% Copyright 2019 by Till Tantau % % This file may be distributed and/or modified % % 1. under the LaTeX Project Public License and/or % 2. under the GNU Public License. % % See the file doc/generic/pgf/licenses/LICENSE for more details. \ProvidesFileRCS{pgflibrarycurvilinear.code.tex} % % This file defines commands for computing points in curvilinear % coordinate systems. % % % Curvilinear coordinate systems defined in terms of a Bezier curve. % % For the following coordinate systems, you must first provide a % Bezier curve (using, as always, four points), which will be set for % the current scope. When the curve is "installed" some expensive % precomputations are done; subsequent calls to % \pgfpointcurvilinearxxx based on this Bezier curve will be % relatively quick. % % Install a Bezier curve % % #1 = start point % #2 = first control point % #3 = second control point % #4 = end point % % Description: % % Subsequent calls to functions like % \pgfpointcurvilinearbezierorthogonal will be relative to the curve % installed using this command. The main job of this macro is a % precomputation for computing length along the curve. % All lengths along the Bezier curve are approximated using a lookup table % of four time/length points. For this, an approximate time for length % 1pt is computed first, and then the length for twice this time, four % times this time, and eight times this time are approximated. A % distance-to-time conversion is then done by a linear interpolation % of distance-to-time for these four points. Note that all of these % computations are not particularly precise, but a compromise trading % speed against precision. Also note that the results will only be % best near the start of the curve and may be far off near the end if % that end is degenerate (second control point very near to end % point). % % Example: % % \pgfsetcurvilinearbeziercurve % {\pgfpoint{0mm}{10mm}} % {\pgfpoint{5.5mm}{10mm}} % {\pgfpoint{10mm}{5.5mm}} % {\pgfpoint{10mm}{0mm}} % nearly a quarter circle % \pgfpointcurvilinearbezierorthogonal{5mm}{5mm} % % should be 5mm along the circle, put at % % distance 15mm from the origin (5mm from the circle line). \def\pgfsetcurvilinearbeziercurve#1#2#3#4{% \pgf@process{#1}% \edef\pgf@curvilinear@line@a{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% \pgf@xa=-\pgf@x% \pgf@ya=-\pgf@y% \pgf@process{#2}% \edef\pgf@curvilinear@line@b{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% \pgf@xb=-\pgf@x% \pgf@yb=-\pgf@y% \advance\pgf@x by\pgf@xa% \advance\pgf@y by\pgf@ya% \pgfmathveclen@{\pgf@sys@tonumber\pgf@x}{\pgf@sys@tonumber\pgf@y}% \let\pgf@curvilinear@lenab\pgfmathresult% \pgf@process{#3}% \edef\pgf@curvilinear@line@c{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% \pgf@xc=-\pgf@x% \pgf@yc=-\pgf@y% \advance\pgf@x by\pgf@xb% \advance\pgf@y by\pgf@yb% \pgfmathveclen@{\pgf@sys@tonumber\pgf@x}{\pgf@sys@tonumber\pgf@y}% \let\pgf@curvilinear@lenbc\pgfmathresult% \pgf@process{#4}% \edef\pgf@curvilinear@line@d{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% \advance\pgf@x by\pgf@xc% \advance\pgf@y by\pgf@yc% \pgfmathveclen@{\pgf@sys@tonumber\pgf@x}{\pgf@sys@tonumber\pgf@y}% \let\pgf@curvilinear@lencd\pgfmathresult % \pgf@x=\pgf@curvilinear@lenab pt% \advance\pgf@x by\pgf@curvilinear@lenbc pt% \advance\pgf@x by\pgf@curvilinear@lencd pt% \pgfmathreciprocal@{\pgf@sys@tonumber\pgf@x}% \pgf@curvilinear@time@a\pgfmathresult