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% Copyright 2019 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.
% Note: at the time of this writing, the library has quadratic runtime.
% Experimentally, it performed well while computing ~12 intersections of two
% plots, each with 600 samples. It failed when the number of samples exceeded 700.
\usepgflibrary{fpu}%
\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%
}%
% #1: the index. It starts at 1 and ends with \pgfintersectionsolutions (inclusive).
% Invalid values will implicitly result in the origin.
\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%
}%
% Gets the segment indices of solution #1.
%
% #1: the solution index (i.e. the same argument as in \pgfpointintersectionsolution)
% #2: [output] a macro name which will contain the segment index of the first path which contains the solution
% #3: [output] a macro name which will contain the segment index of the second path which contains the solution
%
% Example: \pgfintersectiongetsolutionsegmentindices{0}{\first}{\second}%
%
% -> \first may be 0 if point #0 is in the 0'th segment
% -> \second may be 42 if point #0 is in the 42'th segment
%
% The "segment index" is actually close to the "time" of the solution.
% If a solution is at "time" 42.2, it will have segment index 42.
\def\pgfintersectiongetsolutionsegmentindices#1#2#3{%
\ifnum#1<1\relax%
\let#2=\pgfutil@empty
\let#3=\pgfutil@empty
\else%
\ifnum#1>\pgfintersectionsolutions\relax%
\let#2=\pgfutil@empty
\let#3=\pgfutil@empty
\else%
\def\pgf@temp##1##2##3##4{%
\edef#2{##1}%
\edef#3{##2}%
}%
\expandafter\let\expandafter\pgf@tempb\csname pgf@intersect@solution@props@#1\endcsname
\expandafter\pgf@temp\pgf@tempb
\fi%
\fi%
}%
% Gets the time indices of solution #1.
%
% #1: the solution index (i.e. the same argument as in \pgfpointintersectionsolution)
% #2: [output] a macro name which will contain the time of the first path which contains the solution
% It will never be empty.
% #3: [output] a macro name which will contain the time of the second path which contains the solution
% It will never be empty.
%
% Example: \pgfintersectiongetsolutiontimes{0}{\first}{\second}%
%
% -> \first may be 0.5 if point #0 is in just in the middle of the path
% -> \second may be 42.8 if point #0 is in the 42'th segment (compare
% \pgfintersectiongetsolutionsegmentindices) and is at 80% of the
% 42'th segment
%
% Note that the precise time inside of a segment may be unavailable
% (currently, it is only computed for curveto paths and not
% necessarily for lineto). If the precise time is unavailable, this
% call will return the value of
% \pgfintersectiongetsolutionsegmentindices (which is a
% "coarse-grained" time).
\def\pgfintersectiongetsolutiontimes#1#2#3{%
\ifnum#1<1\relax%
\let#2=\pgfutil@empty
\let#3=\pgfutil@empty
\else%
\ifnum#1>\pgfintersectionsolutions\relax%
\let#2=\pgfutil@empty
\let#3=\pgfutil@empty
\else%
\def\pgf@temp##1##2##3##4{%
\edef#2{##3}%
\edef#3{##4}%
%
% check for fallback to segment indices:
\ifx#2\pgfutil@empty \edef#2{##1}\fi
\ifx#3\pgfutil@empty \edef#3{##2}\fi
}%
\expandafter\let\expandafter\pgf@tempb\csname pgf@intersect@solution@props@#1\endcsname
\expandafter\pgf@temp\pgf@tempb
\fi%
\fi%
}%
\def\pgfpoint@intersect@solution@orgin{%
\begingroup%
\pgftransforminvert%
\pgfpointorigin%
\pgf@pos@transform@glob
\global\pgf@x=\pgf@x%
\global\pgf@y=\pgf@y%
\endgroup%
}%
% #1 code which assigns the first path using \pgfsetpath.
% #2 code which assigns the second path using \pgfsetpath.
