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+% Copyright 2008 by Mark Wibrow
+%
+% This file may be distributed and/or modified
+%
+% 1. under the LaTeX Project Public License and/or
+% 2. under the GNU Free Documentation License.
+%
+% See the file doc/generic/pgf/licenses/LICENSE for more details.
+
+
+\newcount\pgf@intersect@solutions
+
+\newif\ifpgf@intersect@sort
+\newif\ifpgf@intersect@sort@by@second@path
+
+\def\pgfintersectionsortbyfirstpath{%
+ \pgf@intersect@sorttrue%
+ \pgf@intersect@sort@by@second@pathfalse%
+}
+
+\def\pgfintersectionsortbysecondpath{%
+ \pgf@intersect@sorttrue%
+ \pgf@intersect@sort@by@second@pathtrue%
+}
+
+\def\pgfpointintersectionsolution#1{%
+ \ifnum#1<1\relax%
+ \pgfpoint@intersect@solution@orgin%
+ \else%
+ \ifnum#1>\pgfintersectionsolutions\relax%
+ \pgfpoint@intersect@solution@orgin%
+ \else%
+ \csname pgfpoint@intersect@solution@#1\endcsname%
+ \fi%
+ \fi%
+}
+
+\def\pgfpoint@intersect@solution@orgin{%
+ \begingroup%
+ \pgftransforminvert%
+ \pgfpointorigin%
+ \pgf@pos@transform{\pgf@x}{\pgf@y}%
+ \global\pgf@x=\pgf@x%
+ \global\pgf@y=\pgf@y%
+ \endgroup%
+}
+
+\long\def\pgfintersectionofpaths#1#2{%
+ \begingroup%
+ \pgfinterruptpath%
+ #1%
+ \pgfgetpath\pgf@intersect@path@a%
+ \global\let\pgf@intersect@path@temp=\pgf@intersect@path@a%
+ \endpgfinterruptpath%
+ \endgroup%
+ \let\pgf@intersect@path@a=\pgf@intersect@path@temp%
+ %
+ \begingroup%
+ \pgfinterruptpath%
+ #2%
+ \pgfgetpath\pgf@intersect@path@b%
+ \global\let\pgf@intersect@path@temp=\pgf@intersect@path@b%
+ \endpgfinterruptpath%
+ \endgroup%
+ \let\pgf@intersect@path@b=\pgf@intersect@path@temp%
+ %
+ \pgf@intersect@solutions=0\relax%
+ \def\pgf@intersect@time@offset{0}%
+ %
+ \ifpgf@intersect@sort@by@second@path%
+ \let\pgf@intersect@temp=\pgf@intersect@path@a%
+ \let\pgf@intersect@path@a=\pgf@intersect@path@b%
+ \let\pgf@intersect@path@b=\pgf@intersect@temp%
+ \fi%
+ \let\pgf@intersect@token@after=\pgf@intersect@path@process@a%
+ \expandafter\pgf@intersectionofpaths\pgf@intersect@path@a\pgf@stop%
+ \edef\pgfintersectionsolutions{\the\pgf@intersect@solutions}%
+ \pgfmathloop%
+ \ifnum\pgfmathcounter>\pgfintersectionsolutions\relax%
+ \else%
+ \pgfutil@namelet{pgfpoint@intersect@solution@\pgfmathcounter}%
+ {pgfpoint@g@intersect@solution@\pgfmathcounter}%
+ \ifpgf@intersect@sort%
+ \pgfutil@namelet{pgf@intersect@solution@\pgfmathcounter @time@a}%
+ {pgf@g@intersect@solution@\pgfmathcounter @time@a}%
+ \fi%
+ \repeatpgfmathloop%
+ \ifpgf@intersect@sort%
+ \pgfintersectionsolutionsortbytime%
+ \fi%
+}
+
+\def\pgf@intersectionofpaths#1{%
+ \ifx#1\pgf@stop%
+ \let\pgf@intersect@next=\relax%
+ \else%
+ \ifx#1\pgfsyssoftpath@movetotoken%
+ \let\pgf@intersect@next=\pgf@intersect@token@moveto%
+ \else%
+ \ifx#1\pgfsyssoftpath@linetotoken%
+ \let\pgf@intersect@next=\pgf@intersect@token@lineto%
+ \else%
+ \ifx#1\pgfsyssoftpath@closepathtoken%
+ \let\pgf@intersect@next=\pgf@intersect@token@lineto%
+ \else%
