<`ldZddlmZmZmZddlmZddlZddlm Z e dgdZ gdZ d;d Z d Z d Zd;d ZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZ dZ!d Z"dd!lm#Z#m$Z$m%Z%m&Z&e#fd"Z'd#Z(d$Z)d%Z*d&Z+d'Z,d(Z-d)Z.d*Z/d+Z0d,Z1d-Z2d.Z3d/Z4d0Z5d1Z6d2Z7d3Z8 dd:kr,ddl?Z?ddl@Z@e?jAe@jBjCdSdS)=zNfontTools.misc.bezierTools.py -- tools for working with Bezier path segments. ) calcBoundssectRectrectArea)IdentityN) namedtuple Intersectionptt1t2)approximateCubicArcLengthapproximateCubicArcLengthCapproximateQuadraticArcLengthapproximateQuadraticArcLengthCcalcCubicArcLengthcalcCubicArcLengthCcalcQuadraticArcLengthcalcQuadraticArcLengthCcalcCubicBoundscalcQuadraticBounds splitLinesplitQuadratic splitCubicsplitQuadraticAtT splitCubicAtTsolveQuadratic solveCubicquadraticPointAtT cubicPointAtT linePointAtTsegmentPointAtTlineLineIntersectionscurveLineIntersectionscurveCurveIntersectionssegmentSegmentIntersections{Gzt?c`tt|t|t|t||S)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. )rcomplex)pt1pt2pt3pt4 tolerances :lib/python3.11/site-packages/fontTools/misc/bezierTools.pyrr*s1  w}gsmWc]I  c||d||zzz|zdz}||z|z |z dz}|||zdz||z |f|||z||zdz|ffS)Ng??)p0p1p2p3midderiv3s r._split_cubic_into_twor:=so R"W  "e +C2glR5 (F b2g_cFlC0 cFlR"WOR0 r/ct||z }t||z t||z zt||z z}||z|kr||zdzSt||||\}}t|g|Rt|g|RzSNr2)absr:_calcCubicArcLengthCRecurse) multr4r5r6r7archboxonetwos r.r>r>Fs rBw<)r)r*r+r,r-r?s r.rrRs( y D &tS#sC @ @@r/g|=c:||zjSN) conjugatereal)v1v2s r._dotrMds   %%r/cr|tj|dzdzzdz tj|dz zS)N)mathsqrtasinh)xs r. _intSecAtanrUhs8 tya!$$ $q (4:a==1+< <>> 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 )rr(r)r*r+s r.rrns#@ #7C='3-# O OOr/c8||z }||z }||z }|dz}t|}|dkrt||z St||}t|tkrUt||dkrt||z St|t|} } | | z| | zz| | zz St|||z } t|||z } tdt| t| z z|z|| | z zz } | S)a$Calculates 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?rrO)r=rMepsilonrU)r)r*r+d0d1dnscaleorigDistabx0x1Lens r.rrs sB sB RA BA FFE ||39~~Ar{{H 8}}w B<<1  sSy>> !2wwB1AA !a%(( ax B ax B a;r??[__45@ERRTWDUV W WC Jr/cNtt|t|t|S)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. )rr(rWs r.rrs# *'3-#QT V VVr/ctd|zd|zzd|zz}t||z dz}td|zd|zz d|zz}||z|zS)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?gFVWg̔xb?r=)r)r*r+v0rKrLs r.rrs, S #4s#::=ORU=UU  B S3Y, ,B c!