\hk@SSKJrJrJrJrJr SSKJr SSKJ r J r SSK J r J r Jr SSKJr SSKJr SSKJrJrJr SS KJr SS KJrJr SS KJr SS KJrJ r J!r! SS K"J#r# SSK$J%r% SSK&J'r' SSK(J)r) SSK*J+r+ SSK,J-r-J.r.J/r/J0r0J1r1 SSK2J3r3J4r4 SSK5J6r6 SSK7r7\8"S5Vs/sH n\ "SSS9PM snur9r:r;"SS\5r<"SS\<5r="SS\<5r>S r?S!r@S"rAS#rBS$rCS%rDgs snf)&)ExprSoopisympify)N)default_sort_keyordered)_symbolDummySymbol)sign) Piecewise)cossintan)Circle)GeometryEntity GeometrySet) GeometryError)LineSegmentRay)Point)And)Matrix)simplify)solve)has_dups has_varietyuniq rotate_leftleast_rotation)as_int func_name) prec_to_dpsN polygon_dummyTrealc<\rSrSrSrSrSS.Sjr\S5r\ S5r \S 5r \S 5r \S 5r \S 5r\S 5rS"SjrS"SjrSrS"Sjr\S5r\S5rSrSrS#SjrSrS#SjrSrSrSrSrS$SjrSr S r!S"S!jr"Sr#g)%PolygonaA 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 A sequence of points. n : int, optional If $> 0$, an n-sided RegularPolygon is created. Default value is $0$. 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 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 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) r)ncU(aX[U5n[U5S:XaURU5 O![U5S:XaURSU5 [ U0UD6$UVs/sHn[ U4SS0UD6PM nn/nUH%nU(a XvS:XaMURU5 M' [U5S:aUSUS:XaUR 5 SnU[U5S- :a[U5S:aXhXhS-XhS-pn[ R"XIU 5(a+UR US-5 XJ:XaUR U5 OUS- nU[U5S- :a[U5S:aM[U5n[U5S:a[R"U/UQ70UD6$[U5S:Xa [U0UD6$[U5S:Xa [U0UD6$[ U0UD6$s snf)Nr(dimrr) listlenappendinsertRegularPolygonrpop is_collinearr__new__Triangler) clsr0argskwargsaverticesnoduppibcs N/var/www/auris/envauris/lib/python3.13/site-packages/sympy/geometry/polygon.pyr=Polygon.__new__zs :D4yA~ ATa Aq!!4262 27;eBi583 IIK #e*q. SZ!^h!e eEl!A!!!** !a% 6IIaLQ#e*q. SZ!^; x=1 !))#CCFC C ]a X00 0 ]a H// /(-f- -?=s$G,cSnURn[[U55H2nX#S- RupEX#RupgXU-Xe-- - nM4 [U5S- $)a+ The area of the polygon. Notes ===== The area calculation can be positive or negative based on the orientation of the points. If any side of the polygon crosses any other side, there will be areas having opposite signs. See Also ======== sympy.geometry.ellipse.Ellipse.area Examples ======== >>> from sympy import Point, Polygon >>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]) >>> poly = Polygon(p1, p2, p3, p4) >>> poly.area 3 In the Z shaped polygon (with the lower right connecting back to the upper left) the areas cancel out: >>> Z = Polygon((0, 1), (1, 1), (0, 0), (1, 0)) >>> Z.area 0 In the M shaped polygon, areas do not cancel because no side crosses any other (though there is a point of contact). >>> M = Polygon((0, 0), (0, 1), (2, 0), (3, 1), (3, 0)) >>> M.area -3/2 rrr2)r@ranger7r)selfarear@rFx1y1x2y2s rIrN Polygon.areasiRyys4y!Aa%[%%FBW\\FB rEBEM !D"~!!cX- nX - n[URUR-URUR-- 5nURnUc [ S5eU$)zReturn 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._is_clockwise(a, b, c) True >>> Polygon._is_clockwise(a, c, b) False zCan't determine orientation)rxyis_nonpositive ValueError)rBrGrHbacat_arearess rI _is_clockwisePolygon._is_clockwisesZU U"$$rtt)bdd244i/0## ;:; ; rTc&URn[U5n0n[U5HlnXS- XS- Xpvn[Xe5R [Xg55nUR XVU5(aS[ R-U- X6'MhXU'Mn [UR55[ R- S- US- - S- Rn U (a'S[ R-n UH nXU- X6'M U$U c [S5eU$)aThe internal angle at each vertex. Returns ======= angles : dict A dictionary where each key is a vertex and each value is the internal angle at that vertex. The vertices are represented as Points. See Also ======== sympy.geometry.point.Point, sympy.geometry.line.LinearEntity.angle_between 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.angles[p1] pi/2 >>> poly.angles[p2] acos(-4*sqrt(17)/17) r2rz(could not determine Polygon orientation.) rCr7rLr angle_betweenr^rPisumvalues is_positiverY) rMr@r0retrFrBrGrH reflex_angwrongtwo_pis rIanglesPolygon.angless<}} IqAq5k4A;!AQ00Q;J!!!**144*,#A cjjl#ADD(*QU3a7DD qttVFa& ]GH H rTc4URSR$)Nr)rCambient_dimensionrMs rIrmPolygon.ambient_dimension s}}Q111rTcSnURn[[U55HnXUS- RX#5- nM [ U5$)aJThe 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 rr)rCrLr7distancer)rMrEr@rFs rI perimeterPolygon.perimeter$sL. }}s4y!A a!e%%dg. .A"{rTc,[UR5$)aThe 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) )r6r@rns rIrCPolygon.verticesAsDDIIrTc<SSUR-- nSup#URn[[U55HAnXES- RupgXERupXi-X-- n X*Xh--- nX:Xy--- nMC [ [ X-5[ X-55$)aTThe centroid of the polygon. Returns ======= centroid : Point See Also ======== sympy.geometry.point.Point, sympy.geometry.util.centroid Examples ======== >>> from sympy import Point, Polygon >>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]) >>> poly = Polygon(p1, p2, p3, p4) >>> poly.centroid Point2D(31/18, 11/18) rrr)rNr@rLr7rr) rMAcxcyr@rFrOrPrQrRvs rIcentroidPolygon.centroides0 q{Oyys4y!Aa%[%%FBW\\FB A RW+ B RW+ B " Xad^Xad^44rTNcnSup#nURn[[U55HznXVS- RupxXVRupXz-X-- n X(S-X--U S--U -- nX7S-Xy--U S--U -- nXGU -SU-U--SU -U --X--U -- nM| URn UR Sn UR SnUS- XS--- nUS- XS--- nUS- XU--- nUcUUU4$XUSU- S---nUXSU - S---nUXSU - USU- ---nX#U4$)aReturns the second moment and product moment of area of a two dimensional polygon. Parameters ========== point : Point, two-tuple of sympifyable objects, or None(default=None) point is the point about which second moment of area is to be found. If "point=None" it