+ \ifx#1\pgfsyssoftpath@curvetosupportatoken%
+ \let\pgf@intersect@next=\pgf@intersect@token@curveto%
+ \else%
+ \ifx#1\pgfsyssoftpath@rectcornertoken%
+ \let\pgf@intersect@next=\pgf@intersect@token@rect%
+ \fi%
+ \fi%
+ \fi%
+ \fi%
+ \fi%
+ \fi%
+ \pgf@intersect@next}
+
+\def\pgf@intersect@token@moveto#1#2{%
+ \def\pgfpoint@intersect@start{\pgfqpoint{#1}{#2}}%
+ \pgf@intersectionofpaths%
+}
+
+\def\pgf@intersect@token@lineto#1#2{%
+ \def\pgfpoint@intersect@end{\pgfqpoint{#1}{#2}}%
+ \def\pgf@intersect@type{line}%
+ \pgf@intersect@token@after%
+}
+\def\pgf@intersect@token@curveto#1#2\pgfsyssoftpath@curvetosupportbtoken#3#4\pgfsyssoftpath@curvetotoken#5#6{%
+ \def\pgfpoint@intersect@firstsupport{\pgfqpoint{#1}{#2}}%
+ \def\pgfpoint@intersect@secondsupport{\pgfqpoint{#3}{#4}}%
+ \def\pgfpoint@intersect@end{\pgfqpoint{#5}{#6}}%
+ \def\pgf@intersect@type{curve}%
+ \pgf@intersect@token@after%
+}
+
+\def\pgf@intersect@token@rect#1#2\pgfsyssoftpath@rectsizetoken#3#4{%
+ \pgf@xa=#1\relax%
+ \advance\pgf@xa by#3\relax%
+ \pgf@ya=#2\relax%
+ \advance\pgf@ya by#4\relax%
+ \edef\pgf@marshal{%
+ \noexpand\pgfsyssoftpath@movetotoken{#1}{#2}%
+ \noexpand\pgfsyssoftpath@linetotoken{#1}{\the\pgf@ya}%
+ \noexpand\pgfsyssoftpath@linetotoken{\the\pgf@xa}{\the\pgf@ya}%
+ \noexpand\pgfsyssoftpath@linetotoken{\the\pgf@xa}{#2}%
+ \noexpand\pgfsyssoftpath@closepathtoken{#1}{#2}%
+ }%
+ \expandafter\pgf@intersectionofpaths\pgf@marshal%
+}
+
+\def\pgf@intersect@path@process@a{%
+ \pgf@intersect@path@getpoints@a%
+ \let\pgf@intersect@token@after=\pgf@intersect@path@process@b%
+ \expandafter\pgf@intersectionofpaths\pgf@intersect@path@b\pgf@stop%
+ \let\pgfpoint@intersect@start=\pgfpoint@intersect@end@a%
+ \let\pgf@intersect@token@after=\pgf@intersect@path@process@a%
+ \c@pgf@counta=\pgf@intersect@time@offset\relax%
+ \advance\c@pgf@counta by1\relax%
+ \edef\pgf@intersect@time@offset{\the\c@pgf@counta}%
+ \pgf@intersectionofpaths%
+}
+
+\def\pgf@intersect@path@getpoints@a{%
+ \let\pgfpoint@intersect@start@a=\pgfpoint@intersect@start%
+ \let\pgfpoint@intersect@end@a=\pgfpoint@intersect@end%
+ \let\pgfpoint@intersect@firstsupport@a=\pgfpoint@intersect@firstsupport%
+ \let\pgfpoint@intersect@secondsupport@a=\pgfpoint@intersect@secondsupport%
+ \let\pgf@intersect@type@a=\pgf@intersect@type%
+}
+
+\def\pgf@intersect@path@process@b{%
+ \pgf@intersect@path@getpoints@b%
+ \csname pgf@intersect@\pgf@intersect@type@a @and@\pgf@intersect@type@b\endcsname%
+ \let\pgfpoint@intersect@start=\pgfpoint@intersect@end@b%
+ \pgf@intersectionofpaths}
+
+\def\pgf@intersect@path@getpoints@b{%
+ \let\pgfpoint@intersect@start@b=\pgfpoint@intersect@start%
+ \let\pgfpoint@intersect@end@b=\pgfpoint@intersect@end%
+ \let\pgfpoint@intersect@firstsupport@b=\pgfpoint@intersect@firstsupport%
+ \let\pgfpoint@intersect@secondsupport@b=\pgfpoint@intersect@secondsupport%
+ \let\pgf@intersect@type@b=\pgf@intersect@type%
+}
+
+\def\pgf@intersect@line@and@line{%