%
% On output, the points, their properties, and the number of points are set.
% Use \pgfintersectionsolutions which expands to the number of intersections
\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%
\pgf@intersect@path@reset@a
%
\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%
%
\pgfprocessround\pgf@intersect@path@a\pgf@intersect@path@a%
\pgfprocessround\pgf@intersect@path@b\pgf@intersect@path@b%
%
\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}%
\edef\pgf@marshal{\noexpand\pgf@intersection@set@properties{\csname pgfpoint@g@intersect@solution@\pgfmathcounter @props\endcsname}}%
\pgf@marshal
\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@intersection@set@properties#1{%
\pgfutil@namedef{pgf@intersect@solution@props@\pgfmathcounter}{#1}%
}%
% #1 a global name prefix to store properties.
\def\pgf@intersection@store@properties#1{%
% we store the time offsets as well and make them available programmatically:
% note that \pgf@intersect@time@a and \pgf@intersect@time@b may be empty.
%
% However, \pgf@intersect@time@offset and
% \pgf@intersect@time@offset@b are *always* valid. In fact,they
% resemble a part of the time: it holds
% 0 <= \pgf@intersect@time@a < 1
% and \pgf@intersect@time@offset > 0.
%
% If we have an intersection in segment 42 of path A,
% \pgf@intersect@time@offset will be 42. The time inside of that
% segment is given as number in the interval [0,1]. If it is 0.3,
% the total time will be 42.3 and that number will be stored as
% \pgf@intersect@time@a.
%
\expandafter\xdef\csname #1@props\endcsname{{\pgf@intersect@time@offset}{\pgf@intersect@time@offset@b}{\pgf@intersect@time@a}{\pgf@intersect@time@b}}%
}%
\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%
\pgf@intersect@path@reset@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@reset@a{%
\def\pgf@intersect@time@offset{0}%
\def\pgf@intersect@time@a{}%
}%
\def\pgf@intersect@path@reset@b{%
\def\pgf@intersect@time@offset@b{0}%
\def\pgf@intersect@time@b{}%
}%
\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%
\c@pgf@counta=\pgf@intersect@time@offset@b\relax%
\advance\c@pgf@counta by1\relax%
\edef\pgf@intersect@time@offset@b{\the\c@pgf@counta}%
\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}%
{%
\pgfextract@process\pgf@intersect@solution@candidate{%
% pgf@x and pgf@y are already assigned by \pgf@iflinesintersect
}%
\pgf@ifsolution@duplicate{\pgf@intersect@solution@candidate}{%
% ah - we a duplicate. Apparently, we have a hit on an
% endpoint.
}{%
\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%
% save coordinates of the intersection
\pgf@process{\pgf@intersect@solution@candidate}%
\pgf@xc=\pgf@x%
\pgf@yc=\pgf@y%
% determine length of path
\pgf@process{\pgfpointdiff{\pgfpoint@intersect@start@a}{\pgfpoint@intersect@end@a}}%
\edef\pgf@marshal{%
\noexpand\pgfmathveclen@{\pgfmath@tonumber{\pgf@x}}{\pgfmath@tonumber{\pgf@y}}%
}%
\pgf@marshal%
\let\pgf@intersect@length@a=\pgfmathresult%
% determine distance of intersection from start
\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%
% scale the distance to the path length (path time)
\pgfmathdivide@{\pgfmathresult}{\pgf@intersect@length@a}%
\pgf@x=\pgfmathresult pt\relax%
% advance by the path segment (see definition of \pgf@intersection@store@properties)
\advance\pgf@x by\pgf@intersect@time@offset pt\relax%
\edef\pgf@intersect@time@a{\pgfmath@tonumber{\pgf@x}}%
% save numbered
\expandafter\global\expandafter\let\csname pgf@g@intersect@solution@\the\pgf@intersect@solutions @time@a\endcsname=
\pgf@intersect@time@a
\else
\let\pgf@intersect@time@a=\pgfutil@empty
\fi%
\let\pgf@intersect@time@b=\pgfutil@empty
\pgf@intersection@store@properties{pgfpoint@g@intersect@solution@\the\pgf@intersect@solutions}%
}%
%
}{%
}%
}%
% 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%
}%
% queried by pgfplots. Do not delete, only increase.