will be calculated about the axis passing through the centroid of the polygon. Returns ======= I_xx, I_yy, I_xy : number or SymPy expression I_xx, I_yy are second moment of area of a two dimensional polygon. I_xy is product moment of area of a two dimensional polygon. Examples ======== >>> from sympy import Polygon, symbols >>> a, b = symbols('a, b') >>> p1, p2, p3, p4, p5 = [(0, 0), (a, 0), (a, b), (0, b), (a/3, b/3)] >>> rectangle = Polygon(p1, p2, p3, p4) >>> rectangle.second_moment_of_area() (a*b**3/12, a**3*b/12, 0) >>> rectangle.second_moment_of_area(p5) (a*b**3/9, a**3*b/9, a**2*b**2/36) References ========== .. [1] https://en.wikipedia.org/wiki/Second_moment_of_area )rrrrr2r )rCrLr7r@rNr})rMpointI_xxI_yyI_xyr@rFrOrPrQrRr|ryc_xc_yI_xx_cI_yy_cI_xy_cs rIsecond_moment_of_areaPolygon.second_moment_of_areasJ#D}}s4y!AA#Y^^FBW\\FB A URU]RU*A- -D URU]RU*A- -D UQrT"W_qtBw.69 9D " IImmAmmAr'aaj)r'aaj)r'aSk* =66) )U1Xc\A-..1Xc\A-..1Xc\E!HSL9::4rTcU(aURup#OURnUup#[USS9n[U[RS9nUR U5nUR U5nUSR USR ::aUSOUSnUSR USR ::aUSOUSn URR U- UR -n U RRU- U R -n X4$)ac Returns the first moment of area of a two-dimensional polygon with respect to a certain point of interest. First moment of area is a measure of the distribution of the area of a polygon in relation to an axis. The first moment of area of the entire polygon about its own centroid is always zero. Therefore, here it is calculated for an area, above or below a certain point of interest, that makes up a smaller portion of the polygon. This area is bounded by the point of interest and the extreme end (top or bottom) of the polygon. The first moment for this area is is then determined about the centroidal axis of the initial polygon. References ========== .. [1] https://skyciv.com/docs/tutorials/section-tutorials/calculating-the-statical-or-first-moment-of-area-of-beam-sections/?cc=BMD .. [2] https://mechanicalc.com/reference/cross-sections Parameters ========== point: Point, two-tuple of sympifyable objects, or None (default=None) point is the point above or below which the area of interest lies If ``point=None`` then the centroid acts as the point of interest. Returns ======= Q_x, Q_y: number or SymPy expressions Q_x is the first moment of area about the x-axis Q_y is the first moment of area about the y-axis A negative sign indicates that the section modulus is determined for a section below (or left of) the centroidal axis Examples ======== >>> from sympy import Point, Polygon >>> a, b = 50, 10 >>> p1, p2, p3, p4 = [(0, b), (0, 0), (a, 0), (a, b)] >>> p = Polygon(p1, p2, p3, p4) >>> p.first_moment_of_area() (625, 3125) >>> p.first_moment_of_area(point=Point(30, 7)) (525, 3000) rsloper)r}rrInfinity cut_sectionrNrWrV) rMrxcych_linev_lineh_polyv_polypoly_1poly_2Q_xQ_ys rIfirst_moment_of_areaPolygon.first_moment_of_areas` ]]FBMMEFBe1%e1::.!!&)!!&)$Qinnq >F1I$Qinnq >F1I  2%v{{2  2%v{{2xrTc8UR5nUSUS-$)a7Returns the polar modulus of a two-dimensional polygon It is a constituent of the second moment of area, linked through the perpendicular axis theorem. While the planar second moment of area describes an object's resistance to deflection (bending) when subjected to a force applied to a plane parallel to the central axis, the polar second moment of area describes an object's resistance to deflection when subjected to a moment applied in a plane perpendicular to the object's central axis (i.e. parallel to the cross-section) Examples ======== >>> from sympy import Polygon, symbols >>> a, b = symbols('a, b') >>> rectangle = Polygon((0, 0), (a, 0), (a, b), (0, b)) >>> rectangle.polar_second_moment_of_area() a**3*b/12 + a*b**3/12 References ========== .. [1] https://en.wikipedia.org/wiki/Polar_moment_of_inertia rr)r)rM second_moments rIpolar_second_moment_of_area#Polygon.polar_second_moment_of_area s'6224 Q-"222rTcURup#Uc0URupEpg[X5- Xs- 5n[X$- Xb- 5n OURU- nURU- n UR 5n U SU- n U SU - n X4$)aReturns a tuple with the section modulus of a two-dimensional polygon. Section modulus is a geometric property of a polygon defined as the ratio of second moment of area to the distance of the extreme end of the polygon from the centroidal axis. Parameters ========== point : Point, two-tuple of sympifyable objects, or None(default=None) point is the point at which section modulus is to be found. If "point=None" it will be calculated for the point farthest from the centroidal axis of the polygon. Returns ======= S_x, S_y: numbers or SymPy expressions S_x is the section modulus with respect to the x-axis S_y is the section modulus with respect to the y-axis A negative sign indicates that the section modulus is determined for a point below the centroidal axis Examples ======== >>> from sympy import symbols, Polygon, Point >>> a, b = symbols('a, b', positive=True) >>> rectangle = Polygon((0, 0), (a, 0), (a, b), (0, b)) >>> rectangle.section_modulus() (a*b**2/6, a**2*b/6) >>> rectangle.section_modulus(Point(a/4, b/4)) (-a*b**2/3, -a**2*b/3) References ========== .. [1] https://en.wikipedia.org/wiki/Section_modulus