hBF 8SSKr\RrSSKrSSKJr J r SS/r Sr \ "S5r\R\R \R""\R$5\R&"\R(\R(\R$S 9S 5555r\R\R&"\R(\R$S 9\R&"\R$\R$S 9S 555r\R\R \R&"\R(\R(\R(\R(S9\R&"\R(\R(\R(\R(S9S5555r\R\R \R&"\R(\R(\R(\R(S9\R&"\R(\R(\R(\R(S9S5555r\R\R \R&"\R(\R(\R(\R(S9S555r\R&"\R(\R(\R(\R(\R4S9\R&"\R(\R(\R(\R(S9\R&"\R$\R$\R$\R4S9\R&"\R(\R(\R(\R(S9S5555r\R\R \R&"\R(\R(\R(\R(S9\R&"\R(\R(S9S5555r\R\R \R&"\R(\R(\R(\R(S9\R&"\R(\R(\R(\R(S9S5555r\R\R \R""\R(5\R&"\R$\R(\R(\R(\R(S9\R&"\R(\R(S9S55555r\R\R \R""\R(5\R&"\R(\R(\R(\R(S9\R&"\R(\R(\R(\R$S9S 55555r\R\R""\R45\R&"\R$\R(\R(\R(\R(S!9\R&"\R(\R(S9S"5555r \R\R \R&"\R$S#9\R&"\R(\R(\R(\R(\R(S$9S%5555r!\R\R&"\R4\R$S&9\R&"\R4S'9\R&"\R4S(9\R&"\R(\R(\R(\R(S)9\R&"\R(\R(\R(\R(\R(S*9S+555555r"\R&"\R$S,9\R&"\R4S-9\R&"\R4S(9S1S.j555r#\R&"\R4\R4\R4S/9\R&"\R4S(9S1S0j55r$g!\\4a SSKJr G Nf=f)2N)cython)ErrorApproxNotFoundErrorcurve_to_quadraticcurves_to_quadraticdNaNv1v2resultc`XR5-Rn[U5S:aSnU$)zReturn the dot product of two vectors. Args: v1 (complex): First vector. v2 (complex): Second vector. Returns: double: Dot product. gV瞯! ' 'F 6{U M)zden)zrzicRURnURn[X!- X1- 5$)aLDivide complex by real using Python's method (two separate divisions). This ensures bit-exact compatibility with Python's complex division, avoiding C's multiply-by-reciprocal optimization that can cause 1 ULP differences on some platforms/compilers (e.g. clang on macOS arm64). https://github.com/fonttools/fonttools/issues/3928 )rimagcomplex)rrrrs r_complex_div_by_realr?s' B B 28RX &&r)abcd)_1_2_3_4cbUn[US5U-n[X-S5U-nX-U-U-nXEXg4$N@)r)rrr r!r"r#r$r%s rcalc_cubic_pointsr)PsF B a % )B aeS )B .B QB 2>r)p0p1p2p3cDX- S-nX!- S-U- nUnX6- U- U- nXuXF4$r')r*r+r,r-r rr!rs rcalc_cubic_parametersr0^s= CA C!A A  QA :rc US:Xa[[XX#55$US:Xa[[XX#55$US:XaL[XX#5upV[[USUSUSUS5[USUSUSUS5-5$US:XaL[XX#5upV[[USUSUSUS5[USUSUSUS5-5$[XX#U5$)aSplit a cubic Bezier into n equal parts. Splits the curve into `n` equal parts by curve time. (t=0..1/n, t=1/n..2/n, ...) Args: p0 (complex): Start point of curve. p1 (complex): First handle of curve. p2 (complex): Second handle of curve. p3 (complex): End point of curve. Returns: An iterator yielding the control points (four complex values) of the subcurves. rr)itersplit_cubic_into_twosplit_cubic_into_three_split_cubic_into_n_gen)r*r+r,r-nrrs rsplit_cubic_into_n_iterr;ls, Av(899Av*22:;;Av#BB3 1qtQqT1Q4 8"1Q41qtQqT: ;   Av#BB3 "1Q41qtQqT :$QqT1Q41qt< =  #221 55r)r*r+r,r-r:)dtdelta_2delta_3i)a1b1c1d1c## [XX#5upVpxSU- n X-n X-n [U5HXn X-n X-nX[-nSU-U -U-U -nSU-U -U-SU-U--U -nX]-U-Xn--X}--U-n[UUUU5v MZ g7f)Nrr3r2)r0ranger))r*r+r,r-r:rrr r!r<r=r>r?t1t1_2r@rArBrCs rr9r9s'rr6JA! QBgGlG 1X Vw [!ebj1n '!ebj1nq1ut|+r 1 Vd]QX % . 2BB//sBB)midderiv3cpUSX---U-S-nX2-U- U- S-nXU-S-XE- U4XDU-X#-S-U44$)adSplit a cubic Bezier into two equal parts. Splits the curve into two equal parts at t = 0.5 Args: p0 (complex): Start point of curve. p1 (complex): First handle of curve. p2 (complex): Second handle of curve. p3 (complex): End point of curve. Returns: tuple: Two cubic Beziers (each expressed as a tuple of four complex values). r3??r/)r*r+r,r-rHrIs rr7r7se* RW  "e +CglR5 (F 2g_clC0 FlRWOR0 r)mid1deriv1mid2deriv2cSU-SU--SU--U-S-nUSU--SU-- S-nUSU--SU--SU--S-nSU-SU-- U- S-nUSU-U-S- XE- U4XDU-Xg- U4XfU-USU--S- U44$) avSplit a cubic Bezier into three equal parts. Splits the curve into three equal parts at t = 1/3 and t = 2/3 Args: p0 (complex): Start point of curve. p1 (complex): First handle of curve. p2 (complex): Second handle of curve. p3 (complex): End point of curve. Returns: tuple: Three cubic Beziers (each expressed as a tuple of four complex values).  