+ \pgf@intersectionoflines{\pgfpoint@intersect@start@a}{\pgfpoint@intersect@end@a}%
+ {\pgfpoint@intersect@start@b}{\pgfpoint@intersect@end@b}%
+}%
+
+\def\pgf@intersect@line@and@curve{%
+ \pgf@intersectionoflineandcurve%
+ {\pgf@process{\pgfpoint@intersect@start@a}}{\pgf@process{\pgfpoint@intersect@end@a}}%
+ {\pgf@process{\pgfpoint@intersect@start@b}}{\pgf@process{\pgfpoint@intersect@firstsupport@b}}%
+ {\pgf@process{\pgfpoint@intersect@secondsupport@b}}{\pgf@process{\pgfpoint@intersect@end@b}}%
+}
+
+\def\pgf@intersect@curve@and@line{%
+ \pgf@intersectionofcurveandline%
+ {\pgf@process{\pgfpoint@intersect@start@a}}{\pgf@process{\pgfpoint@intersect@firstsupport@a}}%
+ {\pgf@process{\pgfpoint@intersect@secondsupport@a}}{\pgf@process{\pgfpoint@intersect@end@a}}%
+ {\pgf@process{\pgfpoint@intersect@start@b}}{\pgf@process{\pgfpoint@intersect@end@b}}%
+}
+
+\def\pgf@intersect@curve@and@curve{%
+ \pgf@intersectionofcurves%
+ {\pgf@process{\pgfpoint@intersect@start@a}}{\pgf@process{\pgfpoint@intersect@firstsupport@a}}%
+ {\pgf@process{\pgfpoint@intersect@secondsupport@a}}{\pgf@process{\pgfpoint@intersect@end@a}}%
+ {\pgf@process{\pgfpoint@intersect@start@b}}{\pgf@process{\pgfpoint@intersect@firstsupport@b}}%
+ {\pgf@process{\pgfpoint@intersect@secondsupport@b}}{\pgf@process{\pgfpoint@intersect@end@b}}%
+}
+
+
+\def\pgfintersectionoflines#1#2#3#4{%
+ \pgf@intersect@solutions=0\relax%
+ \pgf@intersectionoflines{#1}{#2}{#3}{#4}%
+}
+
+\def\pgf@intersectionoflines#1#2#3#4{%
+ \pgf@iflinesintersect{#1}{#2}{#3}{#4}%
+ {%
+ \global\advance\pgf@intersect@solutions by1\relax%
+ \expandafter\pgfextract@process\csname pgfpoint@g@intersect@solution@\the\pgf@intersect@solutions\endcsname{%
+ \pgfpointintersectionoflines{\pgfpoint@intersect@start@a}{\pgfpoint@intersect@end@a}%
+ {\pgfpoint@intersect@start@b}{\pgfpoint@intersect@end@b}%
+ }%
+ \ifpgf@intersect@sort%
+ \pgf@xc=\pgf@x%
+ \pgf@yc=\pgf@y%
+ \pgf@process{\pgfpointdiff{\pgfpoint@intersect@start@a}{\pgfpoint@intersect@end@a}}%
+ \edef\pgf@marshal{%
+ \noexpand\pgfmathveclen@{\pgfmath@tonumber{\pgf@xa}}{\pgfmath@tonumber{\pgf@ya}}%
+ }%
+ \pgf@marshal%
+ \let\pgf@intersect@length@a=\pgfmathresult%
+ \pgf@process{\pgfpointdiff{\pgfpoint@intersect@start@a}{\pgfqpoint{\pgf@xc}{\pgf@yc}}}%
+ \edef\pgf@marshal{%
+ \noexpand\pgfmathveclen@{\pgfmath@tonumber{\pgf@x}}{\pgfmath@tonumber{\pgf@y}}%
+ }%
+ \pgf@marshal%
+ \pgfmathdivide@{\pgfmathresult}{\pgf@intersect@length@a}%
+ \pgf@x=\pgfmathresult pt\relax%
+ \advance\pgf@x by\pgf@intersect@time@offset pt\relax%
+ \expandafter\xdef\csname pgf@g@intersect@solution@\the\pgf@intersect@solutions @time@a\endcsname%
+ {\pgfmath@tonumber{\pgf@x}}%
+ \fi%
+ }{}%
+}
+
+% Test if two lines L1 and L2 intersect.
+%
+% #1 - first point P1 on L1.
+% #2 - second point P2 on L1.
+% #3 - first point P3 on L2.
+% #2 - second point P4 on L2.
+% #5 - code executed if intersection occurs.
+% #6 - code executed if intersection does no occur.
