~9\c@ s$ddlmZmZddlmZmZmZmZmZm Z ddl m Z m Z m Z ddlmZmZddlmZddlmZddlmZmZmZddlmZdd lmZdd lmZdd lm Z dd l!m"Z"dd l#m$Z$m%Z%m&Z&m'Z'm(Z(ddl)m*Z*ddl+m,Z,m-Z-ddl.m/Z/ddl0m1Z1ddl2m3Z3m4Z4m5Z5ddl6m7Z7ddl8Z8de-fdYZ9de9fdYZ:de9fdYZ;dZ<dZ=dZ>dZ?d Z@d!ZAdS("i(tdivisiontprint_function(tExprtStSymboltootpitsympify(tas_inttrangetordered(t_symboltDummy(tsign(t Piecewise(tcostsinttan(t GeometryError(tAnd(tMatrix(tsimplify(tdefault_sort_key(thas_dupst has_varietytuniqt rotate_lefttleast_rotation(t func_namei(tGeometryEntityt GeometrySet(tPoint(tCircle(tLinetSegmenttRay(tsqrtNtPolygoncB seZdZdZedZedZedZedZ edZ edZ edZ dd Zed Zed Zd Zd ZddZdZddZdZdZdZdddZdZdZRS(sj A two-dimensional polygon. A simple polygon in space. Can be constructed from a sequence of points or from a center, radius, number of sides and rotation angle. Parameters ========== vertices : sequence of Points Attributes ========== area angles perimeter vertices centroid sides Raises ====== GeometryError If all parameters are not Points. See Also ======== sympy.geometry.point.Point, sympy.geometry.line.Segment, Triangle Notes ===== Polygons are treated as closed paths rather than 2D areas so some calculations can be be negative or positive (e.g., area) based on the orientation of the points. Any consecutive identical points are reduced to a single point and any points collinear and between two points will be removed unless they are needed to define an explicit intersection (see examples). A Triangle, Segment or Point will be returned when there are 3 or fewer points provided. Examples ======== >>> from sympy import Point, Polygon, pi >>> p1, p2, p3, p4, p5 = [(0, 0), (1, 0), (5, 1), (0, 1), (3, 0)] >>> Polygon(p1, p2, p3, p4) Polygon(Point2D(0, 0), Point2D(1, 0), Point2D(5, 1), Point2D(0, 1)) >>> Polygon(p1, p2) Segment2D(Point2D(0, 0), Point2D(1, 0)) >>> Polygon(p1, p2, p5) Segment2D(Point2D(0, 0), Point2D(3, 0)) The area of a polygon is calculated as positive when vertices are traversed in a ccw direction. When the sides of a polygon cross the area will have positive and negative contributions. The following defines a Z shape where the bottom right connects back to the top left. >>> Polygon((0, 2), (2, 2), (0, 0), (2, 0)).area 0 When the the keyword `n` is used to define the number of sides of the Polygon then a RegularPolygon is created and the other arguments are interpreted as center, radius and rotation. The unrotated RegularPolygon will always have a vertex at Point(r, 0) where `r` is the radius of the circle that circumscribes the RegularPolygon. Its method `spin` can be used to increment that angle. >>> p = Polygon((0,0), 1, n=3) >>> p RegularPolygon(Point2D(0, 0), 1, 3, 0) >>> p.vertices[0] Point2D(1, 0) >>> p.args[0] Point2D(0, 0) >>> p.spin(pi/2) >>> p.vertices[0] Point2D(0, 1) c O sH|jddr|jd}t|}t|dkrO|j|n%t|dkrt|jd|nt||Sg|D]}t|dd|^q}g}x7|D]/}|r||dkrqn|j|qWt|dkr|d|dkr|jnd}x|t|dkrt|dkr||||d||d}} } tj|| | r|j|d|| kr|j|qq%|d7}q%Wt|}t|dkrt j |||St|dkrt ||St|dkr7t ||St||SdS( Ntniiitdimiii( tgettpoptlisttlentappendtinserttRegularPolygonRt is_collinearRt__new__tTriangleR"( tclstargstkwargsR&tatverticestnoduptptitbtc((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR0rs@  ( & +(    cC szd}|j}xZtt|D]F}||dj\}}||j\}}|||||7}q"Wt|dS(s The area of the polygon. Notes ===== The area calculation can be positive or negative based on the orientation of the points. If any side of the polygon crosses any other side, there will be areas having opposite signs. See Also ======== sympy.geometry.ellipse.Ellipse.area Examples ======== >>> from sympy import Point, Polygon >>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]) >>> poly = Polygon(p1, p2, p3, p4) >>> poly.area 3 In the Z shaped polygon (with the lower right connecting back to the upper left) the areas cancel out: >>> Z = Polygon((0, 1), (1, 1), (0, 0), (1, 0)) >>> Z.area 0 In the M shaped polygon, areas do not cancel because no side crosses any other (though there is a point of contact). >>> M = Polygon((0, 0), (0, 1), (2, 0), (3, 1), (3, 0)) >>> M.area -3/2 iii(R3R R+R(tselftareaR3R9tx1ty1tx2ty2((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR=s) cC s`||}||}t|j|j|j|j}|j}|dkr\tdn|S(s-Return True/False for cw/ccw orientation. Examples ======== >>> from sympy import Point, Polygon >>> a, b, c = [Point(i) for i in [(0, 0), (1, 1), (1, 0)]] >>> Polygon._isright(a, b, c) True >>> Polygon._isright(a, c, b) False sCan't determine orientationN(Rtxtytis_nonpositivetNonet ValueError(R5R:R;tbatcatt_areatres((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt_isrights  $  c C s|j}|j|d|d|d}i}xtt|D]}||d||d||}}}t||jt||}||j|||Ardtj|||>> from sympy import Point, Polygon >>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]) >>> poly = Polygon(p1, p2, p3, p4) >>> poly.angles[p1] pi/2 >>> poly.angles[p2] acos(-4*sqrt(17)/17) iiii(R6RKR R+R#t angle_betweenRtPi( R<R3tcwtretR9R5R:R;tang((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytangless !(!cC s|jdjS(Ni(R6tambient_dimension(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRRscC sUd}|j}x9tt|D]%}|||dj||7}q"Wt|S(sThe perimeter of the polygon. Returns ======= perimeter : number or Basic instance See Also ======== sympy.geometry.line.Segment.length Examples ======== >>> from sympy import Point, Polygon >>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]) >>> poly = Polygon(p1, p2, p3, p4) >>> poly.perimeter sqrt(17) + 7 ii(R6R R+tdistanceR(R<R8R3R9((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt perimeters  #cC s t|jS(sQThe vertices of the polygon. Returns ======= vertices : list of Points Notes ===== When iterating over the vertices, it is more efficient to index self rather than to request the vertices and index them. Only use the vertices when you want to process all of them at once. This is even more important with RegularPolygons that calculate each vertex. See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Point, Polygon >>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]) >>> poly = Polygon(p1, p2, p3, p4) >>> poly.vertices [Point2D(0, 0), Point2D(1, 0), Point2D(5, 1), Point2D(0, 1)] >>> poly.vertices[0] Point2D(0, 0) (R*R3(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR62s"c C sdd|j}d\}}|j}xztt|D]f}||dj\}}||j\}} || ||} || ||7}|| || 7}q9Wtt||t||S(sThe centroid of the polygon. Returns ======= centroid : Point See Also ======== sympy.geometry.point.Point, sympy.geometry.util.centroid Examples ======== >>> from sympy import Point, Polygon >>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]) >>> poly = Polygon(p1, p2, p3, p4) >>> poly.centroid Point2D(31/18, 11/18) iii(ii(R=R3R R+RR( R<tAtcxtcyR3R9R>R?R@RAtv((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytcentroidVs  cC sd\}}}|j}xtt|D]}||dj\}}||j\} } || | |} ||d|| | d| 7}||d|| | d| 7}||| d||d| | | || 7}q+W|j} |jd} |jd}|d| |d}|d| | d}|d| | |}|dkre|||fS|| |d|d}|| |d| d}|| |d| |d|}|||fS(sReturns the second moment and product moment of area of a two dimensional polygon. Parameters ========== point : Point, two-tuple of sympifiable objects, or None(default=None) point is the point about which second moment of area is to be found. If "point=None" it will be calculated about the axis passing through the centroid of the polygon. Returns ======= I_xx, I_yy, I_xy : number or sympy expression I_xx, I_yy are second moment of area of a two dimensional polygon. I_xy is product moment of area of a two dimensional polygon. Examples ======== >>> from sympy import Point, Polygon, symbols >>> a, b = symbols('a, b') >>> p1, p2, p3, p4, p5 = [(0, 0), (a, 0), (a, b), (0, b), (a/3, b/3)] >>> rectangle = Polygon(p1, p2, p3, p4) >>> rectangle.second_moment_of_area() (a*b**3/12, a**3*b/12, 0) >>> rectangle.second_moment_of_area(p5) (a*b**3/9, a**3*b/9, a**2*b**2/36) References ========== https://en.wikipedia.org/wiki/Second_moment_of_area iiii i(iiiN(R3R R+R=RYRE(R<tpointtI_xxtI_yytI_xyR3R9R>R?R@RARXRUtc_xtc_ytI_xx_ctI_yy_ctI_xy_c((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytsecond_moment_of_areazs*% ""6     "cC sVg}|j}x@tt| dD](}|jt||||dq&W|S(sThe directed line segments that form the sides of the polygon. Returns ======= sides : list of sides Each side is a directed Segment. See Also ======== sympy.geometry.point.Point, sympy.geometry.line.Segment Examples ======== >>> from sympy import Point, Polygon >>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]) >>> poly = Polygon(p1, p2, p3, p4) >>> poly.sides [Segment2D(Point2D(0, 0), Point2D(1, 0)), Segment2D(Point2D(1, 0), Point2D(5, 1)), Segment2D(Point2D(5, 1), Point2D(0, 1)), Segment2D(Point2D(0, 1), Point2D(0, 0))] ii(R6R R+R,R"(R<RJR3R9((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytsidess  &cC si|j}g|D]}|j^q}g|D]}|j^q,}t|t|t|t|fS(swReturn a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure. (R6RBRCtmintmax(R<tvertsR8txstys((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytboundss c C s,|j}|j|d|d|d}xNtdt|D]7}||j||d||d||Ar@tSq@W|j}xt|D]\}}|j}x{t|t|dkrdnd|dD]J}||}|j|kr|j |kr|j |} | r tSqqWqWt S(s_Is the polygon convex? A polygon is convex if all its interior angles are less than 180 degrees and there are no intersections between sides. Returns ======= is_convex : boolean True if this polygon is convex, False otherwise. See Also ======== sympy.geometry.util.convex_hull Examples ======== >>> from sympy import Point, Polygon >>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]) >>> poly = Polygon(p1, p2, p3, p4) >>> poly.is_convex() True iiiii( R6RKR R+tFalseRdt enumerateR3tp1tp2t intersectiontTrue( R<R3RNR9Rdtsitptstjtsjthit((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt is_convexs !-  6 c s,tdd|jks@tfd|jDrDtSg}x3|jD](}|j||djrTdSqTWt|}|j }t t t | d}|j rFd}x||D]t}||} ||d} | j | j| j| j | j| jj} |dkr.| }q| |k rtSqWtSt} |dj \} }x|dD]}||j \}}dt||krdt||krdt| |kr||kr| || ||| }| |ksd|kr| } qqqqn||} }qjW| S(sT Return True if p is enclosed by (is inside of) self. Notes ===== Being on the border of self is considered False. Parameters ========== p : Point Returns ======= encloses_point : True, False or None See Also ======== sympy.geometry.point.Point, sympy.geometry.ellipse.Ellipse.encloses_point Examples ======== >>> from sympy import Polygon, Point >>> from sympy.abc import t >>> p = Polygon((0, 0), (4, 0), (4, 4)) >>> p.encloses_point(Point(2, 1)) True >>> p.encloses_point(Point(2, 2)) False >>> p.encloses_point(Point(5, 5)) False References ========== [1] http://paulbourke.net/geometry/polygonmesh/#insidepoly R'ic3 s|]}|kVqdS(N((t.0ts(R8(s5lib/python2.7/site-packages/sympy/geometry/polygon.pys @siiiN(RR6tanyRdRkR,t free_symbolsRER%R3R*R R+RvRCRBt is_negativeRpReRf(R<R8tlitRXtpolyR3tindicest orientationR9R5R:ttestthit_oddtp1xtp1ytp2xtp2ytxinters((R8s5lib/python2.7/site-packages/sympy/geometry/polygon.pytencloses_pointsF+.      