\def\pgf@intersections@version{2}%
% #1,#2: line 1
% #3,#4: line 2
\def\pgf@iflinesintersect#1#2#3#4{%
% first: check bounding boxes -- but somewhat increased such that we do not
% exclude "visible" hits due to rounding issues (i.e. use an upper bound):
\pgf@intersect@boundingbox@reset%
\pgf@intersect@boundingbox@update{#1}%
\pgf@intersect@boundingbox@update{#2}%
\pgf@intersect@boundingbox@assign@b%
%
\pgf@intersect@boundingbox@reset%
\pgf@intersect@boundingbox@update{#3}%
\pgf@intersect@boundingbox@update{#4}%
\pgf@intersect@boundingbox@assign@a%
%
\pgf@intersect@boundingbox@a%
\pgf@intersect@boundingbox@b%
%
\pgf@intersect@boundingbox@ifoverlap@upperbound{%
\pgf@iflinesintersect@{#1}{#2}{#3}{#4}%
}{%
\let\pgf@intersect@next=\pgfutil@secondoftwo%
}%
\pgf@intersect@next%
}%
% a helper routine which simply defines \pgf@intersect@next.
%
% In principle, this routine is capable of computing the entire intersection... but we only invoke it after checking for bounding box overlaps. This has two reasons:
% 1. robustness. almost-parallel lines could cause "dimension too large" when solving the linear equation system
% XXX : I still needed to replace the linear solver by one using the FPU. Perhaps I do not need the BB check anymore?
% 2. performance. I hope it is faster to first check for BB (but this is not sure in TeX)
%
% #1,#2: line 1
% #3,#4: line 2
\def\pgf@iflinesintersect@#1#2#3#4{%
% we have two lines of sorts
% l_1(s) := #1 + s * (#2 - #1), 0<= s <= 1
% and
% l_2(t) := #3 + t * (#4 - #3), 0<= t <= 1
% ->
% set up LGS
% ( #2 - #1 ) *s + (#3-#4) * t = #3-#1
% we have a hit if 0<= s,t <= 1 .
#1\relax%
\pgf@xa=\pgf@x%
\pgf@ya=\pgf@y%
#2\relax%
\pgf@xb=\pgf@x%
\pgf@yb=\pgf@y%
#3\relax%
\pgf@xc=\pgf@x%
\pgf@yc=\pgf@y%
#4\relax%
%
% will be overwritten, remember it:
\edef\pgf@intersect@A{%
\pgf@xa=\the\pgf@xa\space
\pgf@ya=\the\pgf@ya\space
}%
%
% B := (2-1)
\advance\pgf@xb by-\pgf@xa
\advance\pgf@yb by-\pgf@ya
%
% A := (3-1)
\advance\pgf@xa by-\pgf@xc
\advance\pgf@ya by-\pgf@yc
\pgf@xa=-\pgf@xa
\pgf@ya=-\pgf@ya
%
% C := (3-4)
\advance\pgf@xc by-\pgf@x
\advance\pgf@yc by-\pgf@y
%
\begingroup
% compute the |.|_1 norm of each of lines. We need to compute
% tolerance factors in order to decide if we have an intersection.