pt% \pgf@process{\pgfpointcurveattime{\pgf@curvilinear@time@a}{\pgf@curvilinear@line@a}{\pgf@curvilinear@line@b}{\pgf@curvilinear@line@c}{\pgf@curvilinear@line@d}}% \pgf@xb=-\pgf@x% \pgf@yb=-\pgf@y% \advance\pgf@x by\pgf@xa% \advance\pgf@y by\pgf@ya% \pgfmathveclen@{\pgf@sys@tonumber\pgf@x}{\pgf@sys@tonumber\pgf@y}% \pgf@curvilinear@length@a\pgfmathresult pt% \ifdim\pgf@curvilinear@length@a>1pt\relax% % Ok, too large, let us make this smaller \pgfmathdivide@{\pgf@sys@tonumber\pgf@curvilinear@time@a}{\pgf@sys@tonumber\pgf@curvilinear@length@a}% \pgf@curvilinear@time@a\pgfmathresult pt% \pgf@process{\pgfpointcurveattime{\pgf@curvilinear@time@a}{\pgf@curvilinear@line@a}{\pgf@curvilinear@line@b}{\pgf@curvilinear@line@c}{\pgf@curvilinear@line@d}} \pgf@xb=-\pgf@x% \pgf@yb=-\pgf@y% \advance\pgf@x by\pgf@xa% \advance\pgf@y by\pgf@ya% \pgfmathveclen@{\pgf@sys@tonumber\pgf@x}{\pgf@sys@tonumber\pgf@y}% \pgf@curvilinear@length@a\pgfmathresult pt% \fi% % Compute three positions: \pgf@process{\pgfpointcurveattime{2\pgf@curvilinear@time@a}{\pgf@curvilinear@line@a}{\pgf@curvilinear@line@b}{\pgf@curvilinear@line@c}{\pgf@curvilinear@line@d}} \pgf@xa=-\pgf@x% \pgf@ya=-\pgf@y% \advance\pgf@x by\pgf@xb% \advance\pgf@y by\pgf@yb% \pgfmathveclen@{\pgf@sys@tonumber\pgf@x}{\pgf@sys@tonumber\pgf@y}% \pgf@curvilinear@length@b\pgfmathresult pt% \advance\pgf@curvilinear@length@b by\pgf@curvilinear@length@a% \pgf@process{\pgfpointcurveattime{4\pgf@curvilinear@time@a}{\pgf@curvilinear@line@a}{\pgf@curvilinear@line@b}{\pgf@curvilinear@line@c}{\pgf@curvilinear@line@d}} \pgf@xb=-\pgf@x% \pgf@yb=-\pgf@y% \advance\pgf@x by\pgf@xa% \advance\pgf@y by\pgf@ya% \pgfmathveclen@{\pgf@sys@tonumber\pgf@x}{\pgf@sys@tonumber\pgf@y}% \pgf@curvilinear@length@c\pgfmathresult pt% \advance\pgf@curvilinear@length@c by\pgf@curvilinear@length@b% \pgf@process{\pgfpointcurveattime{8\pgf@curvilinear@time@a}{\pgf@curvilinear@line@a}{\pgf@curvilinear@line@b}{\pgf@curvilinear@line@c}{\pgf@curvilinear@line@d}} \advance\pgf@x by\pgf@xb% \advance\pgf@y by\pgf@yb% \pgfmathveclen@{\pgf@sys@tonumber\pgf@x}{\pgf@sys@tonumber\pgf@y}% \pgf@curvilinear@length@d\pgfmathresult pt% \advance\pgf@curvilinear@length@d by\pgf@curvilinear@length@c% \let\pgf@curvilinear@comp@a\pgf@curvilinear@comp@a@initial% \let\pgf@curvilinear@comp@b\pgf@curvilinear@comp@b@initial% \let\pgf@curvilinear@comp@c\pgf@curvilinear@comp@c@initial% \let\pgf@curvilinear@comp@d\pgf@curvilinear@comp@d@initial% \let\pgf@curvilinear@comp@e\pgf@curvilinear@comp@e@initial% \let\pgf@curvilinear@point\pgf@curvilinear@curve@point% }% \newdimen\pgf@curvilinear@time@a \newdimen\pgf@curvilinear@length@a \newdimen\pgf@curvilinear@length@b \newdimen\pgf@curvilinear@length@c \newdimen\pgf@curvilinear@length@d \def\pgf@curvilinear@comp@a@initial{% \pgfmathdivide@{\pgf@sys@tonumber\pgf@curvilinear@time@a}{\pgf@sys@tonumber\pgf@curvilinear@length@a}% \let\pgf@curvilinear@quot@a\pgfmathresult% \let\pgf@curvilinear@comp@a\pgf@curvilinear@comp@a@cont% \pgf@curvilinear@comp@a@cont% }% \def\pgf@curvilinear@comp@a@cont{% \pgf@x\pgf@curvilinear@quot@a\pgf@x% }% \def\pgf@curvilinear@comp@b@initial{% \pgf@y=\pgf@curvilinear@length@b% \advance\pgf@y by-\pgf@curvilinear@length@a% \pgfmathdivide@{\pgf@sys@tonumber\pgf@curvilinear@time@a}{\pgf@sys@tonumber\pgf@y}% \let\pgf@curvilinear@quot@b\pgfmathresult% \pgf@y\pgfmathresult\pgf@curvilinear@length@a% \pgf@y-\pgf@y% \advance\pgf@y