+%
+% Let L1 be represented by P1+(P2-P1)s where 0<=s<=1
+% Let L2 be represented by P3+(P4-P3)t where 0<=t<=1
+%
+% Then L1 and L2 intersect at
+%
+% s = |x2-x1 x3-x1| / |x4-x3 x2-x1|
+% |y2-y1 y3-y1| |y4-y3 y2-y1|
+%
+% t = |x4-x3 x3-x1| / |x4-x3 x2-x1|
+% |y4-y3 y3-y1| |y4-y3 y2-y1|
+%
+% with 0<=s,t<=1
+%
+% s and t do not need to be calculated:
+%
+% Let s = A / C and t = B / C
+%
+% Then 0<=s<=1 if !(C=0) && ((A=0) || ((A>0) && !(C<A)) || ((A<0) && !(C>A)))
+% 0<=t<=1 if !(C=0) && ((B=0) || ((B>0) && !(C<B)) || ((B<0) && !(C>B)))
+%
+\newif\ifpgf@s
+\newif\ifpgf@t
+\def\pgfiflinesintersect#1#2#3#4{%
+ \begingroup%
+ \pgf@iflinesintersect{\pgf@process{#1}}{\pgf@process{#2}}{\pgf@process{#3}}{\pgf@process{#4}}%
+ {\aftergroup\pgfutil@firstoftwo}{\aftergroup\pgfutil@secondoftwo}%
+ \endgroup%
+}
+
+\def\pgf@iflinesintersect#1#2#3#4{%
+ #4\relax%
+ \pgf@xc=\pgf@x%
+ \pgf@yc=\pgf@y%
+ #3\relax%
+ \advance\pgf@xc by-\pgf@x%
+ \advance\pgf@yc by-\pgf@y%
+ \pgf@xb=\pgf@x%
+ \pgf@yb=\pgf@y%
+ #2\relax%
+ \pgf@xa=\pgf@x%
+ \pgf@ya=\pgf@y%
+ #1\relax%
+ \advance\pgf@xa by-\pgf@x%
+ \advance\pgf@ya by-\pgf@y%
+ \advance\pgf@xb by-\pgf@x%
+ \advance\pgf@yb by-\pgf@y%
+ %
+ % xc = x4-x3; yc=y4-y3;
+ % xb = x3-x1; yb=y3-y1;
+ % xa = x2-x1; ya=y2-y1;
+ %
+ %
+ % Normalise a little. 16384 may not be a robust choice.
+ %
+ \c@pgf@counta=\pgf@xa\divide\c@pgf@counta by16384\relax%
+ \c@pgf@countb=\pgf@xb\divide\c@pgf@countb by16384\relax%
+ \c@pgf@countc=\pgf@ya\divide\c@pgf@countc by16384\relax%
+ \c@pgf@countd=\pgf@yb\divide\c@pgf@countd by16384\relax%
+ \multiply\c@pgf@counta by\c@pgf@countd%
+ \multiply\c@pgf@countc by\c@pgf@countb%
+ \advance\c@pgf@counta by-\c@pgf@countc%
+ \pgfutil@tempcnta=\c@pgf@counta%
+ %
+ \c@pgf@counta=\pgf@xc\divide\c@pgf@counta by16384\relax%
+ \c@pgf@countc=\pgf@yc\divide\c@pgf@countc by16384\relax%
+ \multiply\c@pgf@countd by\c@pgf@counta%
+ \multiply\c@pgf@countb by\c@pgf@countc%
+ \advance\c@pgf@countd by-\c@pgf@countb%
+ \pgfutil@tempcntb=\c@pgf@countd%
+ %
+ \c@pgf@countb=\pgf@xa\divide\c@pgf@countb by16384\relax%
+ \c@pgf@countd=\pgf@ya\divide\c@pgf@countd by16384\relax%
+ \multiply\c@pgf@counta by\c@pgf@countd%
+ \multiply\c@pgf@countc by\c@pgf@countb%
+ \advance\c@pgf@counta by-\c@pgf@countc%
+ %
+ \pgf@sfalse%
+ \pgf@tfalse%
+ \ifnum\c@pgf@counta=0\relax%
+ \else%
+ \ifnum\pgfutil@tempcnta=0\relax%
+ \pgf@strue%
+ \else%
+ \ifnum\pgfutil@tempcnta>0\relax%
+ \ifnum\c@pgf@counta<\pgfutil@tempcnta%
+ \else%
+ \pgf@strue%
+ \fi%
+ \else%
+ \ifnum\c@pgf@counta>\pgfutil@tempcnta%
+ \else%
+ \pgf@strue%
+ \fi%
+ \fi%
+ \fi%
+ \ifnum\pgfutil@tempcntb=0\relax%
+ \pgf@ttrue%
+ \else%
+ \ifnum\pgfutil@tempcntb>0\relax%
+ \ifnum\c@pgf@counta<\pgfutil@tempcntb%
+ \else%
+ \pgf@ttrue%
+ \fi%
+ \else%
+ \ifnum\c@pgf@counta>\pgfutil@tempcntb%
+ \else%
+ \pgf@ttrue%
+ \fi%
+ \fi%
+ \fi%