% line 1: compute |#2 - #1|_1 :
\ifdim\pgf@xb<0sp \pgf@xb=-\pgf@xb\fi
\ifdim\pgf@yb<0sp \pgf@yb=-\pgf@yb\fi
\advance\pgf@xb by\pgf@yb
\xdef\pgf@intersect@len@a{\pgf@sys@tonumber\pgf@xb}%
%
% line 2: compute |#3 - #4|_1 :
\ifdim\pgf@xc<0sp \pgf@xc=-\pgf@xc\fi
\ifdim\pgf@yc<0sp \pgf@yc=-\pgf@yc\fi
\advance\pgf@xc by\pgf@yc
\xdef\pgf@intersect@len@b{\pgf@sys@tonumber\pgf@xc}%
\endgroup
%
\edef\pgf@marshal{%
\noexpand\pgfutilsolvetwotwoleqfloat{%
{\pgf@sys@tonumber\pgf@xb}{\pgf@sys@tonumber\pgf@xc}%
{\pgf@sys@tonumber\pgf@yb}{\pgf@sys@tonumber\pgf@yc}%
}{%
{\pgf@sys@tonumber\pgf@xa}%
{\pgf@sys@tonumber\pgf@ya}%
}%
}%
\pgf@marshal
%
\let\pgf@intersect@next=\pgfutil@secondoftwo%
\ifx\pgfmathresult\pgfutil@empty
% matrix was singular.
\else
\def\pgf@marshal##1##2{%
\global\pgf@x=##1pt %
\global\pgf@y=##2pt %
}%
\expandafter\pgf@marshal\pgfmathresult
%
\def\pgf@marshal{XXXX}% this should never be read
% FIRST: check line 1:
\ifdim\pgf@x<0sp
% let it count as hit if
% || l_1(s) - l_1(0) || < eps
% <=> |s| * ||#2 - #1|| < eps
% and, since s< 0 here:
% <=> -s * ||#2 - #1|| < eps
\pgf@xa=-\pgf@intersect@len@a\pgf@x
\ifdim\pgf@xa<\pgfintersectiontolerance\relax
% close enough to first endpoint of line 1:
\def\pgf@marshal{1}%
\else
\def\pgf@marshal{0}%
\fi
\else
\ifdim\pgf@x>1pt
% let it count as hit if
% || l_1(s) - l_1(1) || < eps
% <=> |s-1| * ||#2 - #1|| < eps
% and, since s > 1 here:
% <=> s * ||#2 - #1|| - ||#2 - #1|| < eps
\pgf@xa=\pgf@intersect@len@a\pgf@x
\advance\pgf@xa by-\pgf@intersect@len@a pt %
\ifdim\pgf@xa<\pgfintersectiontolerance\relax
% close enough to second endpoint of line 1:
\def\pgf@marshal{1}%
\else
\def\pgf@marshal{0}%
\fi
\else
% 0<= s <= 1: we have an intersection within line 1.
\def\pgf@marshal{1}%
\fi
\fi
%
% SECOND: check line 2:
\if1\pgf@marshal
\ifdim\pgf@y<0sp
% see remarks for line 1. same applies here.
\pgf@xa=-\pgf@intersect@len@b\pgf@y
\ifdim\pgf@xa<\pgfintersectiontolerance\relax
% close enough to first endpoint of line 2:
\def\pgf@marshal{1}%
\else
\def\pgf@marshal{0}%
\fi
\else
\ifdim\pgf@y>1pt
% see remarks for line 1. same applies here.
\pgf@xa=\pgf@intersect@len@b\pgf@y
\advance\pgf@xa by-\pgf@intersect@len@b pt %
\ifdim\pgf@xa<\pgfintersectiontolerance\relax
% close enough to second endpoint of line 2:
\def\pgf@marshal{1}%
\else
\def\pgf@marshal{0}%
\fi
\else
% 0<= t <= 1: we have an intersection within line 2.
