% 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{pgfcorepoints.code.tex} \newdimen\pgf@picminx \newdimen\pgf@picmaxx \newdimen\pgf@picminy \newdimen\pgf@picmaxy \newdimen\pgf@pathminx \newdimen\pgf@pathmaxx \newdimen\pgf@pathminy \newdimen\pgf@pathmaxy \newif\ifpgf@relevantforpicturesize \def\pgf@process#1{{#1\global\pgf@x=\pgf@x\global\pgf@y=\pgf@y}} % Save a point. % % #1 = macro for storing point. % #2 = code for point (should define x and y) % % Example: % % \pgfextract@process\mypoint{\pgf@x=10pt \pgf@y10pt} % \pgfextract@process\myarcpoint{\pgfpointpolar{30}{5cm and 2cm}} \def\pgfextract@process#1#2{% \pgf@process{#2}% \edef#1{\noexpand\global\pgf@x=\the\pgf@x\noexpand\relax\noexpand\global\pgf@y=\the\pgf@y\noexpand\relax}% } % This needed until old shapes code changed. \let\pgfsavepgf@process\pgfextract@process% % Return a point % % #1 = x-coordinate of the point % #2 = y-coordinate of the point % % x = #1 % y = #2 % % Example: % % \pgfpathmoveto{\pgfpoint{2pt+3cm}{3cm}} \def\pgfpoint#1#2{% \pgfmathsetlength\pgf@x{#1}% \pgfmathsetlength\pgf@y{#2}\ignorespaces} % Quickly a point % % #1 = x-coordinate of the point (no calculations done) % #2 = y-coordinate of the point (no calculations done) % % x = #1 % y = #2 % % Example: % % \pgfpathmoveto{\pgfqpoint{2pt}{3cm}} \def\pgfqpoint#1#2{\global\pgf@x=#1\relax\global\pgf@y=#2\relax} % Return the origin. % % x = 0 % y = 0 % % Example: % % \pgfpathmoveto{\pgfpointorigin} \def\pgfpointorigin{\global\pgf@x=0pt \global\pgf@y=\pgf@x\ignorespaces} % Return a transformed point % % #1 = a point % % