+ iRt^RIHtHtHt^RIHt^RIt^RIH t ^RI t ] PtRt] !R.RQO4t.RROtRSRltR t] P*!] P,4] P.!] P0] P0] P0] P0R 7] P.!] P,] P,] P,R 7R 444t] P*!] P,4] P.!] P0] P0] P0] P0R 7] P.!] P,] P,R7RSRl444t^tRt] P:] P<] P*!] P,4] P.!] P0] P0R7R4444t] P:] P<] P*!] P,4] P.!] P,R7R4444t Rt!] P*!] P,4] P.!] P0] P0] P0] P0] P0] P0] P0R7] P.!] P,] P,] P,] P,] P,] P,] P,R7R444t"Rt#] P*!] P,4] P.!] P0] P0] P0R7] P.!] P,] P,] P,R7R444t$Rt%Rt&] P*!] P,4] P.!] P0] P0] P0] P0R 7] P.!] P,] P,] P,] P,] P,R7R 444t'R!t(R"t)R#t*R$t+R%t,R&t-] P.!] P0] P0] P0] P0] P0] P0] P0] P0R'7R(4t.] P*!] P04] P.!] P,] P0] P0] P0] P0] P0] P0] P0R)7] P.!] P,] P,] P,] P,R*7R+444t/R,t0R-t1] P.!] P0] P0] P0] P0] P,] P,] P,] P,] P,] P0] P0] P0] P0R.7 R/4t2^R0IH3t3H4t4H5t5H6t6]33R1lt7R2t8R3t9R4t:] P:] P<] P.!] P0] P0] P0] P0] P0] P0] P0R57R6444t;R7tR;t?R<t@R=tA] P*!] P04] P.!] P,] P0] P0] P0] P0R>7] P.!] P,] P,] P,R?7R@444tBRAtCRBtDRCtERDtFREtGRFtHRGtIRHtJRItKRTRJltLRKtMRLtNRMtORNtPROtQ]RRP8Xd6^RIStS^RITtT]SP!]TP!4P4R#R# ] ] 3d ^RI H t ELi;i)UzNfontTools.misc.bezierTools.py -- tools for working with Bezier path segments. ) calcBoundssectRectrectArea)IdentityN) namedtuple)cythong& .> Intersectionc X\\V!\V!\V!\V!V4#)aCalculates the arc length for a cubic Bezier segment. Whereas :func:`approximateCubicArcLength` approximates the length, this function calculates it by "measuring", recursively dividing the curve until the divided segments are shorter than ``tolerance``. Args: pt1,pt2,pt3,pt4: Control points of the Bezier as 2D tuples. tolerance: Controls the precision of the calcuation. Returns: Arc length value. )calcCubicArcLengthCcomplex)pt1pt2pt3pt4 tolerances&&&&&z/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/fontTools/misc/bezierTools.pycalcCubicArcLengthr8s,  w}gsmWc]I cV^W,,,V,R,pW2,V, V, R,pWV,R,WE, V3WDV,W#,R,V33#)g??)p0p1p2p3midderiv3s&&&& r_split_cubic_into_tworKsc RW  "e +CglR5 (F 2g_clC0 FlRWOR0 r)rrrr)multarchboxcF\W, 4p\W, 4\W#, 4,\W4, 4,pWP,\,V8dWV,R,#\WW44wrx\V.VO5!\V.VO5!,#r)absEPSILONr_calcCubicArcLengthCRecurse) rrrrrr r!onetwos &&&&& rr&r&Ts rw>> calcQuadraticArcLength((0, 0), (0, 0), (0, 0)) # empty segment 0.0 >>> calcQuadraticArcLength((0, 0), (50, 0), (80, 0)) # collinear points 80.0 >>> calcQuadraticArcLength((0, 0), (0, 50), (0, 80)) # collinear points vertical 80.0 >>> calcQuadraticArcLength((0, 0), (50, 20), (100, 40)) # collinear points 107.70329614269008 >>> calcQuadraticArcLength((0, 0), (0, 100), (100, 0)) 154.02976155645263 >>> calcQuadraticArcLength((0, 0), (0, 50), (100, 0)) 120.21581243984076 >>> calcQuadraticArcLength((0, 0), (50, -10), (80, 50)) 102.53273816445825 >>> calcQuadraticArcLength((0, 0), (40, 0), (-40, 0)) # collinear points, control point outside 66.66666666666667 >>> calcQuadraticArcLength((0, 0), (40, 0), (0, 0)) # collinear points, looping back 40.0 )calcQuadraticArcLengthCr r r rs&&&rcalcQuadraticArcLengthr?s @ #7C='3-# OOr)r r rd0d1dn)scaleorigDistabx0x1LencPW, pW!, pWC, pVR,p\V4pVR8Xd\W , 4#\Wc4p\V4\8dY\W44^8d\W , 4#\V4\V4rW,W,,W,, #\WS4V, p \WT4V, p \^\V 4\V 4, ,V,W|V , ,, 4p V #)aCalculates the arc length for a quadratic Bezier segment. Args: pt1: Start point of the Bezier as a complex number. pt2: Handle point of the Bezier as a complex number. pt3: End point of the Bezier as a complex number. Returns: Arc length value. y?)r$r3epsilonr;)r r rr@rArBrCrDrErFrGrHrIrJs&&& rr=r=s@ B B A BA FE |39~A{H 8}w <1 sy> !2wB1 !%(( ax B ax B a;r?[_45@ERTWDUV WC JrcF\\V!\V!\V!4#)aCalculates the arc length for a quadratic Bezier segment. Uses Gauss-Legendre quadrature for a branch-free approximation. See :func:`calcQuadraticArcLength` for a slower but more accurate result. Args: pt1: Start point of the Bezier as 2D tuple. pt2: Handle point of the Bezier as 2D tuple. pt3: End point of the Bezier as 2D tuple. Returns: Approximate arc length value. )approximateQuadraticArcLengthCr r>s&&&rapproximateQuadraticArcLengthrPs  *'3-#QT VVrr>)v0r-r.c\RV,RV,,RV,,4p\W , 4R,p\RV,RV,, RV,,4pW4,V,#)aCalculates the arc length for a quadratic Bezier segment. Uses Gauss-Legendre quadrature for a branch-free approximation. See :func:`calcQuadraticArcLength` for a slower but more accurate result. Args: pt1: Start point of the Bezier as a complex number. pt2: Handle point of the Bezier as a complex number. pt3: End point of the Bezier as a complex number. Returns: Approximate arc length value. g̔xb?gb?gFVW?gqq?g̔xb߿gFVWr$)r r rrQr-r.s&&& rrOrOsxB S #4s#::=ORU=UU B SY, ,B c!$5$;;>ORU>UU B 7R<rc\WV4wwr4wrVwrxVR,p VR,p .p V ^8wdV PV)V , 4V ^8wdV PV)V , 4V U u.uF^p ^T u;8:d ^8gKMKW<,V ,W\,,V,WL,V ,Wl,,V,3NK` up W.,p \V 4#uup i)aCalculates the bounding rectangle for a quadratic Bezier segment. Args: pt1: Start point of the Bezier as a 2D tuple. pt2: Handle point of the Bezier as a 2D tuple. pt3: End point of the Bezier as a 2D tuple. Returns: A four-item tuple representing the bounding rectangle ``(xMin, yMin, xMax, yMax)``. Example:: >>> calcQuadraticBounds((0, 0), (50, 100), (100, 0)) (0, 0, 100, 50.0) >>> calcQuadraticBounds((0, 0), (100, 0), (100, 100)) (0.0, 0.0, 100, 100) @)calcQuadraticParametersappendr)r r raxaybxbycxcyax2ay2rootstpointss&&& rcalcQuadraticBoundsrc*s$$;3S#I HRhr s(C s(C E ax bS3Y ax bS3YA :A: =  =!bf r !26A:#6#;< F f  s/C)C) A C)c V\\V!\V!\V!\V!4#)afApproximates the arc length for a cubic Bezier segment. Uses Gauss-Lobatto quadrature with n=5 points to approximate arc length. See :func:`calcCubicArcLength` for a slower but more accurate result. Args: pt1,pt2,pt3,pt4: Control points of the Bezier as 2D tuples. Returns: Arc length value. Example:: >>> approximateCubicArcLength((0, 0), (25, 100), (75, 100), (100, 0)) 190.04332968932817 >>> approximateCubicArcLength((0, 0), (50, 0), (100, 50), (100, 100)) 154.8852074945903 >>> approximateCubicArcLength((0, 0), (50, 0), (100, 0), (150, 0)) # line; exact result should be 150. 149.99999999999991 >>> approximateCubicArcLength((0, 0), (50, 0), (100, 0), (-50, 0)) # cusp; exact result should be 150. 136.9267662156362 >>> approximateCubicArcLength((0, 0), (50, 0), (100, -50), (-50, 0)) # cusp 154.80848416537057 )approximateCubicArcLengthCr r)s&&&&rapproximateCubicArcLengthrfLs*2 & w}gsmWc] r)rQr-r.v3v4c\W, 4R,p\RV,RV,,RV,,RV,,4p\W0, V,V, 4R,p\RV,RV,, RV,, RV,,4p\W2, 4R,pWE,V,V,V,#) zApproximates the arc length for a cubic Bezier segment. Args: pt1,pt2,pt3,pt4: Control points of the Bezier as complex numbers. Returns: Arc length value. g333333?gc1?g85$t?gu|Y?g#$?g?gc1g#$rS) r r rrrQr-r.rgrhs &&&& rrerejs> SY$ B S c ! " c ! " c ! " B SY_s " #&9 9B S c ! " c ! " c ! " B SY$ B 7R<" r !!rc\WW#4wwrEwrgwrwrVR,p VR,p VR,pVR,p\WV4Uu.uFp^Tu;8:d ^8gKMKVNK pp\WV 4Uu.uFp^Tu;8:d ^8gKMKVNK ppVV,pVUu.uFpVV,V,V,VV,V,,VV,,V ,VV,V,V,VV,V,,V V,,V ,3NK upW.,p\V4#uupiuupiuupi)a(Calculates the bounding rectangle for a quadratic Bezier segment. Args: pt1,pt2,pt3,pt4: Control points of the Bezier as 2D tuples. Returns: A four-item tuple representing the bounding rectangle ``(xMin, yMin, xMax, yMax)``. Example:: >>> calcCubicBounds((0, 0), (25, 100), (75, 100), (100, 0)) (0, 0, 100, 75.0) >>> calcCubicBounds((0, 0), (50, 0), (100, 50), (100, 100)) (0.0, 0.0, 100, 100) >>> print("%f %f %f %f" % calcCubicBounds((50, 0), (0, 100), (100, 100), (50, 0))) 35.566243 0.000000 64.433757 75.000000 @rU)calcCubicParameterssolveQuadraticr)r r rrrXrYrZr[r\r]dxdyax3ay3bx2by2raxRootsyRootsr`rbs&&&& rcalcCubicBoundsrvsE$.A3-T*HRhr(2 s(C s(C s(C s(C'"5 D5Aa!aa5F D'"5 D5Aa!aa5F D VOE  A FQJNR!VaZ '"q& 02 5 FQJNR!VaZ '"q& 02 5   F f E Ds+ E "E &E <EEE+B EcVwrEVwrgWd, pWu, p Tp Tp W3V,p V ^8XdW3.#W*V 3V,, V , p ^T u;8:d^8d(MM$W,V ,W,V ,3pW3W3.#W3.#)aSplit a line at a given coordinate. Args: pt1: Start point of line as 2D tuple. pt2: End point of line as 2D tuple. where: Position at which to split the line. isHorizontal: Direction of the ray splitting the line. If true, ``where`` is interpreted as a Y coordinate; if false, then ``where`` is interpreted as an X coordinate. Returns: A list of two line segments (each line segment being two 2D tuples) if the line was successfully split, or a list containing the original line. Example:: >>> printSegments(splitLine((0, 0), (100, 100), 50, True)) ((0, 0), (50, 50)) ((50, 50), (100, 100)) >>> printSegments(splitLine((0, 0), (100, 100), 100, True)) ((0, 0), (100, 100)) >>> printSegments(splitLine((0, 0), (100, 100), 0, True)) ((0, 0), (0, 0)) ((0, 0), (100, 100)) >>> printSegments(splitLine((0, 0), (100, 100), 0, False)) ((0, 0), (0, 0)) ((0, 0), (100, 100)) >>> printSegments(splitLine((100, 0), (0, 0), 50, False)) ((100, 0), (50, 0)) ((50, 0), (0, 0)) >>> printSegments(splitLine((0, 100), (0, 0), 50, True)) ((0, 100), (0, 50)) ((0, 50), (0, 0)) r)r r where isHorizontalpt1xpt1ypt2xpt2yrXrYrZr[rFramidPts&&&& r splitLinersHJDJD B B B B AAv | b,' '1,AAzz RVb[( ul++ |rc\WV4wrVp\WT,Wd,Wt,V, 4p\RV44pV'gWV3.#\WVV.VO5!#)aSplit a quadratic Bezier curve at a given coordinate. Args: pt1,pt2,pt3: Control points of the Bezier as 2D tuples. where: Position at which to split the curve. isHorizontal: Direction of the ray splitting the curve. If true, ``where`` is interpreted as a Y coordinate; if false, then ``where`` is interpreted as an X coordinate. Returns: A list of two curve segments (each curve segment being three 2D tuples) if the curve was successfully split, or a list containing the original curve. Example:: >>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 150, False)) ((0, 0), (50, 100), (100, 0)) >>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 50, False)) ((0, 0), (25, 50), (50, 50)) ((50, 50), (75, 50), (100, 0)) >>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 25, False)) ((0, 0), (12.5, 25), (25, 37.5)) ((25, 37.5), (62.5, 75), (100, 0)) >>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 25, True)) ((0, 