r5gh/?r3r4r2r(r/)r*r+r,r-rMrNrOrPs rr8r8s: FR"W q2v % *v 6D1r6kAF"v .F RK"r' !AF *v 6D"fq2vo"v .F a"frkS $-6 f}dmT2 f}rAF{c126 r)tr*r+r,r-)_p1_p2c>XU- S--nXCU- S--nXVU- U--$)aTApproximate a cubic Bezier using a quadratic one. Args: t (double): Position of control point. p0 (complex): Start point of curve. p1 (complex): First handle of curve. p2 (complex): Second handle of curve. p3 (complex): End point of curve. Returns: complex: Location of candidate control point on quadratic curve. g?r/)rTr*r+r,r-rUrVs rcubic_approx_controlrXs50 R3 C R3 C )q  r)abcdphcX- nX2- nUS-n[X`U- 5[Xe5- nX%U--$![a* X:XaX:XdX#:XaUs$[[[5s$f=f)aUCalculate the intersection of two lines. Args: a (complex): Start point of first line. b (complex): End point of first line. c (complex): Start point of second line. d (complex): End point of second line. Returns: complex: Location of intersection if one present, ``complex(NaN,NaN)`` if no intersection was found. y?)rZeroDivisionErrorrNAN)rrr r!rYrZr[r\s rcalc_intersectr`sv$ B B RA ! q5MCJ & Av: ! 6qvHsC  !s0A$ A$#A$) tolerancer*r+r,r-c[U5U::a[U5U::agUSX---U-S-n[U5U:agX2-U- U- S-n[XU-S-XV- XT5=(a [XUU-X#-S-X45$)a\Check if a cubic Bezier lies within a given distance of the origin. "Origin" means *the* origin (0,0), not the start of the curve. Note that no checks are made on the start and end positions of the curve; this function only checks the inside of the curve. Args: p0 (complex): Start point of curve. p1 (complex): First handle of curve. p2 (complex): Second handle of curve. p3 (complex): End point of curve. tolerance (double): Distance from origin. Returns: bool: True if the cubic Bezier ``p`` entirely lies within a distance ``tolerance`` of the origin, False otherwise. Tr3rKFrL)rcubic_farthest_fit_inside)r*r+r,r-rarHrIs rrcrc8s: 2w)B9 4 RW  "e +C 3x)glR5 (F $ "WOS\3  W #Cv3 VWr)ra)q1c0rBc2c3c[USUSUSUS5n[R"UR5(agUSnUSnX2U- S--nXBU- S--n[ SXPS- X`S- SU5(dgX2U4$)aApproximate a cubic Bezier with a single quadratic within a given tolerance. Args: cubic (sequence): Four complex numbers representing control points of the cubic Bezier curve. tolerance (double): Permitted deviation from the original curve. Returns: Three complex numbers representing control points of the quadratic curve if it fits within the given tolerance, or ``None`` if no suitable curve could be calculated. rrr2r3NUUUUUU?)r`mathisnanrrc)cubicrardrergrBrfs rcubic_approx_quadraticrmbs0 a%(E!HeAh ?B zz"'' qB qB Bw5! !B Bw5! !B $Q1X r!H}a S S 2:r)r:ra)r?) all_quadratic)rerBrfrg)q0rdnext_q1q2rCc fUS:Xa [X5$US:XaUS:XaU$[USUSUSUSU5n[U5n[SUSUSUSUS5nUSnSnUSU/n [ SUS-5Hn UuppUnUnX:aE[U5n[XS- - USUSUSUS5nU R U5 UU-S-nOUnUnX~- n[ U5U:d.