$5$;;>ORU>UU  B 7R<r/c, t|||\\\ \ dz}dz}g}|dkr| |z |dkr| |z  fd|D||gz}t|S)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) @rctg|]4}d|cxkrdknn"|z|z|zzz|z|z|zzzf5SrrPr3).0taxaybxbycxcys r. z'calcQuadraticBounds..sk ::::A::::: a!b1f r !26A:Q#6#;< ::r/)calcQuadraticParametersappendr) r)r*r+ax2ay2rootspointsrprqrrrsrtrus @@@@@@r.rrs$$;3S#I#I HRhr2R s(C s(C E axx bS3Y axx bS3Y c F f  r/c^tt|t|t|t|S)aApproximates 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 )rr()r)r*r+r,s r.r r s/2 & w}gsmWc]  r/c8t||z dz}td|zd|zzd|zzd|zz}t||z |z|z dz}td|zd|zz d|zz d|zz}t||z dz}||z|z|z|zS) 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?gc1g85$t?gu|Y?g#$?g?g#$gc1?rh) r)r*r+r,rirKrLv3v4s r.rr#s" S3Y$ B S c ! " c ! " c ! "  B S3Y_s " # #&9 9B S c ! " c ! " c ! "  B S3Y$ B 7R<" r !!r/cH t||||\\ \\\ dz} dz}dz}dz}dt||D}dt||D} || z} fd| D||gz} t| S)aXCalculates 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 @rkc2g|]}d|cxkrdknn|Srmr3rnros r.rvz#calcCubicBounds.._- D D DAa!ar/c2g|]}d|cxkrdknn|Srmr3rs r.rvz#calcCubicBounds..`rr/cg|]<}|z|z|z|z|zz|zzz|z|z|z|z|zz|zz zf=Sr3r3) rnrorprqrrrsrtrudxdys r.rvz#calcCubicBounds..cs  FQJNR!VaZ '"q& 02 5 FQJNR!VaZ '"q& 02 5 r/)calcCubicParametersrr)r)r*r+r,ax3ay3bx2by2xRootsyRootsr{r|rprqrrrsrtrurrs @@@@@@@@r.rrGs$.Ac3PS-T-T*HRhr2R(2r s(C s(C s(C s(C D DS"55 D D DF D DS"55 D D DF VOE   c F f  r/c|\}}|\}}||z }||z } |} |} || f|} | dkr||fgS|| | f|z | z } d| cxkrdkrnn|| z| z| | z| zf}||f||fgS||fgS)a Split a line at a given coordinate. Args: pt1: Start point of line as 2D tuple. pt2: End point of line as 2D tuple. where: Position at which to split the line. isHorizontal: Direction of the ray splitting the line. If true, ``where`` is interpreted as a Y coordinate; if false, then ``where`` is interpreted as an X coordinate. Returns: A list of two line segments (each line segment being two 2D tuples) if the line was successfully split, or a list containing the original line. Example:: >>> printSegments(splitLine((0, 0), (100, 100), 50, True)) ((0, 0), (50, 50)) ((50, 50), (100, 100)) >>> printSegments(splitLine((0, 0), (100, 100), 100, True)) ((0, 0), (100, 100)) >>> printSegments(splitLine((0, 0), (100, 100), 0, True)) ((0, 0), (0, 0)) ((0, 0), (100, 100)) >>> printSegments(splitLine((0, 0), (100, 100), 0, False)) ((0, 0), (0, 0)) ((0, 0), (100, 100)) >>> printSegments(splitLine((100, 0), (0, 0), 50, False)) ((100, 0), (50, 0)) ((50, 0), (0, 0)) >>> printSegments(splitLine((0, 100), (0, 0), 50, True)) ((0, 100), (0, 50)) ((0, 50), (0, 0)) rrPr3)r)r*where isHorizontalpt1xpt1ypt2xpt2yrprqrrrsraromidPts r.rrmsHJD$JD$ B B B B RAAvvc | "b,' '1,AAzzzzzzzzzQ R!Vb[(e ucl++c |r/ct|||\}}}t|||||||z }td|D}|s|||fgSt|||g|RS)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)) c3:K|]}d|cxkrdknn|VdSrrPNr3rs