0), (7.32233, 14.6447), (14.6447, 25)) ((14.6447, 25), (50, 75), (85.3553, 25)) ((85.3553, 25), (92.6777, 14.6447), (100, -7.10543e-15)) >>> # XXX I'm not at all sure if the following behavior is desirable: >>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 50, True)) ((0, 0), (25, 50), (50, 50)) ((50, 50), (50, 50), (50, 50)) ((50, 50), (75, 50), (100, 0)) c3P"TFp^Tu;8:d ^8gKMKVxK R#5iNr.0ras& r !splitQuadratic..":)QqAzzqzq) && &)rVrmsorted_splitQuadraticAtT) r r rrxryrFrGc solutionss &&&&& rsplitQuadraticrseF&c4GA! !/E*AI:)::I 3  aA 2 22rc\WW#4wrgr\We,Wu,W,W,V, 4p \RV 44p V 'gWW#3.#\WgW.V O5!#)aSplit a cubic Bezier curve at a given coordinate. Args: pt1,pt2,pt3,pt4: Control points of the Bezier as 2D tuples. where: Position at which to split the curve. isHorizontal: Direction of the ray splitting the curve. If true, ``where`` is interpreted as a Y coordinate; if false, then ``where`` is interpreted as an X coordinate. Returns: A list of two curve segments (each curve segment being four 2D tuples) if the curve was successfully split, or a list containing the original curve. Example:: >>> printSegments(splitCubic((0, 0), (25, 100), (75, 100), (100, 0), 150, False)) ((0, 0), (25, 100), (75, 100), (100, 0)) >>> printSegments(splitCubic((0, 0), (25, 100), (75, 100), (100, 0), 50, False)) ((0, 0), (12.5, 50), (31.25, 75), (50, 75)) ((50, 75), (68.75, 75), (87.5, 50), (100, 0)) >>> printSegments(splitCubic((0, 0), (25, 100), (75, 100), (100, 0), 25, True)) ((0, 0), (2.29379, 9.17517), (4.79804, 17.5085), (7.47414, 25)) ((7.47414, 25), (31.2886, 91.6667), (68.7114, 91.6667), (92.5259, 25)) ((92.5259, 25), (95.202, 17.5085), (97.7062, 9.17517), (100, 1.77636e-15)) c3P"TFp^Tu;8:d ^8gKMKVxK R#5irrrs& rrsplitCubic..Grr)rl solveCubicr_splitCubicAtT) r r rrrxryrFrGrrBrs &&&&&& r splitCubicr(si6%Ss8JA! !/1?U;RI:)::I 3$%% ! 1y 11rc<\WV4wrEp\WEV.VO5!#)a]Split a quadratic Bezier curve at one or more values of t. Args: pt1,pt2,pt3: Control points of the Bezier as 2D tuples. *ts: Positions at which to split the curve. Returns: A list of curve segments (each curve segment being three 2D tuples). Examples:: >>> printSegments(splitQuadraticAtT((0, 0), (50, 100), (100, 0), 0.5)) ((0, 0), (25, 50), (50, 50)) ((50, 50), (75, 50), (100, 0)) >>> printSegments(splitQuadraticAtT((0, 0), (50, 100), (100, 0), 0.5, 0.75)) ((0, 0), (25, 50), (50, 50)) ((50, 50), (62.5, 50), (75, 37.5)) ((75, 37.5), (87.5, 25), (100, 0)) )rVr)r r rtsrFrGrs&&&* rsplitQuadraticAtTrMs&(&c4GA! aA + ++rc\WW#4wrVrx\WVWx.VO5!p V.V ^,R,O5V ^&.V R,RROVN5V R&V #)aSplit a cubic Bezier curve at one or more values of t. Args: pt1,pt2,pt3,pt4: Control points of the Bezier as 2D tuples. *ts: Positions at which to split the curve. Returns: A list of curve segments (each curve segment being four 2D tuples). Examples:: >>> printSegments(splitCubicAtT((0, 0), (25, 100), (75, 100), (100, 0), 0.5)) ((0, 0), (12.5, 50), (31.25, 75), (50, 75)) ((50, 75), (68.75, 75), (87.5, 50), (100, 0)) >>> printSegments(splitCubicAtT((0, 0), (25, 100), (75, 100), (100, 0), 0.5, 0.75)) ((0, 0), (12.5, 50), (31.25, 75), (50, 75)) ((50, 75), (59.375, 75), (68.75, 68.75), (77.3438, 56.25)) ((77.3438, 56.25), (85.9375, 43.75), (93.75, 25), (100, 0)) :NNN)rlr) r r rrrrFrGrrBsplits &&&&* r splitCubicAtTresb(%Ss8JA! 1 + +E #eAhrl#E!H&%)CR.&#&E"I Lr)r r rrrFrGrrBc'\"\WW#4wrVrx\WVWx.VO5!RjxL R#L5i)aSplit a cubic Bezier curve at one or more values of t. Args: pt1,pt2,pt3,pt4: Control points of the Bezier as complex numbers.. *ts: Positions at which to split the curve. Yields: Curve segments (each curve segment being four complex numbers). N)calcCubicParametersC_splitCubicAtTC) r r rrrrFrGrrBs &&&&* rsplitCubicAtTCrs,(&c9JA!qQ/B///s !,*,)rar r rrpointAtToff1off2)t2_1_t_1_t_2_2_t_1_tcWD,p^V, pWf,p^V,V,pWv,V,^Wt,V,We,V,,,,WT,V,,p Wp,W,,WR,,p Wq,W,,WS,,p WV, V,,pW2V, V,,pWW3WW#33#)zSplit a cubic Bezier curve at t. Args: pt1,pt2,pt3,pt4: Control points of the Bezier as complex numbers. t: Position at which to split the curve. Returns: A tuple of two curve segments (each curve segment being four complex numbers). r) r r rrrarrrrrrrs &&&&& rsplitCubicIntoTwoAtTCrs0 B q5D [F1ut|H a6:#3di#o#EFFRUU  <(. (28 3D <(. (28 3D sa C sd" "C t &(B CCrc\V4p.pVP^R4VPR4VwrVVwrxVwr\\ V4^, 4Fp W;,p W;^,,p W, pW,pW_,pWo,p^V,V ,V,V,p^V,V ,V,V,pW,pVV,W|,,V ,pVV,W,,V ,p\ VV3VV3VV34wpppVPVVV34K V#rrLr+)listinsertrWrangelencalcQuadraticPoints)rFrGrrsegmentsrXrYrZr[r\r]it1rdeltadelta_2a1xa1yb1xb1yt1_2c1xc1yr r rs&&&* rrrs bBHIIaIIcN FB FB FB 3r7Q;  U AY-ll2v{R5(2v{R5(w4i"'!B&4i"'!B&+S#Jc S#JO S#c3( Orc\V4pVP^R4VPR4.pVwrgVwrVwrVwr\\ V4^, 4EFpWN,pWN^,,pVV, pVV,pVV,pW,pVV,pVV,pVV,p^V,V,V,V,p^V,V,V ,V,p^V,V,V ,^V,V,,V,p^V ,V,V ,^V,V,,V,pVV,VV,,W,,V ,pVV,V V,,W,,V ,p\ VV3VV3VV3VV34wppp p!VPVVV V!34EK V#r)rrrWrrcalcCubicPoints)"rFrGrrBrrrXrYrZr[r\r]rnrorrrrrdelta_3rt1_3rrrrrrd1xd1yr r rrs"&&&&* rrrs bBIIaIIcNH FB FB FB FB 3r7Q;  U AYR%-'/wDy7l7l2v{R7*2v{R7*2v{R!b&4-/582v{R!b&4-/584i"t)#bg-24i"t)#bg-2, #Jc S#Jc  S#s c3,-- . Or) rFrGrrBrrrrra1b1c1rAc'd"\V4pVP^R4VPR4\\ V4^, 4FpWE,pWE^,,pWv, pW,p W,p Wf,p Wk,p W ,p ^V,V,V,V ,p^V,V,V,^V,V ,,V,pW ,W,,W&,,V,p\ WVV4wppppVVVV3xK R#5i)rrLr+N)rrrWrrcalcCubicPointsC)rFrGrrBrrrrrrrrrrrrrAr r rrs&&&&* rrrs bBIIaIIcN 3r7Q;  U AY-/wy[!ebj1n '!ebj1nq1ut|+u 4 X 16 )A --bb"=S#sCc""! sD.D0)r9acoscospic:\V4\8d'\V4\8d.pV#V)V, .pV#W,RV,V,, pVR8d;V!V4pV)V,R, V, V)V, R, V, .pV#.pV#)u'Solve a quadratic equation. Solves *a*x*x + b*x + c = 0* where a, b and c are real. Args: a: coefficient of *x²* b: coefficient of *x* c: constant term Returns: A list of roots. Note that the returned list is neither guaranteed to be sorted nor to contain unique values! @rLrUr$rM)rFrGrr9r`DDrDDs&&&& rrmrm/s 1v q6G E LR!VHE LUS1Wq[  9r(Cb3h#%)QBH+;a+?@E LE Lrc \V4\8d \WV4#\V4pW, pW , pW0, pWD,RV,, R, pRV,V,V,RV,V,, RV,,R, pW,p Ww,V,p V \8d^MT p \V 4\8d^MT p W, p V R8Xd$V R8Xd\ V)R, \ 4p WV .#V \R,8:Ed\ \\V\V 4, R4R 44p R \V4,pVR, pV\V R, 4,V, pV\V R\,,R, 4,V, pV\V R \,,R, 4,V, p\VVV.4wpppVV, \8d=VV, \8d+\ VV,V,R, \ 4;p;ppMVV, \8d2\ VV,R, \ 4;pp\ V\ 4pMsVV, \8d2\ V\ 4p\ VV,R, \ 4;ppM0\ V\ 4p\ V\ 4p\ V\ 4pVVV.#\\V 4\V4,R 4p WV , ,p VR8dV )p \ WR, , \ 4p V .#) uSolve a cubic equation. Solves *a*x*x*x + b*x*x + c*x + d = 0* where a, b, c and d are real. Args: a: coefficient of *x³* b: coefficient of *x²* c: coefficient of *x* d: constant term Returns: A list of roots. Note that the returned list is neither guaranteed to be sorted nor to contain unique values! Examples:: >>> solveCubic(1, 1, -6, 0) [-3.0, -0.0, 2.0] >>> solveCubic(-10.0, -9.0, 48.0, -29.0) [-2.9, 1.0, 1.0] >>> solveCubic(-9.875, -9.0, 47.625, -28.75) [-2.911392, 1.0, 1.0] >>> solveCubic(1.0, -4.5, 6.75, -3.375) [1.5, 1.5, 1.5] >>> solveCubic(-12.0, 18.0, -9.0, 1.50023651123) [0.5, 0.5, 0.5] >>> solveCubic( ... 