rr)r}boundsmaxrWrVr) rMrx_cy_cx_miny_minx_maxy_maxrWrVrS_xS_ys rIsection_modulusPolygon.section_modulus,sT== =)- &E%CK-ACK-A# A# A113 Aq Aq xrTc /nURn[[U5*S5H%nUR[ X#X#S-55 M' U$)a+The 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))] rr)rCrLr7r8r)rMr]r@rFs rIsides Polygon.sideshsI6}}D z1%A JJwtwU 4 5& rTcURnUVs/sHo"RPM nnUVs/sHo"RPM nn[U5[U5[ U5[ U54$s snfs snf)zgReturn a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure. )rCrVrWminr)rMvertsrExsyss rIrPolygon.boundss`  !5acc5 ! !5acc5 !BR#b'3r733" !s A-A2cURnURUSUSUS5n[S[U55H+nX RXS- XS- X5- (dM+ g URn[ U5H~up5UR n[U[U5S- :XaSOSUS- 5HFnXGnURU;dMURU;dM+URU5n U (dME g M g)aIs the polygon convex? A polygon is convex if all its interior angles are less than 180 degrees and there are no intersections between sides. Returns ======= is_convex : boolean True if this polygon is convex, False otherwise. See Also ======== sympy.geometry.util.convex_hull Examples ======== >>> from sympy import Point, Polygon >>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]) >>> poly = Polygon(p1, p2, p3, p4) >>> poly.is_convex() True r4rrr2FT) rCr^rLr7r enumerater@p1p2 intersection) rMr@cwrFrsiptsjsjhits rI is_convexPolygon.is_convexs8}}   R$r(DG <q#d)$A&&tE{DQKIII% u%EA''CSZ!^ 31AEBX55#S(8//"-Cs$ C&rTc^[TSS9mTUR;d$[U4SjUR55(ag/nURH-nUR UT- 5 USR (dM- g [ U6nURn[[[U5*S55nUR5(a}SnUHtnXXn XXS-n U R*U RU R- -U R*U RU R- -- Rn UcU nMnXLdMt g gSn US RupUSSHonXXRunnS [UU5:aHS [!UU5::a8S [!X5::a)UU:wa#U*X- -UU- - U -nX:XdS U::aU (+n UUpMq U $) a\ Return True if p is enclosed by (is inside of) self. Notes ===== Being on the border of self is considered False. Parameters ========== p : Point Returns ======= encloses_point : True, False or None See Also ======== sympy.geometry.point.Point, sympy.geometry.ellipse.Ellipse.encloses_point Examples ======== >>> from sympy import Polygon, Point >>> p = Polygon((0, 0), (4, 0), (4, 4)) >>> p.encloses_point(Point(2, 1)) True >>> p.encloses_point(Point(2, 2)) False >>> p.encloses_point(Point(5, 5)) False References ========== .. [1] https://paulbourke.net/geometry/polygonmesh/#insidepoly r2r3c3.># UH nTU;v M g7fNr/).0srEs rI )Polygon.encloses_point..s$@ZQ!VZFr4NrTr)rrCanyrr8 free_symbolsr-r@r6rLr7rrWrV is_negativerr)rMrElitr|polyr@indices orientationrFrBrGtesthit_oddp1xp1yp2xp2yxinterss ` rIencloses_pointPolygon.encloses_pointsT !O  $@TZZ$@!@!@A JJq1u 2w### } yyuc$iZ+, >>  KGQK##acc *qssdQSS133Y-??LL&"&K, 7<<Aw||HC3sC= C %CM)#:(+tci&8#)&Ds&JG"zQ'\.5+CrTc [USS9nURSUR5;a[SUR-5e/nURnSnUR H_nUR U- nXW-nURU5RX"U- U- 5n URU [XR:*X(:545 UnMa [U6$)aCA 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, 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) Tr*c38# UHoRv M g7fr)name)rfs rIr*Polygon.arbitrary_point..Fs8&7ff&7zFSymbol %s already appears in object and cannot be used as a parameter.r) r rrrYrrrlengtharbitrary_pointsubsr8rr) rM parametertrrrperim_fraction_startrside_perim_fractionperim_fraction_endpts rIrPolygon.arbitrary_pointsV ID ) 668d&7&78 8ehihnhnno oNN  A"#((9"4 !5!K ""9-22,,.AACB LLc.3Q5KLN P#5 %  rTc,[U[5(d[XRS9n[U[5(d [ S5eUR (a [ S5eSnUR[5nURHUupV[XQ- [SS9nU(dMUS[n[UR[U55S:XaX(0s $SnMW U(a[ S[U5-5e[ S [U5-5e) Nrzother must be a pointznon-numeric coordinatesFT)dictrzGiven point may not be on %szGiven point is not on %s) isinstancerrrmrYrNotImplementedErrorrTr@rrrr&) rMotherrunknownrErcondsolvalues rIparameter_valuePolygon.parameter_valueUs%//%%;%;>> from sympy import Polygon >>> p = Polygon((0, 0), (1, 0), (1, 1)) >>> p.plot_interval() [t, 0, 1] Tr*rr)r )rMrrs rI plot_intervalPolygon.plot_intervaljs0 94 (1ayrTc /n[U[5(a UROU/nURH,nUH#nURUR U55 M% M. [ [ U55nUVs/sHn[U[5(dMUPM nnUVs/sHn[U[5(dMUPM nnU(auU(an[ [ UV V s/sHoH oU ;dM U PM M sn n 55n U (aU Hn URU 5 M [ [X-55$[ [U55$s snfs snfs sn n f)a-The 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)] ) rr-rextendrr6r"rrremover ) rMointersection_resultksideside1entitypointssegmentsrsegmentpoints_in_segmentsrFs rIrPolygon.intersections(J!!!W--AGGA3JJD#**4+<+W&':X)<\)>> from sympy import Polygon, Line >>> a, b = 20, 10 >>> p1, p2, p3, p4 = [(0, b), (0, 0), (a, 0), (a, b)] >>> rectangle = Polygon(p1, p2, p3, p4) >>> t = rectangle.cut_section(Line((0, 5), slope=0)) >>> t (Polygon(Point2D(0, 10), Point2D(0, 5), Point2D(20, 5), Point2D(20, 10)), Polygon(Point2D(0, 5), Point2D(0, 0), Point2D(20, 0), Point2D(20, 5))) >>> upper_segment, lower_segment = t >>> upper_segment.area 100 >>> upper_segment.centroid Point2D(10, 15/2) >>> lower_segment.centroid Point2D(10, 5/2) References ========== .. [1] https://github.com/sympy/sympy/wiki/A-method-to-return-a-cut-section-of-any-polygon-geometry z(This line does not intersect the polygonrTNF)NN) rrYr6rCr8equationrVrWcoeffrrrr-)rMlineintersection_pointsreqrBrGupper_verticeslower_verticesprev prev_pointrcompareedge new_point upper_polygon lower_polygons rIrPolygon.cut_sectionsf#//5"GH Hdmm$ fQi ]]1a  HHQK HHQK E>?bggq%''1egg679EGG,Q. { 2D $ 1 1$ 7I")))A,7")))A,7%%e,J2D $ 1 1$ 7I")))A,7")))A,7%%e,J58(2$ } j.)A7KK#^4M j.)A7KK#^4Mm++rTcz[U[5(aM[nURH5nUR U5nUS:Xa[ R s $XB:dM3UnM7 U$[U[5(a;UR5(a&UR5(aURU5$[5e)ag Returns the shortest distance between self and o. If o is a point, then self does not need to be convex. If o is another polygon self and o must be convex. Examples ======== >>> from sympy import Point, Polygon, RegularPolygon >>> p1, p2 = map(Point, [(0, 0), (7, 5)]) >>> poly = Polygon(*RegularPolygon(p1, 1, 3).vertices) >>> poly.distance(p2) sqrt(61) r) rrrrrqrZeror-r_do_poly_distancer)rMrdistrcurrents rIrqPolygon.distance(s a  D --*a<66M^"D # K 7 # #(8(8Q[[]]))!, ,!