by\pgf@curvilinear@time@a% \edef\pgf@curvilinear@correct@b{\pgf@sys@tonumber\pgf@y}% \let\pgf@curvilinear@comp@b\pgf@curvilinear@comp@b@cont% \pgf@curvilinear@comp@b@cont% }% \def\pgf@curvilinear@comp@b@cont{% \pgf@x\pgf@curvilinear@quot@b\pgf@x% \advance\pgf@x by\pgf@curvilinear@correct@b pt% }% \def\pgf@curvilinear@comp@c@initial{% \pgf@y=\pgf@curvilinear@length@c% \advance\pgf@y by-\pgf@curvilinear@length@b% \pgf@y.5\pgf@y% \pgfmathdivide@{\pgf@sys@tonumber\pgf@curvilinear@time@a}{\pgf@sys@tonumber\pgf@y}% \let\pgf@curvilinear@quot@c\pgfmathresult% \pgf@y\pgf@curvilinear@quot@c\pgf@curvilinear@length@b% \pgf@y-\pgf@y% \advance\pgf@y by2\pgf@curvilinear@time@a% \edef\pgf@curvilinear@correct@c{\pgf@sys@tonumber\pgf@y}% \let\pgf@curvilinear@comp@c\pgf@curvilinear@comp@c@cont% \pgf@curvilinear@comp@c@cont% }% \def\pgf@curvilinear@comp@c@cont{% \pgf@x\pgf@curvilinear@quot@c\pgf@x% \advance\pgf@x by\pgf@curvilinear@correct@c pt% }% \def\pgf@curvilinear@comp@d@initial{% \pgf@y=\pgf@curvilinear@length@d% \advance\pgf@y by-\pgf@curvilinear@length@c% \pgf@y.25\pgf@y% \pgfmathdivide@{\pgf@sys@tonumber\pgf@curvilinear@time@a}{\pgf@sys@tonumber\pgf@y}% \let\pgf@curvilinear@quot@d\pgfmathresult% \pgf@y\pgf@curvilinear@quot@d\pgf@curvilinear@length@c% \pgf@y-\pgf@y% \advance\pgf@y by4\pgf@curvilinear@time@a% \edef\pgf@curvilinear@correct@d{\pgf@sys@tonumber\pgf@y}% \let\pgf@curvilinear@comp@d\pgf@curvilinear@comp@d@cont% \pgf@curvilinear@comp@d@cont% }% \def\pgf@curvilinear@comp@d@cont{% \pgf@x\pgf@curvilinear@quot@d\pgf@x% \advance\pgf@x by\pgf@curvilinear@correct@d pt% }% \def\pgf@curvilinear@comp@e@initial{% \pgfmathmultiply@{8}{\pgf@sys@tonumber\pgf@curvilinear@time@a}% \expandafter\pgfmathdivide@\expandafter{\pgfmathresult}{\pgf@sys@tonumber\pgf@curvilinear@length@d}% \let\pgf@curvilinear@quot@e\pgfmathresult% \let\pgf@curvilinear@comp@e\pgf@curvilinear@comp@e@cont% \pgf@curvilinear@comp@e@cont% }% \def\pgf@curvilinear@comp@e@cont{% \pgf@x\pgf@curvilinear@quot@e\pgf@x% }% % Convert a distance into a time % % #1 = a distance % % Description: % % After having called \pgfsetcurvilinearbeziercurve, you can use this % macro to convert a distance into a time along the curve set in that % command. The result will be stored in \pgf@x. It will only be % reasonably precise for small nonnegative #1 (in particular, #1 % should not be more than about half the length of the curve). \def\pgfcurvilineardistancetotime#1{% \pgfmathsetlength{\pgf@x}{#1}% \ifdim\pgf@x<\pgf@curvilinear@length@c\relax% \ifdim\pgf@x<\pgf@curvilinear@length@a\relax% \pgf@curvilinear@comp@a% \else\ifdim\pgf@x<\pgf@curvilinear@length@b\relax% \pgf@curvilinear@comp@b% \else% \pgf@curvilinear@comp@c% \fi\fi% \else\ifdim\pgf@x<\pgf@curvilinear@length@d\relax% \pgf@curvilinear@comp@d% \else% \pgf@curvilinear@comp@e% \fi\fi% }% % Compute a "Bezier-orthogonal" point for use in a nonlinear transformation. % % #1 = x % #2 = y % % Description: % % In this coordinate system, the x-axis "goes along" the Bezier curve % installed using \pgfsetcurvilinearbeziercurve % and the y-axis is always perpendicular to the (Bezier) curve at the % given x-coordinate. Formally, given a pair (x,y), let B(x) be the point on % the Bezier curve B at distance x from the start of the curve. Let % T(x) be the tangent of B at B(x) and let P(x) be the (normalized) % vector perpendicular