r. z!splitQuadratic..6::QqAzzzzzzzzzqzzzz::r/)rwrsorted_splitQuadraticAtT) r)r*r+rrrarbc solutionss r.rrsF&c344GAq! ,<!L/E*AI::):::::I !c3  aA 2 2 2 22r/ct||||\}}}} t||||||| ||z } td| D} | s||||fgSt|||| g| RS)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)) c3:K|]}d|cxkrdknn|VdSrr3rs r.rzsplitCubic..rr/)rrr_splitCubicAtT) r)r*r+r,rrrarbrr]rs r.rrs6%S#sC88JAq!Q ,<!L/1\?U;RI::):::::I &c3$%% !Q1 1y 1 1 11r/cJt|||\}}}t|||g|RS)aSplit 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)) )rwr)r)r*r+tsrarbrs r.rrs5(&c344GAq! aA + + + ++r/cPt||||\}}}}t||||g|RS)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)) )rr) r)r*r+r,rrarbrr]s r.rrs;(%S#sC88JAq!Q !Q1 *r * * **r/ct|}g}|dd|d|\}}|\}}|\} } tt |dz D]} || } || dz} | | z }||z}||z}||z}d|z| z|z|z}d|z| z|z|z}| | z}||z|| zz| z}||z|| zz| z}t ||f||f||f\}}}||||f|S)NrrYrErPrO)listinsertrxrangelencalcQuadraticPoints)rarbrrsegmentsrprqrrrsrtruir r deltadelta_2a1xa1yb1xb1yt1_2c1xc1yr)r*r+s r.rr(sS bBHIIaIIcNNN FB FB FB 3r77Q;  )) U AYR%-7l7l2v{R5(2v{R5(Bw4i"r'!B&4i"r'!B&+S#Jc S#JOO S#c3(((( Or/ct|}|dd|dg}|\}}|\}} |\} } |\} } tt |dz D]}||}||dz}||z }||z}||z}||z}||z}||z}||z}d|z|z|z|z}d|z|z| z|z}d|z|z| zd|z|zz|z}d| z|z| zd|z|zz|z}||z||zz| |zz| z}||z| |zz| |zz| z}t ||f||f||f||f\}}} }!|||| |!f|S)NrrYrErPr1rO)rrrxrrcalcCubicPoints)"rarbrr]rrrprqrrrsrtrurrrr r rrdelta_3rt1_3rrrrrrd1xd1yr)r*r+r,s" r.rrCs bBIIaIIcNNNH FB FB FB FB 3r77Q;  .. U AYR%-'/BwDy7l7l2v{R7*2v{R7*2v{R!b&4-/582v{R!b&4-/584i"t)#b2g-24i"t)#b2g-2, #Jc S#Jc   S#s c3,---- Or/)rRacoscospict|tkr#t|tkrg}nB| |z g}n:||zd|z|zz }|dkr$||}| |zdz |z | |z dz |z g}ng}|S)uKSolve 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! @rYrk)r=rZ)rarbrrRr{DDrDDs r.rrms 1vv q66G  EER!VHEEUS1Wq[  99$r((Cb3h#%)QBH+;a+?@EEE Lr/c  t|tkrt|||St|}||z }||z }||z }||zd|zz dz }d|z|z|zd|z|zz d|zzdz }||z} ||z|z} | tkrdn| } t| tkrdn| } | | z } | dkr$| dkrt | dz t } | | | gS| tdzkrt tt|t| z d d } d t|z}|dz }|t| dz z|z }|t| dtzzdz z|z }|t| d tzzdz z|z }t|||g\}}}||z tkr1||z tkr#t ||z|zdz t x}x}}n||z tkr3t ||zdz t x}}t |t }n||z tkr3t |t }t ||zdz t x}}n?t |t }t |t }t |t }|||gStt| t|zd } | || z z} |dkr| } t | |dz z t } | gS)utSolve 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 rg"@rkg;@gK@rrYr2rEggrgUUUUUU?)r=rZrfloatround epsilonDigitsrmaxminrRrrrpow)rarbrr]a1a2a3QRR2Q3R2_Q3rTthetarQ2a1_3rcrdx2s r.rrs0L 1vvaA&&& aA QB QB QB b38 s"A rB cBhm +dRi 74?A QB QB7llB"ggRB GE SyyR3YY 2#)] + +1ay 'C-  SQb\3//6677T!WWnCx 3us{## #d * 3b(C/00 04 7 3b(C/00 04 7RRL)) B 7W  