##rTcl UnURnURn[Rn[RnURH"n[R "X75nXX:dM UnM$ URH"n[R "XG5nXh:dM UnM$ [R "X45n XU-::a[ R"SSS9 [ S[*5n [ S[5n URHWnURU R:d8URU R:XdM9URU R:dMUUn MY URHWnURU R:d8URU R:XdM9URU R:dMUUn MY [R "X5n 0n 0nURHnURU ;a(XRRUR5 OUR/XR'URU ;a)XRRUR5 MUR/XR'M URHnURU;a(XRRUR5 OUR/XR'URU;a)XRRUR5 MUR/XR'M U nU n[[ [R[R5[ [R [R55nXSnXSnUR#[U U55nUR#[U U55nUU:aUnO>UU:aUnO5[R "U U5[R "U U5:aUnOUnXSnXSnUR#[U U55nUR#[U U55nUU:aUnO>UU:aUnO5[R "U U5[R "U U5:aUnOUnUR#[UU55n[$UR#[UU55- nUU:SLaq[UU5n['UU5nUR U5nUR)5U R)5:aUn U USU:wa UnU USnGO;UnU USnGO/UU:SLao[UU5n['UU5nUR U5nUR)5U R)5:aUn UUSU:wa UnUUSnOUnUUSnO[UU5n['UU5n['UU5nUR U5nUR U5n[+UU5nUR)5U R)5:aUn U USU:wa UnU USnO UnU USnUUSU:wa UnUUSnO UnUUSnUU :Xa UU :XaU $GM)a Calculates the least distance between the exteriors of two convex polygons e1 and e2. Does not check for the convexity of the polygons as this is checked by Polygon.distance. Notes ===== - Prints a warning if the two polygons possibly intersect as the return value will not be valid in such a case. For a more through test of intersection use intersection(). See Also ======== sympy.geometry.point.Point.distance Examples ======== >>> from sympy import Point, Polygon >>> square = Polygon(Point(0, 0), Point(0, 1), Point(1, 1), Point(1, 0)) >>> triangle = Polygon(Point(1, 2), Point(2, 2), Point(2, 1)) >>> square._do_poly_distance(triangle) sqrt(2)/2 Description of method used ========================== Method: [1] https://web.archive.org/web/20150509035744/http://cgm.cs.mcgill.ca/~orm/mind2p.html Uses rotating calipers: [2] https://en.wikipedia.org/wiki/Rotating_calipers and antipodal points: [3] https://en.wikipedia.org/wiki/Antipodal_point z1Polygons may intersect producing erroneous outputr() stacklevelrrT)r}rrrCrrqwarningswarnrrWrVrrr8rrOnerarrevalfr) rMe2e1 e1_center e2_center e1_max_radius e2_max_radiusvertexr center_diste1_ymaxe2_yminmin_diste1_connectionse2_connectionsr e1_current e2_current support_linepoint1point2angle1angle2e1_nexte2_nexte1_anglee2_angle e1_segmentmin_dist_current e2_segmentmin1min2s rIrPolygon._do_poly_distanceEsJZKK KK   kkFy1A ! "kkFy1A ! "nnY: -7 7 MMM%& ( B3-2,kkFxx'))#GII(=&((WYYBV "kkFxx'))#GII(=&((WYYBV ">>'3 HHDww.(ww'..tww7+/77)ww'ww.(ww'..tww7+/77)ww'HHDww.(ww'..tww7+/77)ww'ww.(ww'..tww7+/77)ww'  E!&&!&&153GH   (+(+++D&,AB++D&,AB F?G f_G ^^GV ,u~~gv/N NGG(+(+++D&,AB++D&,AB F?G f_G ^^GV ,u~~gv/N NGG #11$z72KLHL66tG8%&&H8#,#J8 $Z9 #-#6#6z#B #))+hnn.>>/H!'*1-;!(J,W5a8G!(J,W5a8GX%$.#GZ8 $Z9 #-#6#6z#B #))+hnn.>>/H!'*1-;!(J,W5a8G!(J,W5a8G#J8 $Z9 $Z9 !**73!**73#&tT? #))+hnn.>>/H!'*1-;!(J,W5a8G!(J,W5a8G!'*1-;!(J,W5a8G!(J,W5a8GW$w)>wrTc$[[UR5nUVs/sH)nSRURUR 5PM+ nnSRUSSR USS55nSRSU-Xb5$s snf) zReturns 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". z{},{}z M {} L {} zrz L rNzag@)maprrCformatrVrWjoin)rM scale_factor fill_colorrrEcoordspaths rI_svg Polygon._svg sAt}}%49:Eq'..acc*E:##F1Iuzz&*/EF :fR,&9 :;s0B c2^ 0m U 4SjnU"UR5n[U[U55nU"[[ UR555n[U[U55nX5:aUnOUnUVs/sHnT UPM nn[ U5$s snf)Nc>0n[[[U555Hup#X!U'UTU'M UVs/sHo1UPM sn$s snfr)rr set) point_listkeerFrEDs rIref_list+Polygon._hashable_content..ref_list#sNC!'#j/":;A!<%//JqFJ/ //sA)r@r#r$r6reversedtuple) rMrWS1r_norS2r_revr/ordercanonical_argsrVs @rI_hashable_contentPolygon._hashable_content s  0dii Br 23 d8DII./ 0Br 23 =AA1241U84^$$5s8Bc$^[T[5(aUT:H$[T[5(a[U4SjUR55$[T[ 5(a,TUR ;agURH nTU;dM g g)a Return True if o is contained within the boundary lines of self.altitudes Parameters ========== other : GeometryEntity Returns ======= contained in : bool The points (and sides, if applicable) are contained in self. See Also ======== sympy.geometry.entity.GeometryEntity.encloses Examples ======== >>> from sympy import Line, Segment, Point >>> p = Point(0, 0) >>> q = Point(1, 1) >>> s = Segment(p, q*2) >>> l = Line(p, q) >>> p in q False >>> p in s True >>> q*3 in s False >>> s in l True c3.># UH nTU;v M g7frr/)rrrs rIr'Polygon.__contains__.._s2z!qAvzrTF)rr-rrrrrC)rMrrs ` rI __contains__Polygon.__contains__5sxN a ! !19  7 # #2tzz22 2 5 ! !DMM! 9#rTc 0n[UR5nURUS5 [R"USS6nU(a[[ U55nUR R5HupVURU5n[R"X7X7S-5up[X5RUS- U5n U Rn [U RU RXRS5- -5n Ub/[U RU R R#U55n XU'M U$)a1Returns angle bisectors of a polygon. If prec is given then approximate the point defining the ray to that precision. The distance between the points defining the bisector ray is 1. Examples ======== >>> from sympy import Polygon, Point >>> p = Polygon(Point(0, 0), Point(2, 0), Point(1, 1), Point(0, 3)) >>> p.bisectors(2) {Point2D(0, 0): Ray2D(Point2D(0, 0), Point2D(0.71, 0.71)), Point2D(0, 3): Ray2D(Point2D(0, 3), Point2D(0.23, 2.0)), Point2D(1, 1): Ray2D(Point2D(1, 1), Point2D(0.19, 0.42)), Point2D(2, 0): Ray2D(Point2D(2, 0), Point2D(1.1, 0.38))} rNr(rr2rx)r6r@r8r-r^rYrjitemsindexr_normalize_dimensionrrotate directionrrqrr0) rEprecrGrrr|rBrFrrraydirs rI bisectorsPolygon.bisectorsis" 166l 3q6  " "CG , x}%CHHNN$DA ! A//E CFBb+$$QqS!,C--Ccffcffs<<+?'