1    ttc C st|dt}|jd|jDkrDtd|jng}|j}d}xy|jD]n}|j|}||}|j|j ||||} |j | t ||k||kf|}qcWt |S(s.A parameterized point on the polygon. The parameter, varying from 0 to 1, assigns points to the position on the perimeter that is that fraction of the total perimeter. So the point evaluated at t=1/2 would return the point from the first vertex that is 1/2 way around the polygon. Parameters ========== parameter : str, optional Default value is 't'. Returns ======= arbitrary_point : Point Raises ====== ValueError When `parameter` already appears in the Polygon's definition. See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Polygon, S, Symbol >>> t = Symbol('t', real=True) >>> tri = Polygon((0, 0), (1, 0), (1, 1)) >>> p = tri.arbitrary_point('t') >>> perimeter = tri.perimeter >>> s1, s2 = [s.length for s in tri.sides[:2]] >>> p.subs(t, (s1 + s2/2)/perimeter) Point2D(1, 1/2) trealcs s|]}|jVqdS(N(tname(Rwtf((s5lib/python2.7/site-packages/sympy/geometry/polygon.pys ssFSymbol %s already appears in object and cannot be used as a parameter.i( R RpRRzRFRTRdtlengthtarbitrary_pointtsubsR,RR( R<t parameterRRdRTtperim_fraction_startRxtside_perim_fractiontperim_fraction_endtpt((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRls+   " c C sIddlm}t|ts7t|d|j}nt|tsUtdn|jrmtdnt }t ddt }|j |}xy|j D]n\}}||||dt } | sqn| d |} t|j|| t kri| |6St }qW|r/td t|ntd t|dS( Ni(tsolveR'sother must be a pointsnon-numeric coordinatesRRtdictisGiven point may not be on %ssGiven point is not on %s(tsympy.solvers.solversRt isinstanceRRRRRFRztNotImplementedErrorRkR RpRR3RRR( R<totherRRtunknowntTR8Rtcondtsoltvalue((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytparameter_values*   cC st|dt}|ddgS(sThe plot interval for the default geometric plot of the polygon. Parameters ========== parameter : str, optional Default value is 't'. Returns ======= plot_interval : list (plot interval) [parameter, lower_bound, upper_bound] Examples ======== >>> from sympy import Polygon >>> p = Polygon((0, 0), (1, 0), (1, 1)) >>> p.plot_interval() [t, 0, 1] Rii(RRp(R<RR((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt plot_intervalsc C s]g}t|tr|jn|g}x8|jD]-}x$|D]}|j|j|q>Wq1Wtt|}g|D]}t|tr{|^q{}g|D]}t|tr|^q}|rI|rIttg|D]%} |D]} | | kr| ^qq} | r5x| D]} |j | qWntt ||Stt |SdS(sThe intersection of polygon and geometry entity. The intersection may be empty and can contain individual Points and complete Line Segments. Parameters ========== other: GeometryEntity Returns ======= intersection : list The list of Segments and Points See Also ======== sympy.geometry.point.Point, sympy.geometry.line.Segment Examples ======== >>> from sympy import Point, Polygon, Line >>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]) >>> poly1 = Polygon(p1, p2, p3, p4) >>> p5, p6, p7 = map(Point, [(3, 2), (1, -1), (0, 2)]) >>> poly2 = Polygon(p5, p6, p7) >>> poly1.intersection(poly2) [Point2D(1/3, 1), Point2D(2/3, 0), Point2D(9/5, 1/5), Point2D(7/3, 1)] >>> poly1.intersection(Line(p1, p2)) [Segment2D(Point2D(0, 0), Point2D(1, 0))] >>> poly1.intersection(p1) [Point2D(0, 0)] N( RR%RdtextendRoR*RRR"tremoveR ( R<totintersection_resulttktsidetside1tentitytpointstsegmentsRZtsegmenttpoints_in_segmentsR9((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRos%! (( > cC st|trdt}xH|jD]=}|j|}|dkrGtjS||kr|}qqW|St|tr|jr|jr|j |St dS(s Returns the shortest distance between self and o. If o is a point, then self does not need to be convex. If o is another polygon self and o must be convex. Examples ======== >>> from sympy import Point, Polygon, RegularPolygon >>> p1, p2 = map(Point, [(0, 0), (7, 5)]) >>> poly = Polygon(*RegularPolygon(p1, 1, 3).vertices) >>> poly.distance(p2) sqrt(61) iN( RRRRdRSRtZeroR%Rvt_do_poly_distanceR(R<RtdistRtcurrent((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRSs   ' c C s|}|j}|j}tj}tj}x8|jD]-}tj||}||kr4|}q4q4Wx8|jD]-}tj||}||kro|}qoqoWtj||} | ||krtjdntdt } tdt} xP|jD]E}|j | j ks7|j | j kr|j | j kr|} qqWxP|jD]E}|j | j ks|j | j krN|j | j krN|} qNqNWtj| | } i} i}x|j D]~}|j | kr| |j j |jn|jg| |j <|j| kr*| |jj |j q|j g| |j>> from sympy.geometry import Point, Polygon >>> square = Polygon(Point(0, 0), Point(0, 1), Point(1, 1), Point(1, 0)) >>> triangle = Polygon(Point(1, 2), Point(2, 2), Point(2, 1)) >>> square._do_poly_distance(triangle) sqrt(2)/2 Description of method used ========================== Method: [1] http://cgm.cs.mcgill.ca/~orm/mind2p.html Uses rotating calipers: [2] https://en.wikipedia.org/wiki/Rotating_calipers and antipodal points: [3] https://en.wikipedia.org/wiki/Antipodal_point s1Polygons may intersect producing erroneous outputii(RYRRR6RRStwarningstwarnRRCRBRdRmR,RnR!tOneRLRpRR"tevalfRe( R<te2te1t e1_centert e2_centert e1_max_radiust e2_max_radiustvertextrt center_distte1_ymaxte2_ymintmin_distte1_connectionste2_connectionsRt e1_currentt e2_currentt support_linetpoint1tpoint2tangle1tangle2te1_nextte2_nextte1_anglete2_anglet e1_segmenttmin_dist_currentt e2_segmenttmin1tmin2((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR/s%        6 6 -    $     $      g?s#66cc99cC