to T(x). Then (x,y) would be mapped to B(x) + % y*P(x). As an example, if B is a circle, then the corresponding % curvilinear coordinate system is (essentially, except for an offset in % the y value) the polar coordinate system. % % In addition to setting \pgf@x and \pgf@y, \pgf@xa/ya will be set to % a tangent along the curve at the given point and \pgf@xb/yb to a % tangent orthogonal to the curve. \def\pgfpointcurvilinearbezierorthogonal#1#2{% \pgfmathsetmacro\pgf@curvilinear@yfactor{#2}% \pgfcurvilineardistancetotime{#1}% \pgfpointcurveattime{\pgf@x}{\pgf@curvilinear@line@a}{\pgf@curvilinear@line@b}{\pgf@curvilinear@line@c}{\pgf@curvilinear@line@d} \pgf@xc\pgf@x% save \pgf@yc\pgf@y% save % compute normal: \advance\pgf@xb by-\pgf@xa% \advance\pgf@yb by-\pgf@ya% \ifdim\pgf@xb<0.0001pt\ifdim\pgf@xb>-0.0001pt\ifdim\pgf@yb<0.0001pt\ifdim\pgf@yb>-0.0001pt\pgf@diff@curvi@ac\fi\fi\fi\fi \pgf@process{\pgfpointnormalised{\pgf@x=\pgf@yb\pgf@y=-\pgf@xb}} \pgf@x\pgf@curvilinear@yfactor\pgf@x% \pgf@y\pgf@curvilinear@yfactor\pgf@y% \advance\pgf@x by\pgf@xc% \advance\pgf@y by\pgf@yc% }% \def\pgf@diff@curvi@ac{% \pgf@curvilinear@line@a% \pgf@xa\pgf@x\pgf@ya\pgf@y% \pgf@curvilinear@line@c% \pgf@xb\pgf@x\pgf@yb\pgf@y% \advance\pgf@xb by-\pgf@xa% \advance\pgf@yb by-\pgf@ya% \ifdim\pgf@xb<0.0001pt\ifdim\pgf@xb>-0.0001pt\ifdim\pgf@yb<0.0001pt\ifdim\pgf@yb>-0.0001pt% still degenerate! \pgf@curvilinear@line@d% \pgf@xb\pgf@x\pgf@yb\pgf@y% \advance\pgf@xb by-\pgf@xa% \advance\pgf@yb by-\pgf@ya% \fi\fi\fi\fi% \pgf@xb-\pgf@xb% \pgf@yb-\pgf@yb% }% % Compute a "Bezier-polar" point. % % #1 = x % #2 = y % % Description: % % Let (r:d) be the point (x,y) in polar coordinates, that is, r is the % angle and d the distance of (x,y) to the origin. The point returned % by \pgfpointcurvilinearbezierpolar is now defined as follows: First, % we compute that point at distance d along the Bezier curve B. Let % B(d) be this point. Then, we rotate this point around the start of % the curve (B(0)) by r degrees. % % As an example, consider a triangle with one tip at the origin and % the other tips as (4cm,3cm) and (4cm,-3cm). Then this triangle would be % transformed as follows: We take the first 5cm of the Bezier curve % and rotate it by roughly 37 degrees to the left and by 37 degrees to % the right. % % Note that this command is pretty expensive. \def\pgfpointcurvilinearbezierpolar#1#2{% \pgfmathsetlength\pgfutil@tempdima{#1}% \pgfmathsetlength\pgfutil@tempdimb{#2}% % Compute angle: \pgfpointnormalised{\pgfqpoint{\pgfutil@tempdima}{\pgfutil@tempdimb}}% \ifdim\pgfutil@tempdimb=0pt\relax\ifdim\pgfutil@tempdima=0pt\pgf@x1pt\pgf@y0pt\fi\fi% \ifdim\pgfutil@tempdima<0pt% \pgf@x-\pgf@x% \pgf@y-\pgf@y% \fi% \pgf@ya=-\pgf@y% \pgfsettransformentries% {\pgf@sys@tonumber\pgf@x}{\pgf@sys@tonumber\pgf@y}% {\pgf@sys@tonumber\pgf@ya}{\pgf@sys@tonumber\pgf@x}{0pt}{0pt}% \pgftransformshift{\pgfpointscale{-1}{\pgf@curvilinear@line@a}}% \pgfmathveclen@{\pgf@sys@tonumber\pgfutil@tempdima}{\pgf@sys@tonumber\pgfutil@tempdimb}% \ifdim\pgfutil@tempdima<0pt% \edef\pgfmathresult{-\pgfmathresult}% \fi% \pgfcurvilineardistancetotime{\pgfmathresult}% % Now, transform: \pgf@process{% \pgfpointadd{% \pgfpointtransformed{% \pgfpointcurveattime% {\pgf@x}% {\pgf@curvilinear@line@a}% {\pgf@curvilinear@line@b}% {\pgf@curvilinear@line@c}% {\pgf@curvilinear@line@d}% }% }% \pgf@curvilinear@line@a% }% }% \endinput