??@C#&&#&&((4.1aD%rTr)r)g?z#66cc99)$__name__ __module__ __qualname____firstlineno____doc__ __slots__r=propertyrN staticmethodr^rjrmrrrCr}rrrrrrrrrrrrrrqrrOrarfrq__static_attributes__r/rTrIr-r-s6YvI !).V."."`*11f228!!F 5 5F< ~BJ3>9x@ 4 4,\Un9!vG*666rg,T$:FP:&%*2h rTr-c^\rSrSrSrSrS SjrS!Sjr\S5r Sr Sr \S 5r \S 5r \S 5r\r\S 5r\S 5r\S5r\S5r\S5r\S5r\S5r\S5r\S5r\S5r\S5rSrSrS"SjrS#SjrSr\S5r Sr!U4Sjr"Sr#U=r$$)$r:ia= A regular polygon. Such a polygon has all internal angles equal and all sides the same length. Parameters ========== center : Point radius : number or Basic instance The distance from the center to a vertex n : int The number of sides Attributes ========== vertices center radius rotation apothem interior_angle exterior_angle circumcircle incircle angles Raises ====== GeometryError If the `center` is not a Point, or the `radius` is not a number or Basic instance, or the number of sides, `n`, is less than three. Notes ===== A RegularPolygon can be instantiated with Polygon with the kwarg n. Regular polygons are instantiated with a center, radius, number of sides and a rotation angle. Whereas the arguments of a Polygon are vertices, the vertices of the RegularPolygon must be obtained with the vertices method. See Also ======== sympy.geometry.point.Point, Polygon Examples ======== >>> from sympy import RegularPolygon, Point >>> r = RegularPolygon(Point(0, 0), 5, 3) >>> r RegularPolygon(Point2D(0, 0), 5, 3, 0) >>> r.vertices[0] Point2D(5, 0) )_n_center_radius_rotc [[X#U45up#n[U4SS0UD6n[U[5(d[ SU-5eUR (a[U5 US:a[ SU-5e[R"XX#40UD6nX6l Xl X&l UR(a US[R-U- -UlU$UUlU$)Nr3r2z r must be an Expr object, not %sr(zn must be a >= 3, not %s)rHrrrrr is_Numberr%rr=r}r~r is_numberrrbr)rMrHr/r0rotrAobjs rIr=RegularPolygon.__new__s!- c ! % %f %!T"" BQ FG G ;; 1I1u#$>$BCC$$Ta=f=  '*}}3!ADD&(# ;> rTc URup4pV[U5nX4U4Vs/sHoR"SSU0UD6PM snup4nURX4XV5$s snf)Nr0r/)r@r'r'func) rMrnoptionsrHr/r0rBdpsrFs rI _eval_evalfRegularPolygon._eval_evalfs[YY a$78Qi@i77,S,G,i@ayyq$$AsAc^URURURUR4$)z 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) )r~rr}rrns rIr@RegularPolygon.argss#||T\\477DII==rTc2S[UR5-$NzRegularPolygon(%s, %s, %s, %s)rZr@rns rI__str__RegularPolygon.__str__/% 2BBBrTc2S[UR5-$rrrns rI__repr__RegularPolygon.__repr__rrTcURupp4[U5U-URS--S[[U- 5-- $)zReturns the area. Examples ======== >>> from sympy import RegularPolygon >>> square = RegularPolygon((0, 0), 1, 4) >>> square.area 2 >>> _ == square.length**2 True r2)r@rrrr)rMrHr/r0rs rIrNRegularPolygon.areas=yy aAwqya'3r!t955rTcZURS-[[UR- 5-$)aReturns 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 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 r2)radiusrrr}rns rIrRegularPolygon.length s#({{1}SDGG_,,rTcUR$)aVThe 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 import RegularPolygon, Point >>> rp = RegularPolygon(Point(0, 0), 5, 4) >>> rp.center Point2D(0, 0) )r~rns rIcenterRegularPolygon.center s0||rTcUR$)z Alias for center. Examples ======== >>> from sympy import RegularPolygon, Point >>> rp = RegularPolygon(Point(0, 0), 5, 4) >>> rp.circumcenter Point2D(0, 0) )rrns rI circumcenterRegularPolygon.circumcenter<s{{rTcUR$)aRadius 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 import RegularPolygon, Point >>> radius = Symbol('r') >>> rp = RegularPolygon(Point(0, 0), radius, 4) >>> rp.radius r )rrns rIrRegularPolygon.radiusKs6||rTcUR$)z Alias for radius. Examples ======== >>> from sympy import Symbol >>> from sympy import RegularPolygon, Point >>> radius = Symbol('r') >>> rp = RegularPolygon(Point(0, 0), radius, 4) >>> rp.circumradius r )rrns rI circumradiusRegularPolygon.circumradiushs{{rTcUR$)aCCW 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 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 rrns rIrotationRegularPolygon.rotationys4yyrTchUR[[RUR- 5-$)aThe 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 import RegularPolygon, Point >>> radius = Symbol('r') >>> rp = RegularPolygon(Point(0, 0), radius, 4) >>> rp.apothem sqrt(2)*r/2 )rrrrbr}rns rIapothemRegularPolygon.apothems$6{{Sdgg...rTcUR$)z Alias for apothem. Examples ======== >>> from sympy import Symbol >>> from sympy import RegularPolygon, Point >>> radius = Symbol('r') >>> rp = RegularPolygon(Point(0, 0), radius, 4) >>> rp.inradius sqrt(2)*r/2 )rrns rIinradiusRegularPolygon.inradiuss||rTc\URS- [R-UR- $)aMeasure of the interior angles. Returns ======= interior_angle : number See Also ======== sympy.geometry.line.LinearEntity.angle_between Examples ======== >>> from sympy import RegularPolygon, Point >>> rp = RegularPolygon(Point(0, 0), 4, 8) >>> rp.interior_angle 3*pi/4 r2)r}rrbrns rIinterior_angleRegularPolygon.interior_angles$.! QTT!