b7!2!2 "r'B,#!5}EE EB Ebb "Ww  R"WO];; ;Br=))BB "Ww  r=))BR"WO];; ;Br=))Br=))Br=))BB| U c!ff$g . . AI 88A !b3h, . .s r/cv|\}}|\}}|\}}||z dz} ||z dz} ||z | z } ||z | z } | | f| | f||ffS)Nrkr3) r)r*r+ry2x3y3rtrurrrsrprqs r.rwrwsj FB FB FB r'SB r'SB b2B b2B 8b"XBx ''r/c|\}}|\}}|\}} |\} } || z dz} || z dz} ||z dz| z }||z dz| z }|| z | z |z }| | z | z |z }||f||f| | f| | ffSNrr3)r)r*r+r,rrrrx4y4rrrtrurrrsrprqs r.rrs FB FB FB FB r'SB r'SB r'S2 B r'S2 B b2 B b2 B 8b"XBx"b 11r/c~|\}}|\}}|\}}|} |} |dz|z} |dz|z} ||z|z} ||z|z}| | f| | f| |ffSr<r3)rarbrrprqrrrsrtrurdy1rrrrs r.rrst FB FB FB B B s(bB s(bB b2B b2B 8b"XBx ''r/c|\}}|\}}|\}} |\} } | } | } |dz | z}| dz | z}||zdz |z}|| zdz |z}|| z|z|z}|| z| z|z}| | f||f||f||ffSrr3)rarbrr]rprqrrrsrtrurrrdrrrrrrrs r.rrs FB FB FB FB B B s(bB s(bB r'S2 B r'S2 B b2 B b2 B 8b"XBx"b 11r/cj|dd|z z|d|zz|dd|z z|d|zzfS)zFinds the point at time `t` on a line. Args: pt1, pt2: Coordinates of the line as 2D tuples. t: The time along the line. Returns: A 2D tuple with the coordinates of the point. rrPr3)r)r*ros r.r r *sCVq1u A *c!fA.>Q!.K MMr/cd|z d|z z|dzdd|z z|z|dzz||z|dzz}d|z d|z z|dzdd|z z|z|dzz||z|dzz}||fS)zFinds the point at time `t` on a quadratic curve. Args: pt1, pt2, pt3: Coordinates of the curve as 2D tuples. t: The time along the curve. Returns: A 2D tuple with the coordinates of the point. rPrrOr3)r)r*r+rorTys r.rr7s Q1q5CF"Q!a%[1_s1v%==AANA Q1q5CF"Q!a%[1_s1v%==AANA q6Mr/cd|z d|z zd|z z|dzdd|z zd|z z|z|dzzdd|z z|z|z|dzz||z|z|dzz}d|z d|z zd|z z|dzdd|z zd|z z|z|dzzdd|z z|z|z|dzz||z|z|dzz}||fS)zFinds the point at time `t` on a cubic curve. Args: pt1, pt2, pt3, pt4: Coordinates of the curve as 2D tuples. t: The time along the curve. Returns: A 2D tuple with the coordinates of the point. rPrr1r3)r)r*r+r,rorTrs r.rrFs' Q1q5QU#c!f, q1u+Q ! #c!f , - q1u+/A A & ' a%!)c!f   Q1q5QU#c!f, q1u+Q ! #c!f , - q1u+/A A & ' a%!)c!f   q6Mr/ct|dkrtg||RSt|dkrtg||RSt|dkrtg||RSt dNrOr1Unknown curve degree)rr rr ValueError)segros r.r!r!_s 3xx1}}$S$!$$$$ SQ )#)q)))) SQ%c%1%%%% + , ,,r/c|\}}|\}}|\}}tj||s ||z ||z z Stj||s ||z ||z z SdS)N)rQisclose) ser sxsyexeypxpys r. _line_t_of_ptrnsp FB FB FB <B  %RBG$$ <B  %RBG$$ 2r/c|d|dz |d|dz z}|d|dz |d|dz z}|dko|dk S)NrrPrYr3)rarboriginxDiffyDiffs r.'