9.0, 0.0, 0.0, -7.62939453125e-05 ... ) == [-0.0, -0.0, -0.0] True rkg"@rUg;@gK@rLrr+rggUUUUUU?)r$rMrmfloatround epsilonDigitsrmaxminr9rrrpow)rFrGrrBra2a3QRR2Q3R2_Q3r5thetarQ2a1_3rHrIx2s&&&& rrrPsL 1vaA&& aA B B B 38 s"A rB cBhm +dRi 74?A B B7lB"gRB GE SyR3Y 2#)] +ay 'C- SQb\3/67T!WnCx 3us{# #d * 3b(C/0 04 7 3b(C/0 04 7RRL) B 7W b7!2 "r'B,#!5}E EB Eb "Ww R"WO]; ;Br=)B "Ww r=)BR"WO]; ;Br=)Br=)Br=)BB| U c!f$g . AI 8A !3h, .s rcVwr4VwrVVwrxW7, R,p WH, R,p WW, V , p Wh, V , p W3W3Wx33#)rUr) r r rry2x3y3r\r]rZr[rXrYs &&& rrVrVsV FB FB FB 'SB 'SB 2B 2B 8bXx ''rc$VwrEVwrgVwrVwrWJ, R,p W[, R,p Wd, R,V , pWu, R,V , pW, V , V, pW, V , V, pVV3W3W3W33#rkr)r r rrrrrrx4y4rnror\r]rZr[rXrYs&&&& rrlrls FB FB FB FB 'SB 'SB 'S2 B 'S2 B 2 B 2 B 8bXx" 11r)r r rrrFrGrcW, R,pW!, R,V, pW0, V, V, pWeW@3#rr)r r rrrrGrFs&&&& rrrs; cA cAA A A !>rcVwr4VwrVVwrxTp Tp VR,V,p VR,V,p W5,V,p WF,V,pW3W3W33#r#r)rFrGrrXrYrZr[r\r]rIy1rrrrs&&& rrrsd FB FB FB B B s(bB s(bB 2B 2B 8bXx ''rc2VwrEVwrgVwrVwrT p T p VR, V ,pV R, V ,pWh,R, V,pWy,R, V,pWJ,V,V,pW[,V ,V,pW3W3VV3VV33#rr)rFrGrrBrXrYrZr[r\r]rnrorIrrrrrrrs&&&& rrrs FB FB FB FB B B s(bB s(bB 'S2 B 'S2 B 2 B 2 B 8bXBx"b 11rrFrGrrBrrp4cVR,V,pW,R,V,pW,V,V,pW4WV3#)rrrrs&&&& rrrs; eqB %E R B QB 2?rcV^,^V, ,V^,V,,V^,^V, ,V^,V,,3#)zFinds the point at time `t` on a line. Args: pt1, pt2: Coordinates of the line as 2D tuples. t: The time along the line. Returns: A 2D tuple with the coordinates of the point. r)r r ras&&&r linePointAtTrsCVq1u A *c!fA.>Q!.K MMrc^V, ^V, ,V^,,^^V, ,V,V^,,,W3,V^,,,p^V, ^V, ,V^,,^^V, ,V,V^,,,W3,V^,,,pWE3#)zFinds the point at time `t` on a quadratic curve. Args: pt1, pt2, pt3: Coordinates of the curve as 2D tuples. t: The time along the curve. Returns: A 2D tuple with the coordinates of the point. r)r r rrar5ys&&&& rquadraticPointAtTrs Q1q5CF"Q!a%[1_s1v%==ANA Q1q5CF"Q!a%[1_s1v%==ANA 6MrcWD,p^V, pWf,pWv,V^,,^Wt,V^,,We,V^,,,,,WT,V^,,,pWv,V^,,^Wt,V^,,We,V^,,,,,WT,V^,,,p W3#)zFinds the point at time `t` on a cubic curve. Args: pt1, pt2, pt3, pt4: Coordinates of the curve as 2D tuples. t: The time along the curve. Returns: A 2D tuple with the coordinates of the point. r) r r rrrarrrr5rs &&&&& r cubicPointAtTr,s B q5D [F A vzCF"TYQ%77 8 9 &3q6/   A vzCF"TYQ%77 8 9 &3q6/  6Mr)rar r rr)rrrcWD,p^V, pWf,pWv,V,^Wt,V,We,V,,,,WT,V,,#)zFinds the point at time `t` on a cubic curve. Args: pt1, pt2, pt3, pt4: Coordinates of the curve as complex numbers. t: The time along the curve. Returns: A complex number with the coordinates of the point. r)r r rrrarrrs&&&&& rcubicPointAtTCrFsP& B q5D [F =3 fj3&6S&H!I IBFUXL XXrc\V4^8Xd\.VOVN5!#\V4^8Xd\.VOVN5!#\V4^8Xd\.VOVN5!#\ R4hr7Unknown curve degree)rrrr ValueError)segras&&rsegmentPointAtTr_sh 3x1}$S$!$$ SQ )#)q)) SQ%c%1%% + ,,rc$Vwr4VwrVVwrx\W5, 4\8d\WF, 4\8dR#\W5, 4\WF, 48dWs, WS, , #W, Wd, , #)rrr) septsxsyexeypxpys &&& r _line_t_of_ptr nsi FB FB FB 27|g#bg,"8  27|c"'l"BG$$BG$$rcV^,V^,, V^,V^,, ,pV^,V^,, V^,V^,, ,pVR8*;'dVR8*'*#)rrLr)rFrGoriginxDiffyDiffs&&& r'_both_points_are_on_same_side_of_originr|sc qTF1I !A$"2 3E qTF1I !A$"2 3E --# ..rc VwrEVwrgVwrVwr\P!W4'd;\P!WF4'd\P!WH4'g.