$''))rTcBS[R-UR- $)aMeasure of the exterior angles. Returns ======= exterior_angle : number See Also ======== sympy.geometry.line.LinearEntity.angle_between Examples ======== >>> from sympy import RegularPolygon, Point >>> rp = RegularPolygon(Point(0, 0), 4, 8) >>> rp.exterior_angle pi/4 r2)rrbr}rns rIexterior_angleRegularPolygon.exterior_angles.vdgg~rTcB[URUR5$)a)The circumcircle of the RegularPolygon. Returns ======= circumcircle : Circle See Also ======== circumcenter, sympy.geometry.ellipse.Circle Examples ======== >>> from sympy import RegularPolygon, Point >>> rp = RegularPolygon(Point(0, 0), 4, 8) >>> rp.circumcircle Circle(Point2D(0, 0), 4) )rrrrns rI circumcircleRegularPolygon.circumcircles.dkk4;;//rTcB[URUR5$)a#The incircle of the RegularPolygon. Returns ======= incircle : Circle See Also ======== inradius, sympy.geometry.ellipse.Circle Examples ======== >>> from sympy import RegularPolygon, Point >>> rp = RegularPolygon(Point(0, 0), 4, 7) >>> rp.incircle Circle(Point2D(0, 0), 4*cos(pi/7)) )rrrrns rIincircleRegularPolygon.incircles.dkk4<<00rTcP0nURnURHnX!U'M U$)aA Returns a dictionary with keys, the vertices of the Polygon, and values, the interior angle at each vertex. Examples ======== >>> 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} )rrC)rMrfangr|s rIrjRegularPolygon.angles's. !!AF rTcURn[X!5RnX0R:agX0R:ag[ R X5$)aF 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 FT)rrrrrr-r)rMrErHds rIrRegularPolygon.encloses_point=sJ` KK AM    ))$2 2rTc.U=RU- slg)a~Increment *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 Nr)rMangles rIspinRegularPolygon.spinws( U rTc[U5"UR6nU=RU- sl[R"X1U5$)amOverride GeometryEntity.rotate to first rotate the RegularPolygon about its center. >>> from sympy import Point, RegularPolygon, 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 )typer@rrrl)rMrrr/s rIrlRegularPolygon.rotates6& J " %$$Qr22rTc:U(aJ[USS9nUR"U*R6RX5R"UR6$X:wa![ UR 6RX5$URupEpgXQ-nUR XEXg5$)aOverride GeometryEntity.scale since it is the radius that must be scaled (if x == y) or else a new Polygon must be returned. >>> from sympy import RegularPolygon Symmetric scaling returns a RegularPolygon: >>> RegularPolygon((0, 0), 1, 4).scale(2, 2) RegularPolygon(Point2D(0, 0), 2, 4, 0) Asymmetric scaling returns a kite as a Polygon: >>> RegularPolygon((0, 0), 1, 4).scale(2, 1) Polygon(Point2D(2, 0), Point2D(0, 1), Point2D(-2, 0), Point2D(0, -1)) r2r)r translater@scaler-rCr)rMrVrWrrHr/r0rs rIrRegularPolygon.scales" rq!B>>RC::.44Q:DDbggN N 6DMM*006 6yy a yyq&&rTcURup#pEURSnXb- nURU5nURU5n X- n [SU 5n [SU5n U R U 5n X]- nUR X*XE5$)zOverride 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)) rrx)r@rCreflectr closing_angler)rMrrHr/r0rr|rccvvddl1l2rs rIrRegularPolygon.reflectsyy a MM!  E YYt_ YYt_ W_ ^r" yyR((rTc ~URn[UR5nURnS[R -UR - n[UR 5Vs/sHKn[URU[XT-U-5--URU[XT-U-5--5PMM sn$s snf)aLThe vertices of the RegularPolygon. Returns ======= vertices : list Each vertex is a Point. See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import RegularPolygon, Point >>> rp = RegularPolygon(Point(0, 0), 5, 4) >>> rp.vertices [Point2D(5, 0), Point2D(0, 5), Point2D(-5, 0), Point2D(0, -5)] r2) r~absrrrrbr}rLrrVrrWr)rMrHr/rr|rs rIrCRegularPolygon.verticess0 LL  ii addF477Ntww)'AaccAc!#)n,,accAc!#)n4D.DE') ))s%AB:c[U[5(dg[U[5(d[RX5$URUR:H$)NF)rr-r:__eq__r@)rMrs rIrRegularPolygon.__eq__sA!W%%A~..>>!* *yyAFF""rTc >[TU]5$r)super__hash__)rM __class__s rIrRegularPolygon.__hash__sw!!rTr/)r)r)rrN)%rsrtrurvrwrxr=rryr@rrrNrrr}rrrrrrrrrrrjrrrlrrrCrrr{ __classcell__)rs@rIr:r:s;z5I"%  > >CC66 --*2H   8 6//8 **00000110*83t,3.'4)8))>#""rTr:c2\rSrSrSrSr\S5rSrSr Sr Sr S r \S 5r \S 5r\S 5r\S 5r\S5rSr\S5r\S5r\S5r\S5r\S5r\S5r\S5r\S5r\S5rSrg)r>ia] 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 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)) cd[U5S:waSU;a%[USVs/sHn[U5PM sn6$SU;a%[USVs/sHn[U5PM sn6$SU;a%[ USVs/sHn[U5PM sn6$Sn[ U5eUVs/sHn[ U4SS0UD6PM nn/nUH%nU(a XvS:XaMURU5 M' [U5S :aUSUS :XaUR5 S nU[U5S- :a[U5S:a[XhXhS -XhS-/[S 9up9n [ R"X9U 5(aX6U'SXhS -'URUS -5 US - nU[U5S- :a[U5S:aM[[S U55n[U5S:Xa[R"U/UQ70UD6$[U5S:Xa [!U0UD6$[ U0UD6$s snfs snfs snfs snf)Nr(sssasasasz;Triangle instantiates with three points or a valid keyword.r3r2r4rrr5)keyc USL$rr/)rVs rI"Triangle.__new__..`s$rT)r7_sssr_asa_sasrrr8r;sortedr r<r6filterrr=r) r?r@rArBmsgrCrDrErFrGrHs rIr=Triangle.__new__?s( t9>6%=A=ahqk=ABB6%=A=ahqk=ABB6%=A=ahqk=ABBOC$ $7;eBi583 IIK #e*q. SZ!^5Q<1u6? x=A !))