[UUUU- S--U - UUU- S--U - UU5(aM g U R US5 U $) aApproximate a cubic Bezier curve with a spline of n quadratics. Args: cubic (sequence): Four complex numbers representing control points of the cubic Bezier curve. n (int): Number of quadratic Bezier curves in the spline. tolerance (double): Permitted deviation from the original curve. Returns: A list of ``n+2`` complex numbers, representing control points of the quadratic spline if it fits within the given tolerance, or ``None`` if no suitable spline could be calculated. rr2Frr3yrLriN)rmr;nextrXrEappendrrc)rlr:rarncubics next_cubicrprqrCspliner?rerBrfrgrordd0s rcubic_approx_splinerys: Av%e77Av-5( $U1XuQxq58Q OFfJ" :a=*Q-A 1 G qB BAh F 1a!e_#  5fJ*U Z]JqM:a=*UV-G MM' "w,#%BB W r7Y &?  "r'e$ $r ) "r'e$ $r )   ' ' 9: MM%( Mr)max_err)r:cUVs/sH n[U6PM nn[S[S-5H<n[XX5nUcMUVs/sHofRUR 4PM sns $ [ U5es snfs snf)aApproximate a cubic Bezier curve with a spline of n quadratics. Args: cubic (sequence): Four 2D tuples representing control points of the cubic Bezier curve. max_err (double): Permitted deviation from the original curve. all_quadratic (bool): If True (default) returned value is a quadratic spline. If False, it's either a single quadratic curve or a single cubic curve. Returns: If all_quadratic is True: A list of 2D tuples, representing control points of the quadratic spline if it fits within the given tolerance, or ``None`` if no suitable spline could be calculated. If all_quadratic is False: Either a quadratic curve (if length of output is 3), or a cubic curve (if length of output is 4). r)rrEMAX_Nryrrr)curverzrnr[r:rwss rrrsz0#( (%QWa[%E ( 1eai $UwF  .45fVVQVV$f5 5 ! e $$ ) 6s A7!A<)llast_ir?c UVVs/sHo3Vs/sH n[U6PM snPM nnn[U5[U5:Xde[U5nS/U-nS=pxSn [XXUU5n U cU [:XaOVU S- n UnM*XU'US-U-nX:Xa:UV V s/sH*oV s/sHoRU R 4PM sn PM, sn n $Mv[ U5es snfs snnfs sn fs sn n f)aReturn quadratic Bezier splines approximating the input cubic Beziers. Args: curves: A sequence of *n* curves, each curve being a sequence of four 2D tuples. max_errors: A sequence of *n* floats representing the maximum permissible deviation from each of the cubic Bezier curves. all_quadratic (bool): If True (default) returned values are a quadratic spline. If False, they are either a single quadratic curve or a single cubic curve. Example:: >>> curves_to_quadratic( [ ... [ (50,50), (100,100), (150,100), (200,50) ], ... [ (75,50), (120,100), (150,75), (200,60) ] ... ], [1,1] ) [[(50.0, 50.0), (75.0, 75.0), (125.0, 91.66666666666666), (175.0, 75.0), (200.0, 50.0)], [(75.0, 50.0), (97.5, 75.0), (135.41666666666666, 82.08333333333333), (175.0, 67.5), (200.0, 60.0)]] The returned splines have "implied oncurve points" suitable for use in TrueType ``glif`` outlines - i.e. in the first spline returned above, the first quadratic segment runs from (50,50) to ( (75 + 125)/2 , (120 + 91.666..)/2 ) = (100, 83.333...). Returns: If all_quadratic is True, a list of splines, each spline being a list of 2D tuples. If all_quadratic is False, a list of curves, each curve being a quadratic (length 3), or cubic (length 4). Raises: fontTools.cu2qu.Errors.ApproxNotFoundError: if no suitable approximation can be found for all curves with the given parameters. 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