+ \fi%
+ \let\pgf@intersect@next=\pgfutil@secondoftwo%
+ \ifpgf@s%
+ \ifpgf@t%
+ \let\pgf@intersect@next=\pgfutil@firstoftwo%
+ \fi%
+ \fi%
+ \pgf@intersect@next%
+}
+
+
+
+
+\def\pgfintersectionoflineandcurve#1#2#3#4#5#6{%
+ \pgf@intersect@solutions=0\relax%
+ \pgf@intersectionoflineandcurve{#1}{#2}{#3}{#4}{#5}{#6}%
+}
+
+\def\pgf@intersectionoflineandcurve#1#2#3#4#5#6{%
+ \pgf@intersectionofcurves%
+ {\pgf@process{#1}}%
+ {%
+ \pgf@process{%
+ \pgfpointadd{#1\relax\pgf@x=0.666666\pgf@x\pgf@y=0.666666\pgf@y}%
+ {#2\relax\pgf@x=0.333333\pgf@x\pgf@y=0.333333\pgf@y}%
+ }%
+ }%
+ {%
+ \pgf@process{%
+ \pgfpointadd{#1\relax\pgf@x=0.333333\pgf@x\pgf@y=0.333333\pgf@y}%
+ {#2\relax\pgf@x=0.666666\pgf@x\pgf@y=0.666666\pgf@y}%
+ }%
+ }%
+ {\pgf@process{#2}}%
+ {\pgf@process{#3}}%
+ {\pgf@process{#4}}%
+ {\pgf@process{#5}}%
+ {\pgf@process{#6}}%
+}%
+
+\def\pgf@intersectionofcurveandline#1#2#3#4#5#6{%
+ \pgf@intersectionofcurves%
+ {\pgf@process{#1}}%
+ {\pgf@process{#2}}%
+ {\pgf@process{#3}}%
+ {\pgf@process{#4}}%
+ {\pgf@process{#5}}%
+ {%
+ \pgf@process{%
+ \pgfpointadd{#5\relax\pgf@x=0.666666\pgf@x\pgf@y=0.666666\pgf@y}%
+ {#6\relax\pgf@x=0.333333\pgf@x\pgf@y=0.333333\pgf@y}%
+ }%
+ }%
+ {%
+ \pgf@process{%
+ \pgfpointadd{#5\relax\pgf@x=0.333333\pgf@x\pgf@y=0.333333\pgf@y}%
+ {#6\relax\pgf@x=0.666666\pgf@x\pgf@y=0.666666\pgf@y}%
+ }%
+ }%
+ {\pgf@process{#6}}%
+}%
+
+
+
+
+\def\pgfintersectiontolerance{0.1pt}
+\def\pgfintersectiontolerancefactor{0.1}
+
+
+
+% Find the intersections of two bezier curves.
+%
+% #1 - #4 = curve 1.
+% #5 - #8 = curve 2.
+% #9 = the solution number.
+%
+% There is no guarantee of ordering of solutions. If there are
+% no solutions, the origin is returned.
+%
+\def\pgfpointintersectionofcurves#1#2#3#4#5#6#7#8#9{%
+ \pgf@intersect@solutions=0\relax%
+ \pgf@intersectionofcurves%
+ {\pgf@process{#1}}{\pgf@process{#2}}{\pgf@process{#3}}{\pgf@process{#4}}%
+ {\pgf@process{#5}}{\pgf@process{#6}}{\pgf@process{#7}}{\pgf@process{#8}}%
+ \pgfpointintersectionsolution{#9}%
+}
+
+% Return any intersection points of two curves C1 and C2.
+% No order can be guaranteed for the solutions.
+%
+% #1, #2, #3, #4 - the points on C1
+% #5, #6, #7, #8 - the points on C2
+%
+% Returns:
+%
+% \pgf@intersect@solutions - the number of solutions.
+% \pgfpointintersectionsolution{<S>} - the point for soultion S.
+%
+% (Sort of) use:
+%
+% intersection(C1,C2)
+% S = {};
+% intersection'(C1,C2);
+% return S;
+%
+% intersection'(C1,C2)
+% B1 = boundingbox(C1);
+% B2 = boundingbox(C2);
+% if intersect(B1,B2)
+% if (B1.width < q) and (B1.height < q) and
+% (B2.width < q) and (B2.height < q)
+% S = S + {average_of_all_points(B1,B2)}; \\ is there a better choice?
+% else
+% Q = subdivideLeft(C1);
+% R = subdivideRight(C1);
+% intersection'(C2,Q);
+% intersection'(C2,R);
+%
+% where q is a small value (tolerance).