\def\pgf@marshal{1}%
\fi
\fi
\fi
%
\if1\pgf@marshal
% Ok, compute the intersection point and return it:
% we use (x,y) = A + s * (B-A)
% keep in mind that (s,t) == (\pgf@x,\pgf@y)
\pgf@intersect@A
\pgf@yc=\pgf@x
\global\pgf@x=\pgf@sys@tonumber\pgf@xb\pgf@yc
\global\pgf@y=\pgf@sys@tonumber\pgf@yb\pgf@yc
\global\advance\pgf@x by \pgf@xa
\global\advance\pgf@y by \pgf@ya
\let\pgf@intersect@next=\pgfutil@firstoftwo%
\fi
\fi
}%
\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\pgfintersectiontoleranceupperbound{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 solution 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@intersect@boundingbox@assign@a{%
\edef\pgf@intersect@boundingbox@a{%
% lower left:
\noexpand\pgf@xb=\the\pgf@xa\space%
\noexpand\pgf@yb=\the\pgf@ya\space%
% upper right:
\noexpand\pgf@xc=\the\pgf@xb\space%
\noexpand\pgf@yc=\the\pgf@yb\space%
}%
}%
\def\pgf@intersect@boundingbox@assign@b{%
\edef\pgf@intersect@boundingbox@b{%
% lower left:
\noexpand\global\noexpand\pgf@x=\the\pgf@xa\space%
\noexpand\global\noexpand\pgf@y=\the\pgf@ya\space%
% upper right:
\noexpand\pgf@xa=\the\pgf@xb\space%
\noexpand\pgf@ya=\the\pgf@yb\space%
}%
}%
% see \pgf@intersect@boundingbox@assign@a and \pgf@intersect@boundingbox@assign@b for the naming conventions
\def\pgf@intersect@boundingbox@ifoverlap{%
\def\pgf@intersect@next{\pgfutil@secondoftwo}%
%
\ifdim\pgf@xa<\pgf@xb%
\else%
\ifdim\pgf@x>\pgf@xc%
\else%
\ifdim\pgf@ya<\pgf@yb%
\else%
\ifdim\pgf@y>\pgf@yc%
\else%
\def\pgf@intersect@next{\pgfutil@firstoftwo}%
\fi
\fi
\fi
\fi
\pgf@intersect@next
}%
\def\pgf@intersect@boundingbox@ifoverlap@upperbound{%
\begingroup
\def\pgf@intersect@next{\pgfutil@secondoftwo}%
%
\advance\pgf@xa by+\pgfintersectiontolerance\relax
\ifdim\pgf@xa<\pgf@xb%
\else%
\global\advance\pgf@x by-\pgfintersectiontolerance\relax
\ifdim\pgf@x>\pgf@xc%
\else%
\advance\pgf@ya by\pgfintersectiontolerance\relax
\ifdim\pgf@ya<\pgf@yb%
\else%
\global\advance\pgf@y by-\pgfintersectiontolerance\relax
\ifdim\pgf@y>\pgf@yc%
\else%
\def\pgf@intersect@next{\pgfutil@firstoftwo}%
\fi
\fi
\fi
\fi
\expandafter
\endgroup
\pgf@intersect@next
}%
\def\pgf@intersect@boundingbox@ifoverlap@UNUSED{%
\let\pgf@intersect@next=\pgfutil@secondoftwo%
\ifdim\pgf@xa<\pgf@xb%
\else%
\ifdim\pgf@x>\pgf@xc%
\else%
\ifdim\pgf@ya<\pgf@yb%
\else%
\ifdim\pgf@y>\pgf@yc%
\else%
\let\pgf@intersect@next=\pgfutil@firstoftwo%
\fi
\fi
\fi
\fi
\pgf@intersect@next
}%
\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}%
\pgf@intersect@boundingbox@assign@b%
%
\pgf@intersect@boundingbox@reset%
\pgf@intersect@boundingbox@update{#5}%
\pgf@intersect@boundingbox@update{#6}%
\pgf@intersect@boundingbox@update{#7}%
\pgf@intersect@boundingbox@update{#8}%
\pgf@intersect@boundingbox@assign@a%
%
\pgf@intersect@boundingbox@a%
\pgf@intersect@boundingbox@b%
%
\pgf@intersect@boundingbox@ifoverlap{%
\pgf@@@intersectionofcurves{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}%
}{%
% no overlap -- no intersection.
