diff options
Diffstat (limited to 'graphics/pgf/base/tex/generic/libraries/pgflibraryintersections.code.tex')
-rw-r--r-- | graphics/pgf/base/tex/generic/libraries/pgflibraryintersections.code.tex | 1172 |
1 files changed, 1172 insertions, 0 deletions
diff --git a/graphics/pgf/base/tex/generic/libraries/pgflibraryintersections.code.tex b/graphics/pgf/base/tex/generic/libraries/pgflibraryintersections.code.tex new file mode 100644 index 0000000000..2457157690 --- /dev/null +++ b/graphics/pgf/base/tex/generic/libraries/pgflibraryintersections.code.tex @@ -0,0 +1,1172 @@ +% 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 |