#CCFC C ]a H// /(-f- -KBAA=sH H#8H("H-cUR$)aCThe triangle's vertices Returns ======= vertices : tuple Each element in the tuple is a Point See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy 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)) )r@rns rIrCTriangle.verticesis0yyrTc[U[5(dgURVs/sHo"RPM snup4nURVs/sHo"RPM nnSnU"X4U/UQ76=(dN U"X5U/UQ76=(d= U"XCU/UQ76=(d, U"XEU/UQ76=(d U"XSU/UQ76=(d U"XTU/UQ76$s snfs snf)a(Is another triangle similar to this one. Two triangles are similar if one can be uniformly scaled to the other. Parameters ========== other: Triangle Returns ======= is_similar : boolean See Also ======== sympy.geometry.entity.GeometryEntity.is_similar Examples ======== >>> from sympy import Triangle, Point >>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3)) >>> t2 = Triangle(Point(0, 0), Point(-4, 0), Point(-4, -3)) >>> t1.is_similar(t2) True >>> t2 = Triangle(Point(0, 0), Point(-4, 0), Point(-4, -4)) >>> t1.is_similar(t2) False Fc[X- 5n[X- 5n[X%- 5n[Xg:H5=(a [Xx:H5$r)rbool) u1u2u3v1v2v3r)r(e3s rI _are_similar)Triangle.is_similar.._are_similars:"%B"%B"%B>4d28n 4rT)rr-rr)t1t2rs1_1s1_2s1_3s2rs rI is_similarTriangle.is_similarsD"g&&46HH=HDKKH=D&(hh /hdkkh / 5D2r20 T /B /0 T /B /0 T /B /0 T /B / 0 T /B /  0> /s CC cD[SUR55(+$)aAre all the sides the same length? Returns ======= is_equilateral : boolean See Also ======== sympy.geometry.entity.GeometryEntity.is_similar, RegularPolygon is_isosceles, is_right, is_scalene Examples ======== >>> from sympy import Triangle, Point >>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3)) >>> t1.is_equilateral() False >>> from sympy import sqrt >>> t2 = Triangle(Point(0, 0), Point(10, 0), Point(5, 5*sqrt(3))) >>> t2.is_equilateral() True c38# UHoRv M g7frrrrs rIr*Triangle.is_equilateral..s<Axxr)r!rrns rIis_equilateralTriangle.is_equilaterals8<<<<>> from sympy import Triangle, Point >>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(2, 4)) >>> t1.is_isosceles() True c38# UHoRv M g7frrrs rIr(Triangle.is_isosceles..s5*Q*rr rrns rI is_isoscelesTriangle.is_isosceless,5$**555rTcD[SUR55(+$)a'Are all the sides of the triangle of different lengths? Returns ======= is_scalene : boolean See Also ======== is_equilateral, is_isosceles, is_right Examples ======== >>> from sympy import Triangle, Point >>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(1, 4)) >>> t1.is_scalene() True c38# UHoRv M g7frrrs rIr&Triangle.is_scalene.. s9jHHjrrrns rI is_scaleneTriangle.is_scalenes,9djj9999rTcURn[R"USUS5=(dA [R"USUS5=(d [R"USUS5$)a=Is 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 import Triangle, Point >>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3)) >>> t1.is_right() True rrr2)rris_perpendicularrMrs rIis_rightTriangle.is_right sd. JJ''!ad31  $ $QqT1Q4 01  $ $QqT1Q4 0 1rTc URnURnUSUSRUS5USUSRUS5USUSRUS50$)asThe 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 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)) rrr2)rrCperpendicular_segmentrMrr|s rI altitudesTriangle.altitudes# st< JJ MM!ad0016!ad0016!ad00168 8rTcURnURn[XS5R[XS55S$)aThe orthocenter of the triangle. The orthocenter is the intersection of the altitudes of a triangle. It may lie inside, outside or on the triangle. Returns ======= orthocenter : Point See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1) >>> t = Triangle(p1, p2, p3) >>> t.orthocenter Point2D(0, 0) rr)r&rCrr)rMrBr|s rI orthocenterTriangle.orthocenterG s@6 NN MMAdG}))$q1w-8;;rTcURVs/sHoR5PM snup#nURU5S$s snf)a_The circumcenter of the triangle The circumcenter is the center of the circumcircle. Returns ======= circumcenter : Point See Also ======== sympy.geometry.point.Point Examples ======== >>> from sympy 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) r)rperpendicular_bisectorr)rMrVrBrGrHs rIrTriangle.circumcenterf sA28>> from sympy import Symbol >>> from sympy 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) r)rrqrrCrns rIrTriangle.circumradius s$2~~d//q1ABBrTcB[URUR5$)aeThe circle which passes through the three vertices of the triangle. Returns ======= circumcircle : Circle See Also ======== sympy.geometry.ellipse.Circle Examples ======== >>> from sympy 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) )rrrrns rIrTriangle.circumcircle s0d''):):;;rTcURVs/sHn[U5PM nnURnURn[ US[USU5R US5S5n[ US[USU5R US5S5n[ US[USU5R US5S5nUSXSSXcSU0$s snf)a_The angle bisectors of the triangle. An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. Returns ======= bisectors : dict Each key is a vertex (Point) and each value is the corresponding bisector (Segment). See Also ======== sympy.geometry.point.Point, sympy.geometry.line.Segment Examples ======== >>> from sympy import Point, Triangle, Segment >>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1) >>> t = Triangle(p1, p2, p3) >>> from sympy import sqrt >>> t.bisectors()[p2] == Segment(Point(1, 0), Point(0, sqrt(2) - 1)) True rrr2)rrrCincenterrr)rMlrr|rHrrl3s rIrqTriangle.bisectors s@#jj )jT!Wj ) MM MM QqT4!a=55ad;A> ? QqT4!a=55ad;A> ? QqT4!a=55ad;A> ?!bA$aD"-- *sC c URn[SVs/sHo!URPM sn5n[U5nURn[ UR [UVs/sHofRPM sn55U- 5n[ UR [UVs/sHofRPM sn55U- 5n[Xx5$s snfs snfs snf)aThe center of the incircle. The incircle is the circle which lies inside the triangle and touches all three sides. Returns ======= incenter : Point See Also ======== incircle, sympy.geometry.point.Point Examples ======== >>> from sympy import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1) >>> t = Triangle(p1, p2, p3) >>> t.incenter Point2D(1 - sqrt(2)/2, 1 - sqrt(2)/2) )rr2r) rrrrcrCrdotrVrWr) rMrrFr4rEr|virVrWs rIr3Triangle.incenter