+%
+\def\pgfintersectionofcurves#1#2#3#4#5#6#7#8{%
+ \pgf@intersect@solutions=0\relax%
+ \pgf@intersectionofcurves%
+ {\pgf@process{#1}}{\pgf@process{#2}}{\pgf@process{#3}}{\pgf@process{#4}}%
+ {\pgf@process{#5}}{\pgf@process{#6}}{\pgf@process{#7}}{\pgf@process{#8}}%
+}%
+\def\pgf@intersectionofcurves#1#2#3#4#5#6#7#8{%
+ \begingroup%
+ \dimendef\pgf@time@a=2\relax%
+ \dimendef\pgf@time@aa=4\relax%
+ \dimendef\pgf@time@b=6\relax%
+ \dimendef\pgf@time@bb=8\relax%
+ \pgf@time@a=0pt\relax%
+ \pgf@time@aa=1pt\relax%
+ \pgf@time@b=0pt\relax%
+ \pgf@time@bb=1pt\relax%
+ \let\pgf@intersect@subdivide@curve=\pgf@intersect@subdivide@curve@b%
+ \let\pgf@curve@subdivde@after=\pgf@@intersectionofcurves%
+ \pgf@@intersectionofcurves{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}%
+ \endgroup%
+}
+
+\def\pgf@@intersectionofcurves#1#2#3#4#5#6#7#8{%
+ \pgf@intersect@boundingbox@reset%
+ \pgf@intersect@boundingbox@update{#1}%
+ \pgf@intersect@boundingbox@update{#2}%
+ \pgf@intersect@boundingbox@update{#3}%
+ \pgf@intersect@boundingbox@update{#4}%
+ \edef\pgf@intersect@boundingbox@b{%
+ \noexpand\pgf@x=\the\pgf@xa%
+ \noexpand\pgf@y=\the\pgf@ya%
+ \noexpand\pgf@xa=\the\pgf@xb%
+ \noexpand\pgf@ya=\the\pgf@yb%
+ }%
+ \pgf@intersect@boundingbox@reset%
+ \pgf@intersect@boundingbox@update{#5}%
+ \pgf@intersect@boundingbox@update{#6}%
+ \pgf@intersect@boundingbox@update{#7}%
+ \pgf@intersect@boundingbox@update{#8}%
+ \edef\pgf@intersect@boundingbox@a{%
+ \noexpand\pgf@xb=\the\pgf@xa%
+ \noexpand\pgf@yb=\the\pgf@ya%
+ \noexpand\pgf@xc=\the\pgf@xb%
+ \noexpand\pgf@yc=\the\pgf@yb%
+ }%
+ \pgf@intersect@boundingbox@a%
+ \pgf@intersect@boundingbox@b%
+ \ifdim\pgf@xa<\pgf@xb%
+ \else%
+ \ifdim\pgf@x>\pgf@xc%
+ \else%
+ \ifdim\pgf@ya<\pgf@yb%
+ \else%
+ \ifdim\pgf@y>\pgf@yc%
+ \else%
+ \advance\pgf@xc by-\pgf@xb%
+ \advance\pgf@yc by-\pgf@yb%
+ \advance\pgf@xa by-\pgf@x%
+ \advance\pgf@ya by-\pgf@y%
+ \let\pgf@intersect@subdivde=\relax%
+ \ifdim\pgf@xc<\pgfintersectiontolerance\relax%
+ \ifdim\pgf@xa<\pgfintersectiontolerance\relax%
+ \ifdim\pgf@yc<\pgfintersectiontolerance\relax%
+ \ifdim\pgf@ya<\pgfintersectiontolerance\relax%
+ \pgfextract@process\pgf@intersect@solution@candidate{%
+ \pgf@intersect@boundingbox@a%
+ \pgf@intersect@boundingbox@b%
+ \pgf@x=0.25\pgf@x%
+ \advance\pgf@x by0.25\pgf@xa%
+ \advance\pgf@x by0.25\pgf@xb%
+ \advance\pgf@x by0.25\pgf@xc%
+ \pgf@y=0.25\pgf@y%
+ \advance\pgf@y by0.25\pgf@ya%
+ \advance\pgf@y by0.25\pgf@yb%
+ \advance\pgf@y by0.25\pgf@yc%
+ }%
+ % We must avoid duplicate solutions.