sddlm}t||j}g|D]}dj|j|j^q)}dj|ddj|d}djd |||S( s"Returns SVG path element for the Polygon. Parameters ========== scale_factor : float Multiplication factor for the SVG stroke-width. Default is 1. fill_color : str, optional Hex string for fill color. Default is "#66cc99". i(tNs{0},{1}s M {0} L {1} zis L isag@(tsympy.core.evalfRtmapR6tformatRBRCtjoin(R<t scale_factort fill_colorRRgR8tcoordstpath((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt_svgs +#c  sifd}||j}t|t|}|tt|j}t|t|}||kr~|}n|}g|D]}|^q}t|S(Nc s^i}x:ttt|D] \}}|||<||>> from sympy import Line, Segment, Point >>> p = Point(0, 0) >>> q = Point(1, 1) >>> s = Segment(p, q*2) >>> l = Line(p, q) >>> p in q False >>> p in s True >>> q*3 in s False >>> s in l True c3 s|]}|kVqdS(N((RwRx(R(s5lib/python2.7/site-packages/sympy/geometry/polygon.pys Ks( RR%R"RyRdRR6RpRk(R<RR((Rs5lib/python2.7/site-packages/sympy/geometry/polygon.pyt __contains__!s'   N(t__name__t __module__t__doc__R0tpropertyR=t staticmethodRKRQRRRTR6RYRERcRdRjRvRRRRRoRSRRRR(((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR%s.U ,1,$$ ?! . X ;   9   R.cB seZdZddddgZddZedZdZd Zed Z ed Z ed Z e Z ed Z edZedZedZedZedZedZedZedZedZedZdZdZd!dZddd!dZdZedZdZd Z RS("s A regular polygon. Such a polygon has all internal angles equal and all sides the same length. Parameters ========== center : Point radius : number or Basic instance The distance from the center to a vertex n : int The number of sides Attributes ========== vertices center radius rotation apothem interior_angle exterior_angle circumcircle incircle angles Raises ====== GeometryError If the `center` is not a Point, or the `radius` is not a number or Basic instance, or the number of sides, `n`, is less than three. Notes ===== A RegularPolygon can be instantiated with Polygon with the kwarg n. Regular polygons are instantiated with a center, radius, number of sides and a rotation angle. Whereas the arguments of a Polygon are vertices, the vertices of the RegularPolygon must be obtained with the vertices method. See Also ======== sympy.geometry.point.Point, Polygon Examples ======== >>> from sympy.geometry import RegularPolygon, Point >>> r = RegularPolygon(Point(0, 0), 5, 3) >>> r RegularPolygon(Point2D(0, 0), 5, 3, 0) >>> r.vertices[0] Point2D(5, 0) t_nt_centert_radiust_roticK stt|||f\}}}t|dd|}t|tsXtd|n|jrt||dkrtd|qntj |||||}||_ ||_ ||_ |j r|dtj|n||_|S(NR'is r must be an Expr object, not %sisn must be a >= 3, not %s(RRRRRRt is_NumberRRR0RRRt is_numberRRMR(R<R;RR&trotR4tobj((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR0s!      'cC s|j|j|j|jfS(s- Returns the center point, the radius, the number of sides, and the orientation angle. Examples ======== >>> from sympy import RegularPolygon, Point >>> r = RegularPolygon(Point(0, 0), 5, 3) >>> r.args (Point2D(0, 0), 5, 3, 0) (RRRR(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR3scC sdt|jS(NsRegularPolygon(%s, %s, %s, %s)(RR3(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt__str__scC sdt|jS(NsRegularPolygon(%s, %s, %s, %s)(RR3(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt__repr__scC s@|j\}}}}t|||jddtt|S(sReturns the area. Examples ======== >>> from sympy.geometry import RegularPolygon >>> square = RegularPolygon((0, 0), 1, 4) >>> square.area 2 >>> _ == square.length**2 True ii(R3R RRR(R<R;RR&R((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR=scC s|jdtt|jS(sReturns the length of the sides. The half-length of the side and the apothem form two legs of a right triangle whose hypotenuse is the radius of the regular polygon. Examples ======== >>> from sympy.geometry import RegularPolygon >>> from sympy import sqrt >>> s = square_in_unit_circle = RegularPolygon((0, 0), 1, 4) >>> s.length sqrt(2) >>> sqrt((_/2)**2 + s.apothem**2) == s.radius True i(tradiusRRR(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRscC s|jS(sThe center of the RegularPolygon This is also the center of the circumscribing circle. Returns ======= center : Point See Also ======== sympy.geometry.point.Point, sympy.geometry.ellipse.Ellipse.center Examples ======== >>> from sympy.geometry import RegularPolygon, Point >>> rp = RegularPolygon(Point(0, 0), 5, 4) >>> rp.center Point2D(0, 0) (R(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytcenterscC s|jS(s Alias for center. Examples ======== >>> from sympy.geometry import RegularPolygon, Point >>> rp = RegularPolygon(Point(0, 0), 5, 4) >>> rp.circumcenter Point2D(0, 0) (R(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt circumcenters cC s|jS(s(Radius of the RegularPolygon This is also the radius of the circumscribing circle. Returns ======= radius : number or instance of Basic See Also ======== sympy.geometry.line.Segment.length, sympy.geometry.ellipse.Circle.radius Examples ======== >>> from sympy import Symbol >>> from sympy.geometry import RegularPolygon, Point >>> radius = Symbol('r') >>> rp = RegularPolygon(Point(0, 0), radius, 4) >>> rp.radius r (R(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRscC s|jS(s( Alias for radius. Examples ======== >>> from sympy import Symbol >>> from sympy.geometry import RegularPolygon, Point >>> radius = Symbol('r') >>> rp = RegularPolygon(Point(0, 0), radius, 4) >>> rp.circumradius r (R(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt circumradius,scC s|jS(s<CCW angle by which the RegularPolygon is rotated Returns ======= rotation : number or instance of Basic Examples ======== >>> from sympy import pi >>> from sympy.abc import a >>> from sympy.geometry import RegularPolygon, Point >>> RegularPolygon(Point(0, 0), 3, 4, pi/4).rotation pi/4 Numerical rotation angles are made canonical: >>> RegularPolygon(Point(0, 0), 3, 4, a).rotation a >>> RegularPolygon(Point(0, 0), 3, 4, pi).rotation 0 (R(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytrotation=scC s|jttj|jS(sAThe inradius of the RegularPolygon. The apothem/inradius is the radius of the inscribed circle. Returns ======= apothem : number or instance of Basic See Also ======== sympy.geometry.line.Segment.length, sympy.geometry.ellipse.Circle.radius Examples ======== >>> from sympy import Symbol >>> from sympy.geometry import RegularPolygon, Point >>> radius = Symbol('r') >>> rp = RegularPolygon(Point(0, 0), radius, 4) >>> rp.apothem sqrt(2)*r/2 (RRRRMR(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytapothemYscC s|jS(s/ Alias for apothem. Examples ======== >>> from sympy import Symbol >>> from sympy.geometry import RegularPolygon, Point >>> radius = Symbol('r') >>> rp = RegularPolygon(Point(0, 0), radius, 4) >>> rp.inradius sqrt(2)*r/2 (R(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytinradiusvscC s|jdtj|jS(sMeasure of the interior angles. Returns ======= interior_angle : number See Also ======== sympy.geometry.line.LinearEntity.angle_between Examples ======== >>> from sympy.geometry import RegularPolygon, Point >>> rp = RegularPolygon(Point(0, 0), 4, 8) >>> rp.interior_angle 3*pi/4 i(RRRM(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytinterior_anglescC sdtj|jS(sMeasure of the exterior angles. Returns ======= exterior_angle : number See Also ======== sympy.geometry.line.LinearEntity.angle_between Examples ======== >>> from sympy.geometry import RegularPolygon, Point >>> rp = RegularPolygon(Point(0, 0), 4, 8) >>> rp.exterior_angle pi/4 i(RRMR(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytexterior_anglescC st|j|jS(sThe circumcircle of the RegularPolygon. Returns ======= circumcircle : Circle See Also ======== circumcenter, sympy.geometry.ellipse.Circle Examples ======== >>> from sympy.geometry import RegularPolygon, Point >>> rp = RegularPolygon(Point(0, 0), 4, 8) >>> rp.circumcircle Circle(Point2D(0, 0), 4) (R RR(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt circumcirclescC st|j|jS(sThe incircle of the RegularPolygon. Returns ======= incircle : Circle See Also ======== inradius, sympy.geometry.ellipse.Circle Examples ======== >>> from sympy.geometry import RegularPolygon, Point >>> rp = RegularPolygon(Point(0, 0), 4, 7) >>> rp.incircle Circle(Point2D(0, 0), 4*cos(pi/7)) (R RR(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytincirclescC s1i}|j}x|jD]}|||>> from sympy import RegularPolygon, Point >>> r = RegularPolygon(Point(0, 0), 5, 3) >>> r.angles {Point2D(-5/2, -5*sqrt(3)/2): pi/3, Point2D(-5/2, 5*sqrt(3)/2): pi/3, Point2D(5, 0): pi/3} (RR6(R<RORPRX((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRQs  cC sU|j}t||j}||jkr.tS||jkrAtStj||SdS(sN Return True if p is enclosed by (is inside of) self. Notes ===== Being on the border of self is considered False. The general Polygon.encloses_point method is called only if a point is not within or beyond the incircle or circumcircle, respectively. Parameters ========== p : Point Returns ======= encloses_point : True, False or None See Also ======== sympy.geometry.ellipse.Ellipse.encloses_point Examples ======== >>> from sympy import RegularPolygon, S, Point, Symbol >>> p = RegularPolygon((0, 0), 3, 4) >>> p.encloses_point(Point(0, 0)) True >>> r, R = p.inradius, p.circumradius >>> p.encloses_point(Point((r + R)/2, 0)) True >>> p.encloses_point(Point(R/2, R/2 + (R - r)/10)) False >>> t = Symbol('t', real=True) >>> p.encloses_point(p.arbitrary_point().subs(t, S.Half)) False >>> p.encloses_point(Point(5, 5)) False N( RR"RRRkRRpR%R(R<R8R;td((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRs0 cC s|j|7_dS(sIncrement *in place* the virtual Polygon's rotation by ccw angle. See also: rotate method which moves the center. >>> from sympy import Polygon, Point, pi >>> r = Polygon(Point(0,0), 1, n=3) >>> r.vertices[0] Point2D(1, 0) >>> r.spin(pi/6) >>> r.vertices[0] Point2D(sqrt(3)/2, 1/2) See Also ======== rotation rotate : Creates a copy of the RegularPolygon rotated about a Point N(R(R<tangle((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytspin;scC s7t||j}|j|7_tj|||S(sOverride GeometryEntity.rotate to first rotate the RegularPolygon about its center. >>> from sympy import Point, RegularPolygon, Polygon, pi >>> t = RegularPolygon(Point(1, 0), 1, 3) >>> t.vertices[0] # vertex on x-axis Point2D(2, 0) >>> t.rotate(pi/2).vertices[0] # vertex on y axis now Point2D(0, 2) See Also ======== rotation spin : Rotates a RegularPolygon in place (ttypeR3RRtrotate(R<RRR((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR QsicC s|rAt|dd}|j| jj||j|jS||krft|jj||S|j\}}}}||9}|j||||S(sOverride GeometryEntity.scale since it is the radius that must be scaled (if x == y) or else a new Polygon must be returned. >>> from sympy import RegularPolygon Symmetric scaling returns