}%
}%
\def\pgf@@@intersectionofcurves#1#2#3#4#5#6#7#8{%
% compute DIFFERENCE vectors:
\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%
% check if both difference vectors are point wise
% less than tolerance (i.e. |v|_infty < eps ).
% That means that both bounding boxes are "small enough"
\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{%
% set (x,y) = mean(the 4 points of the two bounding boxes):
\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%
\begingroup
\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%
\edef\pgf@intersect@time@a{\pgfmath@tonumber{\pgf@time@a}}%
%
\advance\pgf@time@b by\pgf@time@bb%
\divide\pgf@time@b by2\relax%
\advance\pgf@time@b by\pgf@intersect@time@offset@b pt\relax%
\edef\pgf@intersect@time@b{\pgfmath@tonumber{\pgf@time@b}}%
%
\pgf@intersection@store@properties{pgfpoint@g@intersect@solution@\the\pgf@intersect@solutions}%
\expandafter\global\expandafter\let%
\csname pgfpoint@g@intersect@solution@\the\pgf@intersect@solutions\endcsname=%
\pgf@intersect@solution@candidate%
\ifpgf@intersect@sort%
\expandafter\xdef%
\csname pgf@g@intersect@solution@\the\pgf@intersect@solutions @time@a\endcsname%
{\pgf@intersect@time@a}%
\fi%
\endgroup
}%
\fi%
\fi%
\fi%
\fi%
\ifx\pgf@intersect@subdivde\pgf@stop%
\else%
\pgf@intersect@subdivide@curve{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}%
\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%
}%
% The following subroutines are part of a conversion from pgfbasic
% math to FPU. This transition is necessary due to the restricted
% accuracy of pgfbasic. In order to limit the error rate of the
% transition pgfbasic -> FPU, I chose to
% keep the old "pattern" of sorts \advance\pgf@xa by0.5\pgf@y etc and
% simply adapt to some FPU call.
%
% The following routines constitute the "adapter":
\def\pgf@float@adapter@setxy{%
\pgfmathfloatparsenumber{\pgf@sys@tonumber\pgf@x}\let\pgf@fpu@x=\pgfmathresult
\pgfmathfloatparsenumber{\pgf@sys@tonumber\pgf@y}\let\pgf@fpu@y=\pgfmathresult
}%
\def\pgf@float@adapter@mult#1=#2*#3{%
\pgfmathfloatmultiplyfixed@{#3}{#2}%
\let#1=\pgfmathresult
}%
\def\pgf@float@adapter@advance#1by#2*#3{%
\pgfmathfloatmultiplyfixed@{#3}{#2}%
\let\pgfutil@temp=\pgfmathresult
\pgfmathfloatadd@{#1}{\pgfutil@temp}%
\let#1=\pgfmathresult
}%
\def\pgf@float@adapter@tostring#1{%
\pgfmathfloattofixed{#1}\edef#1{\pgfmathresult pt }%
}%
\def\pgf@curve@subdivide@left#1#2#3#4{%
%
% The left curve (from t=0 to t=.5)
%
\begingroup
#1\relax%
\pgfutil@tempdima=\pgf@x%
