStdd|Dr2dSdS)zsTrue if every coordinate is zero, False if any coordinate is not zero, and None if it cannot be determined.cSsg|] }|jqSr()rrUr(r(r)rJsz!Point.is_zero..Fcss|]}|dkVqdSr+r(rUr(r(r)r*sz Point.is_zero..NT)r>r8)rCZnonzeror(r(r)r$s 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 )rr5r`r(r(r)lengths 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 import Point >>> p1, p2 = Point(1, 1), Point(13, 5) >>> p1.midpoint(p2) Point2D(7, 3) cSs"g|]\}}t||tjqSr()r rZHalfrHr(r(r)rJsz"Point.midpoint..)rrKrLrCrrNr(r(r)midpointszPoint.midpointcCstdgt|ddS)zOA point of all zeros of the same ambient dimension as the current pointrFrF)rr2r`r(r(r)rDsz Point.origincCsp|j}|djr(tdg|ddgS|djrLtddg|ddgSt|d |dg|ddgS)auReturns a non-zero point that is orthogonal to the line containing `self` and the origin. Examples ======== >>> from sympy import Line, Point >>> a = Point(1, 2, 3) >>> a.orthogonal_direction Point3D(-2, 1, 0) >>> b = _ >>> Line(b, b.origin).is_perpendicular(Line(a, a.origin)) True rrr)rlr$r)rCrr(r(r)orthogonal_directions   zPoint.orthogonal_directioncCs>tt|t|\}}|jr&td|||||S)aProject the point `a` onto the line between the origin and point `b` along the normal direction. Parameters ========== a : Point b : Point Returns ======= p : Point See Also ======== sympy.geometry.line.LinearEntity.projection Examples ======== >>> from sympy import Line, Point >>> a = Point(1, 2) >>> b = Point(2, 5) >>> z = a.origin >>> p = Point.project(a, b) >>> Line(p, a).is_perpendicular(Line(p, b)) True >>> Point.is_collinear(z, p, b) True "Cannot project to the zero vector.)rrKr$r6r)r'rIr(r(r)projects"z Point.projectcCs,t|t|\}}tddt||DS)a2The Taxicab Distance from self to point p. Returns the sum of the horizontal and vertical distances to point p. Parameters ========== p : Point Returns ======= taxicab_distance : The sum of the horizontal and vertical distances to point p. See Also ======== sympy.geometry.point.Point.distance Examples ======== >>> from sympy import Point >>> p1, p2 = Point(1, 1), Point(4, 5) >>> p1.taxicab_distance(p2) 7 css|]\}}t||VqdSr+rwrHr(r(r)r*%sz)Point.taxicab_distance..)rrKrrLrr(r(r)taxicab_distanceszPoint.taxicab_distancecCs@t|t|\}}|jr(|jr(tdtddt||DS)a=The Canberra Distance from self to point p. Returns the weighted sum of horizontal and vertical distances to point p. Parameters ========== p : Point Returns ======= canberra_distance : The weighted sum of horizontal and vertical distances to point p. The weight used is the sum of absolute values of the coordinates. Examples ======== >>> from sympy import Point >>> p1, p2 = Point(1, 1), Point(3, 3) >>> p1.canberra_distance(p2) 1 >>> p1, p2 = Point(0, 0), Point(3, 3) >>> p1.canberra_distance(p2) 2 Raises ====== ValueError when both vectors are zero. See Also ======== sympy.geometry.point.Point.distance rcss.|]&\}}t||t|t|VqdSr+rrHr(r(r)r*Ssz*Point.canberra_distance..)rrKr$r6rrLrr(r(r)canberra_distance's) zPoint.canberra_distancecCs |t|S)zdReturn the Point that is in the same direction as `self` and a distance of 1 from the originrr`r(r(r)unitUsz Point.unitN)r),__name__ __module__ __qualname____doc__Zis_Pointr<rErPrTrZr\r^rarbrcrfrgrhri classmethodrK staticmethodr|propertyrlr}rBrrrrrrrrr$rrrDrrrrrr(r(r(r)r*sb@H'    -4 )&8      &!.rc@seZdZdZdZddddZddZed d Zdd d Z dddZ ddZ dddZ eddZ eddZeddZd S)r:aA 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) rFr/cOs&|sd|d<t||}tj|f|S)Nrrrrr<r=r/r>r?r(r(r)r<s zPoint2D.