a RegularPolygon: >>> RegularPolygon((0, 0), 1, 4).scale(2, 2) RegularPolygon(Point2D(0, 0), 2, 4, 0) Asymmetric scaling returns a kite as a Polygon: >>> RegularPolygon((0, 0), 1, 4).scale(2, 1) Polygon(Point2D(2, 0), Point2D(0, 1), Point2D(-2, 0), Point2D(0, -1)) R'i(Rt translateR3tscaleR%R6tfunc(R<RBRCRR;RR&R((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR hs)  cC s|j\}}}}|jd}||}|j|}|j|} | |} td| } td|} | j| } || 7}|j|| ||S(s5Override GeometryEntity.reflect since this is not made of only points. Examples ======== >>> from sympy import RegularPolygon, Line >>> RegularPolygon((0, 0), 1, 4).reflect(Line((0, 1), slope=-2)) RegularPolygon(Point2D(4/5, 2/5), -1, 4, atan(4/3)) i(ii(ii(R3R6treflectR#t closing_angleR (R<tlineR;RR&RRXRtcctvvtddtl1tl2RP((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRs     c C s|j}t|j}|j}dtj|j}gt|jD]G}t|j |t ||||j |t |||^qES(sThe vertices of the RegularPolygon. Returns ======= vertices : list Each vertex is a Point. See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy.geometry import RegularPolygon, Point >>> rp = RegularPolygon(Point(0, 0), 5, 4) >>> rp.vertices [Point2D(5, 0), Point2D(0, 5), Point2D(-5, 0), Point2D(0, -5)] i( RtabsRRRRMRR RRBRRCR(R<R;RRRXR((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR6s   cC sBt|tstSt|ts2tj||S|j|jkS(N(RR%RkR.t__eq__R3(R<R((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRs cC stt|jS(N(tsuperR.t__hash__(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRsN(!RRRt __slots__R0RR3RRR=RRRYRRRRRRRRRRRQRRRER R RR6RR(((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR.Vs:<    :    R1cB seZdZdZedZdZdZdZdZ dZ edZ ed Z ed Z ed Zed Zd ZedZedZedZedZedZedZedZedZRS(s A polygon with three vertices and three sides. Parameters ========== points : sequence of Points keyword: asa, sas, or sss to specify sides/angles of the triangle Attributes ========== vertices altitudes orthocenter circumcenter circumradius circumcircle inradius incircle exradii medians medial nine_point_circle Raises ====== GeometryError If the number of vertices is not equal to three, or one of the vertices is not a Point, or a valid keyword is not given. See Also ======== sympy.geometry.point.Point, Polygon Examples ======== >>> from sympy.geometry import Triangle, Point >>> Triangle(Point(0, 0), Point(4, 0), Point(4, 3)) Triangle(Point2D(0, 0), Point2D(4, 0), Point2D(4, 3)) Keywords sss, sas, or asa can be used to give the desired side lengths (in order) and interior angles (in degrees) that define the triangle: >>> Triangle(sss=(3, 4, 5)) Triangle(Point2D(0, 0), Point2D(3, 0), Point2D(3, 4)) >>> Triangle(asa=(30, 1, 30)) Triangle(Point2D(0, 0), Point2D(1, 0), Point2D(1/2, sqrt(3)/6)) >>> Triangle(sas=(1, 45, 2)) Triangle(Point2D(0, 0), Point2D(2, 0), Point2D(sqrt(2)/2, sqrt(2)/2)) c O st|dkrd|krEtg|dD]}t|^q,Sd|krxtg|dD]}t|^q_Sd|krtg|dD]}t|^qSd}t|ng|D]}t|dd|^q}g}x7|D]/}|r||dkrqn|j|qWt|d kr[|d|d kr[|jnd }x|t|dkrt|dkrt ||||d ||dgd t \}} } tj || | r|||$t(R+t_sssRt_asat_sasRRR,R)tsortedRR/RER*tfilterRR0R"( R2R3R4R5tmsgR6R7R8R9R:R;((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR0s@ ' ' '( & +5  cC s|jS(sThe triangle's vertices Returns ======= vertices : tuple Each element in the tuple is a Point See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy.geometry import Triangle, Point >>> t = Triangle(Point(0, 0), Point(4, 0), Point(4, 3)) >>> t.vertices (Point2D(0, 0), Point2D(4, 0), Point2D(4, 3)) (R3(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR6-scC st|tstSg|jD]}|j^q\}}}g|jD]}|j^qE}d}|||||p|||||p|||||p|||||p|||||p|||||S(sIs another triangle similar to this one. Two triangles are similar if one can be uniformly scaled to the other. Parameters ========== other: Triangle Returns ======= is_similar : boolean See Also ======== sympy.geometry.entity.GeometryEntity.is_similar Examples ======== >>> from sympy.geometry import Triangle, Point >>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3)) >>> t2 = Triangle(Point(0, 0), Point(-4, 0), Point(-4, -3)) >>> t1.is_similar(t2) True >>> t2 = Triangle(Point(0, 0), Point(-4, 0), Point(-4, -4)) >>> t1.is_similar(t2) False c S sRt||}t||}t||}t||koQt||kS(N(Rtbool( tu1tu2tu3tv1tv2tv3RRte3((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt _are_similaros(RR%RkRdR(tt1tt2Rts1_1ts1_2ts1_3ts2R/((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt is_similarGs"( cC std|jD S(skAre all the sides the same length? Returns ======= is_equilateral : boolean See Also ======== sympy.geometry.entity.GeometryEntity.is_similar, RegularPolygon is_isosceles, is_right, is_scalene Examples ======== >>> from sympy.geometry import Triangle, Point >>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3)) >>> t1.is_equilateral() False >>> from sympy import sqrt >>> t2 = Triangle(Point(0, 0), Point(10, 0), Point(5, 5*sqrt(3))) >>> t2.is_equilateral() True cs s|]}|jVqdS(N(R(RwRx((s5lib/python2.7/site-packages/sympy/geometry/polygon.pys s(RRd(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytis_equilateral}scC std|jDS(sAre two or more of the sides the same length? Returns ======= is_isosceles : boolean See Also ======== is_equilateral, is_right, is_scalene Examples ======== >>> from sympy.geometry import Triangle, Point >>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(2, 