\pgfutil@tempdimb=\pgf@y%
\pgf@float@adapter@setxy
\pgf@float@adapter@mult\pgf@fpu@xa=.5*\pgf@fpu@x \pgf@float@adapter@mult\pgf@fpu@ya=.5*\pgf@fpu@y%
\pgf@float@adapter@mult\pgf@fpu@xb=.25*\pgf@fpu@x \pgf@float@adapter@mult\pgf@fpu@yb=.25*\pgf@fpu@y%
\pgf@float@adapter@mult\pgf@fpu@xc=.125*\pgf@fpu@x\pgf@float@adapter@mult\pgf@fpu@yc=.125*\pgf@fpu@y%
#2\relax%
\pgf@float@adapter@setxy
\pgf@float@adapter@advance\pgf@fpu@xa by.5*\pgf@fpu@x\pgf@float@adapter@advance\pgf@fpu@ya by.5*\pgf@fpu@y%
\pgf@float@adapter@advance\pgf@fpu@xb by.5*\pgf@fpu@x\pgf@float@adapter@advance\pgf@fpu@yb by.5*\pgf@fpu@y%
\pgf@float@adapter@advance\pgf@fpu@xc by.375*\pgf@fpu@x\pgf@float@adapter@advance\pgf@fpu@yc by.375*\pgf@fpu@y%
#3\relax%
\pgf@float@adapter@setxy
\pgf@float@adapter@advance\pgf@fpu@xb by.25*\pgf@fpu@x\pgf@float@adapter@advance\pgf@fpu@yb by.25*\pgf@fpu@y%
\pgf@float@adapter@advance\pgf@fpu@xc by.375*\pgf@fpu@x\pgf@float@adapter@advance\pgf@fpu@yc by.375*\pgf@fpu@y%
#4\relax%
\pgf@float@adapter@setxy
\pgf@float@adapter@advance\pgf@fpu@xc by.125*\pgf@fpu@x\pgf@float@adapter@advance\pgf@fpu@yc by.125*\pgf@fpu@y%
%
\pgf@float@adapter@tostring\pgf@fpu@xa
\pgf@float@adapter@tostring\pgf@fpu@ya
\pgf@float@adapter@tostring\pgf@fpu@xb
\pgf@float@adapter@tostring\pgf@fpu@yb
\pgf@float@adapter@tostring\pgf@fpu@xc
\pgf@float@adapter@tostring\pgf@fpu@yc
\edef\pgf@marshal{%
\noexpand\pgf@curve@subdivde@after%
{\noexpand\pgf@x=\the\pgfutil@tempdima\noexpand\pgf@y=\the\pgfutil@tempdimb}%
{\noexpand\pgf@x=\pgf@fpu@xa\noexpand\pgf@y=\pgf@fpu@ya}%
{\noexpand\pgf@x=\pgf@fpu@xb\noexpand\pgf@y=\pgf@fpu@yb}%
{\noexpand\pgf@x=\pgf@fpu@xc\noexpand\pgf@y=\pgf@fpu@yc}%
}%
\expandafter
\endgroup
\pgf@marshal%
}%
\def\pgf@curve@subdivide@right#1#2#3#4{%
%
% The right curve (from t=0.5 to t=1)
%
\begingroup
#1\relax%
\pgf@float@adapter@setxy
\pgf@float@adapter@mult\pgf@float@tmpa=.125*\pgf@fpu@x\pgf@float@adapter@mult\pgf@float@tmpb=.125*\pgf@fpu@y%
#2\relax%
\pgf@float@adapter@setxy
\pgf@float@adapter@advance\pgf@float@tmpa by.375*\pgf@fpu@x\pgf@float@adapter@advance\pgf@float@tmpb by.375*\pgf@fpu@y%
\pgf@float@adapter@mult\pgf@fpu@xa=.25*\pgf@fpu@x\pgf@float@adapter@mult\pgf@fpu@ya=.25*\pgf@fpu@y%
#3\relax%
\pgf@float@adapter@setxy
\pgf@float@adapter@advance\pgf@float@tmpa by.375*\pgf@fpu@x\pgf@float@adapter@advance\pgf@float@tmpb by.375*\pgf@fpu@y%
\pgf@float@adapter@advance\pgf@fpu@xa by.5*\pgf@fpu@x\pgf@float@adapter@advance\pgf@fpu@ya by.5*\pgf@fpu@y%