#\P!W4'd;\P!WW4'd\P!WY4'g.#\P!W4'd\P!W4'd.#\P!WF4'd\P!WW4'd.#\P!Wd4'dTTp W, W, , p WV, ,V ,pW3p\V\WV4\W#V4R7.#\P!W4'dTTp Wu, Wd, , pVW, ,V,pW3p\V\WV4\W#V4R7.#Wu, Wd, , pW, W, , p \P!VV 4'd.#VV,V, W,, V ,VV , , p VW, ,V,pW3p\ WV4'd6\ WV4'd$\V\WV4\W#V4R7.#.#)aFinds intersections between two line segments. Args: s1, e1: Coordinates of the first line as 2D tuples. s2, e2: Coordinates of the second line as 2D tuples. Returns: A list of ``Intersection`` objects, each object having ``pt``, ``t1`` and ``t2`` attributes containing the intersection point, time on first segment and time on second segment respectively. Examples:: >>> a = lineLineIntersections( (310,389), (453, 222), (289, 251), (447, 367)) >>> len(a) 1 >>> intersection = a[0] >>> intersection.pt (374.44882952482897, 313.73458370177315) >>> (intersection.t1, intersection.t2) (0.45069111555824465, 0.5408153767394238) rrr)r8iscloserr r)s1e1s2e2s1xs1ye1xe1ys2xs2ye2xe2yr5slope34rrslope12s&&&& rlineLineIntersectionsr"sE.HCHCHCHC S4<<#9#9$,,sBXBX  S4<<#9#9$,,sBXBX  ||C$,,s"8"8  ||C$,,s"8"8  ||C 9+ 3w # %V -3 bb8Q    ||C 9+ qw # %V -3 bb8Q   ySY'GySY'G ||GW%% 3 w} ,s 2w7HIA17c!A B.  1"" = = -3 bb8Q   IrcV^,pVR,p\P!V^,V^,, V^,V^,, 4p\P!V)4P V^,)V^,)4#)rr)r8atan2rrotate translate)segmentstartendangles& r_alignment_transformationr+sj AJE "+C JJs1va(#a&58*; .s<]cQm!m!m!]r) r+transformPointsrrVrmrlrrr)curveline aligned_curverFrGr intersectionsrBs&& r_curve_line_intersections_tr4s-d3CCEJM 5zQ)=9a&qtQqT1Q48 Uq(-8 a"1Q41qtQqT: /00 <]< <>> curve = [ (100, 240), (30, 60), (210, 230), (160, 30) ] >>> line = [ (25, 260), (230, 20) ] >>> intersections = curveLineIntersections(curve, line) >>> len(intersections) 3 >>> intersections[0].pt (84.9000930760723, 189.87306176459828) rr) rrrrr4r rrWr)r0r1 pointFinderr3rarline_ts&& rcurveLineIntersectionsr8s* 5zQ' Uq# /00M ( 5  #% # #))b)  (4 ( (\R&AB 6 rc|\V4^8Xd \V!#\V4^8Xd \V!#\R4h)rr)rrcrvr)rs&r _curve_boundsr: s: 1v{"A&& Q1"" + ,,rc\V4^8XdVwr#\W#V4pW$3WC3.#\V4^8Xd\.VOVN5!#\V4^8Xd\.VOVN5!#\ R4hr)rrrrr)rrarrmidpoints&& r_split_segment_at_tr=sw 1v{a( }-- 1v{ '!'Q'' Q1#a### + ,,rc Ba\V4p\V4pV'gRpV'gRp\WV4wrxV'g.#Rp \V4S8d"\V4S8dV !V4V !V43.#\VR4wrV^,V !V43p V !V4V^,3p \VR4wrV^,V !V43pV !V4V^,3p.pVP \ WSV VR74VP \ WSV VR74VP \ WSV VR74VP \ WSV VR74V3Rlp\ 4p.pVF6pV!V4pVV9dKVPV4VPV4K8 V#)rLc>RV^,V^,,,#r#r)rs&rr<._curve_curve_intersections_t..midpoint1sadQqTk""rr)range1range2ch<\V^,S, 4\V^,S, 43#)r)int)r precisions&r._curve_curve_intersections_t..Vs&SA!23SA9J5KLr)rLr+) r:rrr=extend_curve_curve_intersections_tsetaddrW)curve1curve2rFrBrCbounds1bounds2 intersects_r<c11c12 c11_range c12_rangec21c22 c21_range c22_rangefound unique_keyseen unique_valuesrkeys&&f&& rrJrJ!sF#GF#G  W.MJ  #9$'):Y)F&!8F#3455"63/HCHV,-I&!6!9-I"63/HCHV,-I&!6!9-I E LL$ i )   LL$ i )   LL$ i )   LL$ i )  MJ 5DMn $;   R  rc\V4PV4p\;QJdRV4F 'dK R# R#!RV44#)c3^"TF#p\P!V^,R4xK% R#5i)rrLN)r8r)rps& rr_is_linelike..fs": 1t||AaD#&& s+-FT)r+r/all)r' maybelines& r _is_linelikerfds@)'2BB7KI 3: :33:3:3: : ::rc 6\V4'dV^,VR,3p\V4'd!V^,VR,3p\.VOVO5!