__new__cCs||kSr+r(rRr(r(r)rTszPoint2D.__contains__cCs|j|j|j|jfS)zwReturn a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure. )rVyr`r(r(r)boundsszPoint2D.boundsNcCspt|}t|}|}|dk r0t|dd}||8}|j\}}t||||||||}|dk rl||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) Nrro)rrrr>)rCZangleptcrNrrVrr(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) rro)r translater>scalerVr)rCrVrrr(r(r)rs z Point2D.scalecCsL|jr|jdkstd|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 )r0r0zmatrix must be a 3x3 matrixrr0rNr) is_Matrixshaper6r>rrtolist)rCmatrixrVrr(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) )rrVr)rCrVrr(r(r)rszPoint2D.translatecCs|jS)z Returns the two coordinates of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.coordinates (0, 1) rQr`r(r(r) coordinatess zPoint2D.coordinatescCs |jdS)z Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.x 0 rrQr`r(r(r)rVs z Point2D.xcCs |jdS)z Returns the Y coordinate of the Point. Examples ======== >>> from sympy import Point2D >>> p = Point2D(0, 1) >>> p.y 1 rrQr`r(r(r)r"s z Point2D.y)N)rrN)rr)rrrrrjr<rTrrrrrrrrVrr(r(r(r)r:\s 2     r:c@seZdZdZdZddddZddZed d Zd d Z d dZ ddZ d"ddZ ddZ d#ddZeddZeddZeddZed d!ZdS)$r;a>A 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) r0FrcOs&|sd|d<t||}tj|f|S)Nr0rrrr(r(r)r<as zPoint3D.__new__cCs||kSr+r(rRr(r(r)rTgszPoint3D.__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)rtr(r(r) are_collinearjs"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|]}|dVqdSr~r(rmr(r(r)r*sz+Point3D.direction_cosine..)direction_ratior rrVrz)rCpointr'rIr(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] )rVrr)rCrr(r(r)rszPoint3D.direction_ratiocCs<t|tst|dd}t|tr2||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)] r0ro)r,rrr;rr[r(r(r)rs   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>rrVrr)rCrVrrrr(r(r)rs z Point3D.scalecCsX|jr|jdkstd|j\}}}t|}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 matrixrrrNr0)rrr6r>rr;rr)rCrrVrrr{r(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) )r;rVrr)rCrVrrr(r(r)rszPoint3D.translatecCs|jS)z Returns the three coordinates of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 2) >>> p.coordinates (0, 1, 2) rQr`r(r(r)r(s zPoint3D.coordinatescCs |jdS)z Returns the X coordinate of the Point. Examples ======== >>> from sympy import Point3D >>> p = Point3D(0, 1, 3) >>> p.x 0 rrQr`r(r(r)rV7s 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 rrQr`r(r(r)rFs 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 rrQr`r(r(r)rUs z Point3D.z)rrrN)rrr)rrrrrjr<rTrrrrrrrrrrrVrrr(r(r(r)r;1s(- #&     r;),rr7Z sympy.corerrrZsympy.core.addrZsympy.core.containersrZsympy.core.numbersrZsympy.core.parametersrZsympy.simplifyr r Zsympy.geometry.exceptionsr Z(sympy.functions.elementary.miscellaneousr Z$sympy.functions.elementary.complexesr Z(sympy.functions.elementary.trigonometricrrZsympy.matricesrZsympy.matrices.expressionsrZsympy.utilities.iterablesrrZsympy.utilities.miscrrrentityrZmpmath.libmp.libmpfrrr:r;r(r(r(r)s6           8V