\pgf@float@adapter@mult\pgf@fpu@xb=.5*\pgf@fpu@x\pgf@float@adapter@mult\pgf@fpu@yb=.5*\pgf@fpu@y%
#4\relax%
\pgf@float@adapter@setxy
\pgf@float@adapter@advance\pgf@float@tmpa by.125*\pgf@fpu@x\pgf@float@adapter@advance\pgf@float@tmpb by.125*\pgf@fpu@y%
\pgf@float@adapter@advance\pgf@fpu@xa by.25*\pgf@fpu@x\pgf@float@adapter@advance\pgf@fpu@ya by.25*\pgf@fpu@y%
\pgf@float@adapter@advance\pgf@fpu@xb by.5*\pgf@fpu@x\pgf@float@adapter@advance\pgf@fpu@yb by.5*\pgf@fpu@y%
\let\pgf@fpu@xc=\pgf@fpu@x\let\pgf@fpu@yc=\pgf@fpu@y%
%
\pgf@float@adapter@tostring\pgf@float@tmpa
\pgf@float@adapter@tostring\pgf@float@tmpb
\pgf@float@adapter@tostring\pgf@fpu@xa
\pgf@float@adapter@tostring\pgf@fpu@ya
\pgf@float@adapter@tostring\pgf@fpu@xb
\pgf@float@adapter@tostring\pgf@fpu@yb
\pgf@float@adapter@tostring\pgf@fpu@xc
\pgf@float@adapter@tostring\pgf@fpu@yc
\edef\pgf@marshal{%
\noexpand\pgf@curve@subdivde@after%
{\noexpand\pgf@x=\pgf@float@tmpa\noexpand\pgf@y=\pgf@float@tmpb}%
{\noexpand\pgf@x=\pgf@fpu@xa\noexpand\pgf@y=\pgf@fpu@ya}%
{\noexpand\pgf@x=\pgf@fpu@xb\noexpand\pgf@y=\pgf@fpu@yb}%
{\noexpand\pgf@x=\pgf@fpu@xc\noexpand\pgf@y=\pgf@fpu@yc}%
}%
\expandafter
\endgroup
\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@ifsolution@duplicate@{\pgfmathcounter}%
\repeatpgfmathloop%
\pgf@intersect@next%
}%
\def\pgf@ifsolution@duplicate@#1{%
\pgf@process{\csname pgfpoint@g@intersect@solution@#1\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%
}%
\newif\ifpgf@intersect@solutions@sortfinish
% Sort solutions according to their time index.
%
\def\pgfintersectionsolutionsortbytime{%
\pgf@intersect@solutions@sortfinishtrue%
\pgfmathloop%
\ifnum\pgfmathcounter<\pgfintersectionsolutions\relax%
\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%
%
\pgfintersectionsolutionsortbytime@swap{pgfpoint@intersect@solution@\pgfmathcounter}%
{pgfpoint@intersect@solution@\the\pgfutil@tempcnta}%
%
\pgfintersectionsolutionsortbytime@swap{pgf@intersect@solution@\pgfmathcounter @time@a}%
{pgf@intersect@solution@\the\pgfutil@tempcnta @time@a}%
%
\pgfintersectionsolutionsortbytime@swap{pgf@intersect@solution@props@\pgfmathcounter}%
{pgf@intersect@solution@props@\the\pgfutil@tempcnta}%
\fi%
\repeatpgfmathloop%
\ifpgf@intersect@solutions@sortfinish%
\else%
\expandafter\pgfintersectionsolutionsortbytime%
\fi%
}%
\def\pgfintersectionsolutionsortbytime@swap#1#2{%
\pgfutil@namelet{pgf@intersect@temp}{#1}%
\pgfutil@namelet{#1}{#2}%
\pgfutil@namelet{#2}{pgf@intersect@temp}%
}%
\endinput