#\W4pVUu.uF/p\VPVP VP R7NK1 up#\V4'dV^,VR,3p\W4#\W4pVUu.uF/p\\W^,4V^,V^,R7NK1 up#uupiuupi)aFinds intersections between a curve and a curve. Args: curve1: List of coordinates of the first curve segment as 2D tuples. curve2: List of coordinates of the second curve segment as 2D tuples. Returns: A list of ``Intersection`` objects, each object having ``pt``, ``t1`` and ``t2`` attributes containing the intersection point, time on first segment and time on second segment respectively. Examples:: >>> curve1 = [ (10,100), (90,30), (40,140), (220,220) ] >>> curve2 = [ (5,150), (180,20), (80,250), (210,190) ] >>> intersections = curveCurveIntersections(curve1, curve2) >>> len(intersections) 3 >>> intersections[0].pt (81.7831487395506, 109.88904552375288) rr) rfr"r8rrrrrJr)rMrNline1line2hitsr5intersection_tsrs&& rcurveCurveIntersectionsrlis*Fq 6":%   1Ivbz)E(8%8%8 8)&8DFJJTLADDQTTadd;TJ J f  q 6":%%f4426BO" !B 162a5RUK!  K  s $5D5Dc Rp\V4\V48dYrRp\V4^8d)\V4^8d \W4pMF\W4pM:\V4^8Xd \V4^8Xd\.VOVO5!pM \ R4hV'gV#VUu.uF/p\ VP VPVPR7NK1 up#uupi)aFinds intersections between two segments. Args: seg1: List of coordinates of the first segment as 2D tuples. seg2: List of coordinates of the second segment as 2D tuples. Returns: A list of ``Intersection`` objects, each object having ``pt``, ``t1`` and ``t2`` attributes containing the intersection point, time on first segment and time on second segment respectively. Examples:: >>> curve1 = [ (10,100), (90,30), (40,140), (220,220) ] >>> curve2 = [ (5,150), (180,20), (80,250), (210,190) ] >>> intersections = segmentSegmentIntersections(curve1, curve2) >>> len(intersections) 3 >>> intersections[0].pt (81.7831487395506, 109.88904552375288) >>> curve3 = [ (100, 240), (30, 60), (210, 230), (160, 30) ] >>> line = [ (25, 260), (230, 20) ] >>> intersections = segmentSegmentIntersections(curve3, line) >>> len(intersections) 3 >>> intersections[0].pt (84.9000930760723, 189.87306176459828) FTz4Couldn't work out which intersection function to user) rrlr8r"rrrrr)seg1seg2swappedr3rs&& rsegmentSegmentIntersectionsrqs<G 4y3t9d 4y1} t9q=3D?M24>M TaCIN-;t;d; OPP =J K]LADDQTTadd 3] KK Ks 5Cc\V4pRRPRV44,# \d RT,u#i;i)zk >>> _segmentrepr([1, [2, 3], [], [[2, [3, 4], [0.1, 2.2]]]]) '(1, (2, 3), (), ((2, (3, 4), (0.1, 2.2))))' z(%s)z, c38"TFp\V4xK R#5ir0) _segmentrepr)rr5s& rr_segmentrepr..s!>2a,q//2sz%g)iterjoin TypeError)objits& rrtrtsG ? #Y !>2!>>>> czs ,AAc@VFp\\V44K R#)zdHelper for the doctests, displaying each segment in a list of segments on a single line as a tuple. N)printrt)rr's& r printSegmentsr}s l7#$r__main__r)rfrerPrOrr r?r=rvrcrrrrrrrrmrrrrrrr"r8rlrq)g{Gzt?)gMbP?NN)X__doc__fontTools.misc.arrayToolsrrrfontTools.misc.transformrr8 collectionsrrAttributeError ImportErrorfontTools.misccompiledCOMPILEDr%r__all__rrreturnsdoublelocalsr r&r rrMcfuncinliner3r;r?r=rPrOrcrfrervrrrrrrrrrrr9rrrrmrrVrlrrrrrrrrrr rr"r+r4r8r:r=rJrfrlrqrtr}__name__sysdoctestexittestmodfailedrrrrsSED- "& ?? .*<=  @&  ~~ ~~ ~~ ~~  FMM 6==I  J      mm  A   A  &..V^^4&5& ; ; PF  ~~ ~~ nn nn -- ]] mm mm }} }} &@W"   }} }} }}   BD<    }} }} }} }} }} !" !"H#L6r*3Z"2J,0> nn nn nn nn  0  0 mm ^^    }}6==D  D46 F nn nn nn nn }} }} -- MM MM ~~ ~~ ~~ ~~##6%$"&BYB( 2 nn nn nn ( 2  nn nn nn nn ~~ ~~ ~~ N 4 mm &--fmmFMMJ YK  Y - %/ K\C =#L- -@F; 'T-L` ?% zHHW__  % %& Y.  $&%%&sb..cc