4)) >>> t1.is_isosceles() True cs s|]}|jVqdS(N(R(RwRx((s5lib/python2.7/site-packages/sympy/geometry/polygon.pys s(RRd(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt is_isoscelesscC std|jD S(sAre all the sides of the triangle of different lengths? Returns ======= is_scalene : boolean See Also ======== is_equilateral, is_isosceles, is_right Examples ======== >>> from sympy.geometry import Triangle, Point >>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(1, 4)) >>> t1.is_scalene() True cs s|]}|jVqdS(N(R(RwRx((s5lib/python2.7/site-packages/sympy/geometry/polygon.pys s(RRd(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt is_scalenescC sU|j}tj|d|dpTtj|d|dpTtj|d|dS(sIs the triangle right-angled. Returns ======= is_right : boolean See Also ======== sympy.geometry.line.LinearEntity.is_perpendicular is_equilateral, is_isosceles, is_scalene Examples ======== >>> from sympy.geometry import Triangle, Point >>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3)) >>> t1.is_right() True iii(RdR"tis_perpendicular(R<Rx((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytis_rights cC sj|j}|j}i|dj|d|d6|dj|d|d6|dj|d|d6S(sThe altitudes of the triangle. An altitude of a triangle is a segment through a vertex, perpendicular to the opposite side, with length being the height of the vertex measured from the line containing the side. Returns ======= altitudes : dict The dictionary consists of keys which are vertices and values which are Segments. See Also ======== sympy.geometry.point.Point, sympy.geometry.line.Segment.length Examples ======== >>> from sympy.geometry import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1) >>> t = Triangle(p1, p2, p3) >>> t.altitudes[p1] Segment2D(Point2D(0, 0), Point2D(1/2, 1/2)) iii(RdR6tperpendicular_segment(R<RxRX((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt altitudess   cC s?|j}|j}t||djt||ddS(s"The orthocenter of the triangle. The orthocenter is the intersection of the altitudes of a triangle. It may lie inside, outside or on the triangle. Returns ======= orthocenter : Point See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy.geometry import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1) >>> t = Triangle(p1, p2, p3) >>> t.orthocenter Point2D(0, 0) ii(R=R6R!Ro(R<R5RX((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt orthocenter s  cC sgg|jD]}|j^q \}}}|j|sVt|||j|n|j|dS(sThe circumcenter of the triangle The circumcenter is the center of the circumcircle. Returns ======= circumcenter : Point See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy.geometry import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1) >>> t = Triangle(p1, p2, p3) >>> t.circumcenter Point2D(1/2, 1/2) i(Rdtperpendicular_bisectorRotprint(R<RBR5R:R;((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR*s+cC stj|j|jdS(sThe radius of the circumcircle of the triangle. Returns ======= circumradius : number of Basic instance See Also ======== sympy.geometry.ellipse.Circle.radius Examples ======== >>> from sympy import Symbol >>> from sympy.geometry import Point, Triangle >>> a = Symbol('a') >>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, a) >>> t = Triangle(p1, p2, p3) >>> t.circumradius sqrt(a**2/4 + 1/4) i(RRSRR6(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRHscC st|j|jS(sThe circle which passes through the three vertices of the triangle. Returns ======= circumcircle : Circle See Also ======== sympy.geometry.ellipse.Circle Examples ======== >>> from sympy.geometry import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1) >>> t = Triangle(p1, p2, p3) >>> t.circumcircle Circle(Point2D(1/2, 1/2), sqrt(2)/2) (R RR(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRcscC s|j}|j}|j}t|dt|d|j|dd}t|dt|d|j|dd}t|dt|d|j|dd}i||d6||d6||d6S(sThe angle bisectors of the triangle. An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. Returns ======= bisectors : dict Each key is a vertex (Point) and each value is the corresponding bisector (Segment). See Also ======== sympy.geometry.point.Point, sympy.geometry.line.Segment Examples ======== >>> from sympy.geometry import Point, Triangle, Segment >>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1) >>> t = Triangle(p1, p2, p3) >>> from sympy import sqrt >>> t.bisectors()[p2] == Segment(Point(1, 0), Point(0, sqrt(2) - 1)) True iii(RdR6tincenterR"R!Ro(R<RxRXR;RRtl3((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt bisectors}s   111c C s|j}tgdddgD]}||j^q}t|}|j}t|jtg|D]}|j^q`|}t|jtg|D]}|j^q|}t ||S(sThe center of the incircle. The incircle is the circle which lies inside the triangle and touches all three sides. Returns ======= incenter : Point See Also ======== incircle, sympy.geometry.point.Point Examples ======== >>> from sympy.geometry import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1) >>> t = Triangle(p1, p2, p3) >>> t.incenter Point2D(1 - sqrt(2)/2, 1 - sqrt(2)/2) iii( RdRRtsumR6RtdotRBRCR( R<RxR9tlR8RXtviRBRC((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRAs /  55cC std|j|jS(sThe radius of the incircle. Returns ======= inradius : number of Basic instance See Also ======== incircle, sympy.geometry.ellipse.Circle.radius Examples ======== >>> from sympy.geometry import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(4, 