+ \let\pgf@intersect@subdivde=\pgf@stop%
+ \pgf@ifsolution@duplicate\pgf@intersect@solution@candidate{}%
+ {%
+ \global\advance\pgf@intersect@solutions by1\relax%
+ \expandafter\global\expandafter\let%
+ \csname pgfpoint@g@intersect@solution@\the\pgf@intersect@solutions\endcsname=%
+ \pgf@intersect@solution@candidate%
+ {%
+ \ifpgf@intersect@sort%
+ \advance\pgf@time@a by\pgf@time@aa%
+ \divide\pgf@time@a by2\relax%
+ \advance\pgf@time@a by\pgf@intersect@time@offset pt\relax%
+ \expandafter\xdef%
+ \csname pgf@g@intersect@solution@\the\pgf@intersect@solutions @time@a\endcsname%
+ {\pgfmath@tonumber{\pgf@time@a}}%
+ \fi%
+ }%
+ }%
+ \fi%
+ \fi%
+ \fi%
+ \fi%
+ \ifx\pgf@intersect@subdivde\pgf@stop%
+ \else%
+ \pgf@intersect@subdivide@curve{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}%
+ \fi%
+ \fi%
+ \fi%
+ \fi%
+ \fi%
+}
+
+\def\pgf@intersect@subdivide@curve@b#1#2#3#4#5#6#7#8{%
+ \begingroup%
+ \advance\pgf@time@bb by\pgf@time@b\relax%
+ \divide\pgf@time@bb by2\relax%
+ \let\pgf@intersect@subdivide@curve=\pgf@intersect@subdivide@curve@a%
+ \pgf@curve@subdivide@left{#5}{#6}{#7}{#8}{#1}{#2}{#3}{#4}%
+ \endgroup%
+ \begingroup%
+ \advance\pgf@time@b by\pgf@time@bb\relax%
+ \divide\pgf@time@b by2\relax%
+ \let\pgf@intersect@subdivide@curve=\pgf@intersect@subdivide@curve@a%
+ \pgf@curve@subdivide@right{#5}{#6}{#7}{#8}{#1}{#2}{#3}{#4}%
+ \endgroup%
+}
+
+\def\pgf@intersect@subdivide@curve@a#1#2#3#4#5#6#7#8{%
+ \begingroup%
+ \advance\pgf@time@aa by\pgf@time@a\relax%
+ \divide\pgf@time@aa by2\relax%
+ \let\pgf@intersect@subdivide@curve=\pgf@intersect@subdivide@curve@b%
+ \pgf@curve@subdivide@left{#5}{#6}{#7}{#8}{#1}{#2}{#3}{#4}%
+ \endgroup%
+ \begingroup%
+ \advance\pgf@time@a by\pgf@time@aa\relax%
+ \divide\pgf@time@a by2\relax%
+ \let\pgf@intersect@subdivide@curve=\pgf@intersect@subdivide@curve@b%
+ \pgf@curve@subdivide@right{#5}{#6}{#7}{#8}{#1}{#2}{#3}{#4}%
+ \endgroup%
+}
+
+\def\pgf@intersect@boundingbox@reset{%
+ \pgf@xa=16000pt\relax%
+ \pgf@ya=16000pt\relax%
+ \pgf@xb=-16000pt\relax%
+ \pgf@yb=-16000pt\relax%
+}
+
+\def\pgf@intersect@boundingbox@update#1{%
+ #1\relax%
+ \ifdim\pgf@x<\pgf@xa\pgf@xa=\pgf@x\fi%
+ \ifdim\pgf@y<\pgf@ya\pgf@ya=\pgf@y\fi%
+ \ifdim\pgf@x>\pgf@xb\pgf@xb=\pgf@x\fi%
+ \ifdim\pgf@y>\pgf@yb\pgf@yb=\pgf@y\fi%
+}
+
+
+\def\pgf@curve@subdivide@left#1#2#3#4{%
+ %
+ % The left curve (from t=0 to t=.5)
+ %
+ #1\relax%
+ \pgfutil@tempdima=\pgf@x%
+ \pgfutil@tempdimb=\pgf@y%
+ \pgf@xa=.5\pgf@x\pgf@ya=.5\pgf@y%
+ \pgf@xb=.25\pgf@x\pgf@yb=.25\pgf@y%
+ \pgf@xc=.125\pgf@x\pgf@yc=.125\pgf@y%
+ #2\relax%
+ \advance\pgf@xa by.5\pgf@x\advance\pgf@ya by.5\pgf@y%
+ \advance\pgf@xb by.5\pgf@x\advance\pgf@yb by.5\pgf@y%
+ \advance\pgf@xc by.375\pgf@x\advance\pgf@yc by.375\pgf@y%
+ #3\relax%
+ \advance\pgf@xb by.25\pgf@x\advance\pgf@yb by.25\pgf@y%
+ \advance\pgf@xc by.375\pgf@x\advance\pgf@yc by.375\pgf@y%
+ #4\relax%
+ \advance\pgf@xc by.125\pgf@x\advance\pgf@yc by.125\pgf@y%
+ \edef\pgf@marshal{%
+ \noexpand\pgf@curve@subdivde@after%
+ {\noexpand\pgf@x=\the\pgfutil@tempdima\noexpand\pgf@y=\the\pgfutil@tempdimb}%
+ {\noexpand\pgf@x=\the\pgf@xa\noexpand\pgf@y\the\pgf@ya}%
+ {\noexpand\pgf@x=\the\pgf@xb\noexpand\pgf@y=\the\pgf@yb}
+ {\noexpand\pgf@x=\the\pgf@xc\noexpand\pgf@y=\the\pgf@yc}%