_both_points_are_on_same_side_of_originrzs` qTF1I !A$"2 3E qTF1I !A$"2 3E -# ..r/c |\}}|\}}|\}} |\} } tj|| r,tj||rtj||sgStj| | r,tj||rtj|| sgStj|| rtj| | rgStj||rtj||rgStj||rM|} | | z | |z z } | | |z z| z}| |f}t|t|||t|||gStj|| rM|} ||z ||z z }|| |z z|z}| |f}t|t|||t|||gS||z ||z z }| | z | |z z } tj|| rgS||z|z | |zz | z|| z z } || |z z|z}| |f}t |||rBt |||r1t|t|||t|||gSgS)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) (0.45069111555824454, 0.5408153767394238) r )rQrrrr)s1e1s2e2s1xs1ye1xe1ys2xs2ye2xe2yrTslope34rr slope12s r.r"r"s.HCHCHCHC S##'<S#9#9BF,sTWBXBX  S##'<S#9#9BF,sTWBXBX  |C$,sC"8"8  |C$,sC"8"8  |C  9s+ q3w # %V -B33 b"b8Q8Q      |C  9s+ q3w # %V -B33 b"b8Q8Q     SyS3Y'GSyS3Y'G |GW%% 3 w} ,s 2w7HIA1s7c!A QB. B 1"b" = =  -B33 b"b8Q8Q     Ir/c|d}|d}tj|d|dz |d|dz }tj| |d |d S)NrrrP)rQatan2rrotate translate)segmentstartendangles r._alignment_transformationrso AJE "+C Js1va(#a&58*; < .s6<<cQmmmm!mmmmm!mmmm<|S)aFinds intersections between a curve and a line. Args: curve: List of coordinates of the curve segment as 2D tuples. line: List of coordinates of the line 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:: >>> curve = [ (100, 240), (30, 60), (210, 230), (160, 30) ] >>> line = [ (25, 260), (230, 20) ] >>> intersections = curveLineIntersections(curve, line) >>> len(intersections) 3 >>> intersections[0].pt (84.90010344084885, 189.87306176459828) r1rrr )rrrrrrxrr)rr pointFinderrror s r.r#r#s* 5zzQ' Uq# /000M ( 5 5UU [ #% # # # #\RA-:R:Rr:R:R:RSSSTTTT r/ct|dkr t|St|dkr t|Std)Nr1rr)rrrr)rs r. _curve_boundsr#sE 1vv{{"A&& Q1"" + , ,,r/ct|dkr|\}}t|||}||f||fgSt|dkrtg||RSt|dkrtg||RSt dr)rr rrr)rrorrmidpoints r._split_segment_at_tr&s 1vv{{11a((H !}-- 1vv{{ '!'Q'''' Q1#a##### + , ,,r/MbP?c t|}t|}|sd}|sd}t||\}}|sgSd} t|kr*t|kr| || |fgSt|d\} } |d| |f} | ||df} t|d\}}|d| |f}| ||df}g}|t | || ||t | || ||t | || ||t | || |fd}t }g}|D]<}||}||vr||||=|S)N)rYrEc*d|d|dzzS)Nr2rrPr3)rs r.r%z._curve_curve_intersections_t..midpoint+sadQqTk""r/r2rrP)range1range2cdt|dz t|dz fS)NrrP)int)r precisions r.z._curve_curve_intersections_t..Ps.SA!233SA9J5K5KLr/) r#rrr&extend_curve_curve_intersections_tsetaddrx)curve1curve2r/r+r,bounds1bounds2 intersects_r%c11c12 c11_range c12_rangec21c22 c21_range c22_rangefound unique_keyseen unique_valuesrkeys ` r.r2r2sF##GF##G  Wg..MJ  ###9$$'):):Y)F)F&!!88F#3#3455"63//HCHHV,,-I&!!6!9-I"63//HCHHV,,-I&!!6!9-I E LL$ i )     LL$ i )     LL$ i )     LL$ i )    MLLLJ 55DM!!jnn $;;   R    r/c@t|}fd|DS)a Finds intersections between a curve and a curve. Args: curve1: List of coordinates of the first curve segment as 2D tuples. curve2: List of coordinates of the second curve segment as 2D tuples. Returns: A list of ``Intersection`` objects, each object having ``pt``, ``t1`` and ``t2`` attributes containing the intersection point, time on first segment and time on second segment respectively. Examples:: >>> curve1 = [ (10,100), (90,30), (40,140), (220,220) ] >>> curve2 = [ (5,150), (180,20), (80,250), (210,190) ] >>> intersections = curveCurveIntersections(curve1, curve2) >>> len(intersections) 3 >>> intersections[0].pt (81.7831487395506, 109.88904552375288) c tg|]4}tt|d|d|d5S)rrPr )rr!)rnrr5s r.rvz+curveCurveIntersections..tsN     1662a5RUKKK   r/)r2)r5r6intersection_tss` r.r$r$^s?*366BBO    !   r/cd}t|t|kr||}}d}t|dkr5t|dkrt||}nUt||}nDt|dkr"t|dkrtg||R}nt d|s|Sd|DS)a*Finds intersections between two segments. Args: seg1: List of coordinates of the first segment as 2D tuples. seg2: List of coordinates of the second segment as 2D tuples. Returns: A list of ``Intersection`` objects, each object having ``pt``, ``t1`` and ``t2`` attributes containing the intersection point, time on first segment and time on second segment respectively. Examples:: >>> curve1 = [ (10,100), (90,30), (40,140), (220,220) ] >>> curve2 = [ (5,150), (180,20), (80,250), (210,190) ] >>> intersections = segmentSegmentIntersections(curve1, curve2) >>> len(intersections) 3 >>> intersections[0].pt (81.7831487395506, 109.88904552375288) >>> curve3 = [ (100, 240), (30, 60), (210, 230), (160, 30) ] >>> line = [ (25, 260), (230, 20) ] >>> intersections = segmentSegmentIntersections(curve3, line) >>> len(intersections) 3 >>> intersections[0].pt (84.90010344084885, 189.87306176459828) FTrOz4Couldn't work out which intersection function to usecPg|]#}t|j|j|j$S)r )rr r r rs r.rvz/segmentSegmentIntersections..s- K K KLADQTad 3 3 3 K K Kr/)rr$r#r"r)seg1seg2swappedrs r.r%r%zs<G 4yy3t994d 4yy1}} t99q==3D$??MM24>>MM TaCIINN-;t;d;;; OPPP  K K] K K KKr/c t|}ddd|DzS#t$rd|zcYSwxYw)zw >>> _segmentrepr([1, [2, 3], [], [[2, [3, 4], [0.1, 2.2]]]]) '(1, (2, 3), (), ((2, (3, 4), (0.1, 2.2))))' z(%s)z, c34K|]}t|VdSrH) _segmentrepr)rnrTs r.rz_segmentrepr..s(!>!>a,q//!>!>!>!>!>!>r/z%g)iterjoin TypeError)objits r.rRrRsg ? #YY !>!>2!>!>!>>>>> czs3AAcH|D]}tt|dS)zlHelper for the doctests, displaying each segment in a list of segments on a single line as a tuple. N)printrR)rrs r. printSegmentsrZs6%% l7##$$$$%%r/__main__)r&)r'NN)D__doc__fontTools.misc.arrayToolsrrrfontTools.misc.transformrrQ collectionsrr__all__rr:r>rrrZrMrUrrrrrr rrrrrrrrrrRrrrrrrwrrrr rrr!rrr"rrr#r#r&r2r$r%rRrZ__name__sysdoctestexittestmodfailedr3r/r.rgsEDDDDDDDDD------ """"""z.*<*<*<==    :&     A A A A &&&=== P P PF@WWW"BD