0), Point(0, 3) >>> t = Triangle(p1, p2, p3) >>> t.inradius 1 i(RR=RT(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRscC st|j|jS(s*The incircle of the triangle. The incircle is the circle which lies inside the triangle and touches all three sides. Returns ======= incircle : Circle See Also ======== sympy.geometry.ellipse.Circle Examples ======== >>> from sympy.geometry import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(2, 0), Point(0, 2) >>> t = Triangle(p1, p2, p3) >>> t.incircle Circle(Point2D(2 - sqrt(2), 2 - sqrt(2)), 2 - sqrt(2)) (R RAR(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRscC s|j}|dj}|dj}|dj}|||d}|j}it||||jd6t||||jd6t||||jd6}|S(sThe radius of excircles of a triangle. An excircle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Returns ======= exradii : dict See Also ======== sympy.geometry.polygon.Triangle.inradius Examples ======== The exradius touches the side of the triangle to which it is keyed, e.g. the exradius touching side 2 is: >>> from sympy.geometry import Point, Triangle, Segment2D, Point2D >>> p1, p2, p3 = Point(0, 0), Point(6, 0), Point(0, 2) >>> t = Triangle(p1, p2, p3) >>> t.exradii[t.sides[2]] -2 + sqrt(10) References ========== [1] http://mathworld.wolfram.com/Exradius.html [2] http://mathworld.wolfram.com/Excircles.html iii(RdRR=R(R<RR5R:R;RxR=texradii((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyRHs&     cC ss|j}|j}it|d|dj|d6t|d|dj|d6t|d|dj|d6S(sThe medians of the triangle. A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas. Returns ======= medians : dict Each key is a vertex (Point) and each value is the median (Segment) at that point. See Also ======== sympy.geometry.point.Point.midpoint, sympy.geometry.line.Segment.midpoint Examples ======== >>> from sympy.geometry import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1) >>> t = Triangle(p1, p2, p3) >>> t.medians[p1] Segment2D(Point2D(0, 0), Point2D(1/2, 1/2)) iii(RdR6R"tmidpoint(R<RxRX((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytmedians. s   "cC s.|j}t|dj|dj|djS(s"The medial triangle of the triangle. The triangle which is formed from the midpoints of the three sides. Returns ======= medial : Triangle See Also ======== sympy.geometry.line.Segment.midpoint Examples ======== >>> from sympy.geometry import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1) >>> t = Triangle(p1, p2, p3) >>> t.medial Triangle(Point2D(1/2, 0), Point2D(1/2, 1/2), Point2D(0, 1/2)) iii(RdR1RI(R<Rx((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytmedialR s cC st|jjS(sThe nine-point circle of the triangle. Nine-point circle is the circumcircle of the medial triangle, which passes through the feet of altitudes and the middle points of segments connecting the vertices and the orthocenter. Returns ======= nine_point_circle : Circle See also ======== sympy.geometry.line.Segment.midpoint sympy.geometry.polygon.Triangle.medial sympy.geometry.polygon.Triangle.orthocenter Examples ======== >>> from sympy.geometry import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1) >>> t = Triangle(p1, p2, p3) >>> t.nine_point_circle Circle(Point2D(1/4, 1/4), sqrt(2)/4) (R RKR6(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytnine_point_circleo scC s&|jr|jSt|j|jS(sThe Euler line of the triangle. The line which passes through circumcenter, centroid and orthocenter. Returns ======= eulerline : Line (or Point for equilateral triangles in which case all centers coincide) Examples ======== >>> from sympy.geometry import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1) >>> t = Triangle(p1, p2, p3) >>> t.eulerline Line2D(Point2D(0, 0), Point2D(1/2, 1/2)) (R7R>R!R(R<((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt eulerline s (RRRR0RR6R6R7R8R9R;R=R>RRRRCRARRRHRJRKRLRM(((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR1s,8 * 6    $ %#2$ cC s |tdS(sAReturn the radian value for the given degrees (pi = 180 degrees).i(R(R((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytrad scC s |tdS(sAReturn the degree value for the given radians (pi = 180 degrees).i(R(R((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytdeg scC stt|}|S(N(RRN(Rtrv((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyt_slope scC sWtddt|jt|dfdtd|d}td|df|S(s8Return triangle having side with length l on the x-axis.itslopei(ii(ii(R!RQRoR1(td1RFtd2txy((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR" s)cC s|td|}t|df|}g|j|D]}|jjr4|^q4}|s\dS|d}td|df|S(s7Return triangle having side of length l1 on the x-axis.i(iiN(ii(R RoRCtis_nonnegativeRER1(RRRBtc1tc2R5tinterR((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR! s. cC s]tdd}t|d}ttt||tt||}t|||S(s9Return triangle having side with length l2 on the x-axis.i(RRRNRR1(RRRRmRntp3((s5lib/python2.7/site-packages/sympy/geometry/polygon.pyR# s/(Bt __future__RRt sympy.coreRRRRRRtsympy.core.compatibilityRR R tsympy.core.symbolR R t$sympy.functions.elementary.complexesR t$sympy.functions.elementary.piecewiseRt(sympy.functions.elementary.trigonometricRRRtsympy.geometry.exceptionsRt sympy.logicRtsympy.matricesRtsympy.simplifyRtsympy.utilitiesRtsympy.utilities.iterablesRRRRRtsympy.utilities.miscRRRRRZRtellipseR RR!R"R#tsympyR$RR%R.R1RNRORQR"R!R#(((s5lib/python2.7/site-packages/sympy/geometry/polygon.pytsH.( ?u