+ }%
+ \pgf@marshal%
+}
+
+\def\pgf@curve@subdivide@right#1#2#3#4{%
+ %
+ % The right curve (from t=0.5 to t=1)
+ %
+ #1\relax%
+ \pgfutil@tempdima=.125\pgf@x\pgfutil@tempdimb=.125\pgf@y%
+ #2\relax%
+ \advance\pgfutil@tempdima by.375\pgf@x\advance\pgfutil@tempdimb by.375\pgf@y%
+ \pgf@xa=.25\pgf@x\pgf@ya=.25\pgf@y%
+ #3\relax%
+ \advance\pgfutil@tempdima by.375\pgf@x\advance\pgfutil@tempdimb by.375\pgf@y%
+ \advance\pgf@xa by.5\pgf@x\advance\pgf@ya by.5\pgf@y%
+ \pgf@xb=.5\pgf@x\pgf@yb=.5\pgf@y%
+ #4\relax%
+ \advance\pgfutil@tempdima by.125\pgf@x\advance\pgfutil@tempdimb by.125\pgf@y%
+ \advance\pgf@xa by.25\pgf@x\advance\pgf@ya by.25\pgf@y%
+ \advance\pgf@xb by.5\pgf@x\advance\pgf@yb by.5\pgf@y%
+ \pgf@xc=\pgf@x\pgf@yc=\pgf@y%
+ \edef\pgf@marshal{%
+ \noexpand\pgf@curve@subdivde@after%
+ {\pgf@x\the\pgfutil@tempdima\pgf@y\the\pgfutil@tempdimb}%
+ {\pgf@x\the\pgf@xa\pgf@y\the\pgf@ya}{\pgf@x\the\pgf@xb\pgf@y\the\pgf@yb}
+ {\pgf@x\the\pgf@xc\pgf@y\the\pgf@yc}%
+ }%
+ \pgf@marshal%
+}
+
+
+% A solution S1 is considered a duplicate of S2, if
+%
+% |x1 - x2|f < q and |y1 - y2|f < q
+%
+% where q is a small value (tolerance).
+%
+% #1 - the solution.
+%
+\def\pgf@ifsolution@duplicate#1{%
+ #1%
+ \pgf@xa=\pgf@x%
+ \pgf@ya=\pgf@y%
+ \let\pgf@intersect@next=\pgfutil@secondoftwo%
+ \pgfmathloop%
+ \ifnum\pgfmathcounter>\pgf@intersect@solutions\relax%
+ \else%
+ \pgf@process{\csname pgfpoint@g@intersect@solution@\pgfmathcounter\endcsname}%
+ \advance\pgf@x by-\pgf@xa%
+ \advance\pgf@y by-\pgf@ya%
+ \ifdim\pgf@x<0pt\relax\pgf@x=-\pgf@x\fi%
+ \ifdim\pgf@y<0pt\relax\pgf@y=-\pgf@y\fi%
+ %
+ \pgf@x=\pgfintersectiontolerancefactor\pgf@x%
+ \pgf@y=\pgfintersectiontolerancefactor\pgf@y%
+ \ifdim\pgf@x<\pgfintersectiontolerance\relax%
+ \ifdim\pgf@y<\pgfintersectiontolerance\relax%
+ \let\pgf@intersect@next=\pgfutil@firstoftwo%
+ \fi%
+ \fi%
+ \repeatpgfmathloop%
+ \pgf@intersect@next%
+}
+
+
+\newif\ifpgf@intersect@solutions@sortfinish
+
+% Sort solutions according to their time index.
+%
+\def\pgfintersectionsolutionsortbytime{%
+ \pgf@intersect@solutions@sortfinishtrue%
+ \pgfmathloop%
+ \ifnum\pgfmathcounter=\pgfintersectionsolutions\relax%
+ \else%
+ \pgfutil@tempcnta=\pgfmathcounter%
+ \advance\pgfutil@tempcnta by1\relax%
+ \ifdim\csname pgf@intersect@solution@\pgfmathcounter @time@a\endcsname pt>%
+ \csname pgf@intersect@solution@\the\pgfutil@tempcnta @time@a\endcsname pt\relax%
+ \pgf@intersect@solutions@sortfinishfalse%
+ %
+ \pgfutil@namelet{pgf@intersect@temp}{pgfpoint@intersect@solution@\pgfmathcounter}%
+ \pgfutil@namelet{pgfpoint@intersect@solution@\pgfmathcounter}%
+ {pgfpoint@intersect@solution@\the\pgfutil@tempcnta}%
+ \pgfutil@namelet{pgfpoint@intersect@solution@\the\pgfutil@tempcnta}{pgf@intersect@temp}%
+ %
+ \pgfutil@namelet{pgf@intersect@temp}{pgf@intersect@solution@\pgfmathcounter @time@a}%
+ \pgfutil@namelet{pgf@intersect@solution@\pgfmathcounter @time@a}%
+ {pgf@intersect@solution@\the\pgfutil@tempcnta @time@a}%
+ \pgfutil@namelet{pgf@intersect@solution@\the\pgfutil@tempcnta @time@a}{pgf@intersect@temp}%
+ \fi%
+ \repeatpgfmathloop%
+ \ifpgf@intersect@solutions@sortfinish%
+ \else%
+ \expandafter\pgfintersectionsolutionsortbytime%
+ \fi%
+}
+
+\endinput