s6 JJ 3AaDKK3 4 F MM QUU6!"4!B44!"456q8 9 QUU6!"4!B44!"456q8 9Q{ 4#5"4sC%C%C!cL[SUR-UR- 5$)a5The radius of the incircle. Returns ======= inradius : number of Basic instance See Also ======== incircle, sympy.geometry.ellipse.Circle.radius Examples ======== >>> from sympy import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(4, 0), Point(0, 3) >>> t = Triangle(p1, p2, p3) >>> t.inradius 1 r2)rrNrrrns rIrTriangle.inradius s 0DII 677rTcB[URUR5$)aThe incircle of the triangle. The incircle is the circle which lies inside the triangle and touches all three sides. Returns ======= incircle : Circle See Also ======== sympy.geometry.ellipse.Circle Examples ======== >>> from sympy import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(2, 0), Point(0, 2) >>> t = Triangle(p1, p2, p3) >>> t.incircle Circle(Point2D(2 - sqrt(2), 2 - sqrt(2)), 2 - sqrt(2)) )rr3rrns rIrTriangle.incircle s6dmmT]]33rTc VURnUSRnUSRnUSRnX#-U-S- nURnURS[XeU- - 5URS[XeU- - 5URS[XeU- - 50nU$)aThe 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 import Point, Triangle >>> 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] https://mathworld.wolfram.com/Exradius.html .. [2] https://mathworld.wolfram.com/Excircles.html rrr2)rrrNr)rMrrBrGrHrrNexradiis rIr@Triangle.exradii9 sLzz GNN GNN GNN SUAIyy::a=(41:"6::a=(41:"6::a=(41:"68rTc URnURnUSRnUSRnUSRnUSRUSRUSR/nUSRUSRUSR/n[ U*US-XFS--XVS-U*U-U-- -5[ X6S-XFS-- XVS-X4- U-- -5[ X6S-XFS--XVS-X4-U- - - 5[ U*US-XGS--XWS-U*U-U-- -5[ X7S-XGS-- XWS-X4- U-- -5[ X7S-XGS--XWS-X4-U- - - 5S.nUS[ USUS5US[ USUS5US[ US US 50n U $) aExcenters of the triangle. An excenter is the center of a circle that is tangent to a side of the triangle and the extensions of the other two sides. Returns ======= excenters : dict Examples ======== The excenters are keyed to the side of the triangle to which their corresponding excircle is tangent: The center is keyed, e.g. the excenter of a circle touching side 0 is: >>> from sympy import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(6, 0), Point(0, 2) >>> t = Triangle(p1, p2, p3) >>> t.excenters[t.sides[0]] Point2D(12*sqrt(10), 2/3 + sqrt(10)/3) See Also ======== sympy.geometry.polygon.Triangle.exradii References ========== .. [1] https://mathworld.wolfram.com/Excircles.html rrr2)rOrQx3rPrRy3rOrPrQrRrCrD)rrCrrVrWrr) rMrr|rBrGrHrVrW exc_coords excenterss rIrFTriangle.excentersk s L JJ MM aDKK aDKK aDKK qTVVQqTVVQqTVV $ qTVVQqTVVQqTVV $A2ad71qT6>!aD&1"Q$q&/9:1qT6!aD&=Q4Q781qT6!aD&=Q4Q78A2ad71qT6>!aD&1"Q$q&/9:1qT6!aD&=Q4Q781qT6!aD&=Q4Q78   aD% 4(*T*:; aD% 4(*T*:; aD% 4(*T*:; rTc URnURnUS[USUSR5US[USUSR5US[USUSR50$)aVThe 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 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)) rrr2)rrCrmidpointr%s rImediansTriangle.medians st< JJ MM!gadAaDMM2!gadAaDMM2!gadAaDMM24 4rTcURn[USRUSRUSR5$)aThe medial triangle of the triangle. The triangle which is formed from the midpoints of the three sides. Returns ======= medial : Triangle See Also ======== sympy.geometry.line.Segment.midpoint Examples ======== >>> from sympy import Point, Triangle >>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1) >>> t = Triangle(p1, p2, p3) >>> t.medial Triangle(Point2D(1/2, 0), Point2D(1/2, 1/2), Point2D(0, 1/2)) rrr2)rr>rIr s rImedialTriangle.medial s44 JJ! qt}}admmDDrTc:[URR6$)aeThe 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 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) )rrMrCrns rInine_point_circleTriangle.nine_point_circle s<t{{++,,rTcUR5(a UR$[URUR5$)aThe 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 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)) )rr)rrrns rI eulerlineTriangle.eulerline s8,    ## #D$$d&7&788rTr/N)rsrtrurvrwr=ryrCr rrrr!r&r)rrrrqr3rrr@rFrJrMrPrSr{r/rTrIr>r>sc7r(.T240l=<60:018!8!8F<<<$$6CC4<<2&.P  D882448//b<<|!4!4FEE8-->99rTr>cU[-S- $)zAReturn the radian value for the given degrees (pi = 180 degrees).r)rs rIradrX%  R48OrTcU[- S-$)zAReturn the degree value for the given radians (pi = 180 degrees).rVrW)r/s rIdegr[* rYrTc.[[U55nU$r)rrX)rrvs rI_sloper^/ s SVB IrTc [S[U5S9R[US4[SU- 5S95Sn[SUS4U5$)z8Return triangle having side with length l on the x-axis.rxrrrV)rr^rr>)d1r4d2xys rIrr4 sO fF2J ' 4 4 aV6#(+, ../ 1B FQFB ''rTc[SU5n[US4U5nURU5Vs/sH!oURR(dMUPM# nnU(dgUSn[ SUS4U5$s snf)z7Return triangle having side of length l1 on the x-axis.rxrN)rrrWis_nonnegativer>)rrr5c1c2rBinterrs rIrr; so  B Q B+ B+1ss/A/AQ+E B  qB FRGR (( Cs A4A4c[SS5n[US5n[[[U55U-[[U55U-5n[ X4U5$)z9Return triangle having side with length l2 on the x-axis.r)rrrXrr>)rrrrrp3s rIrrF sI q!B r1B s3q6{2~s3q6{2~ .B BB rT)E sympy.corerrrrrsympy.core.evalfrsympy.core.sortingr r sympy.core.symbolr r r $sympy.functions.elementary.complexesr$sympy.functions.elementary.piecewiser(sympy.functions.elementary.trigonometricrrrellipserrrr exceptionsrrrrrrr sympy.logicrsympy.matricesrsympy.simplify.simplifyrsympy.solvers.solversrsympy.utilities.iterablesr r!r"r#r$sympy.utilities.miscr%r&mpmath.libmp.libmpfr'r$rLrVrWrr-r:r>rXr[r^rrr)rFs0rIrzs//8445:BB/%$$!,'^^2+7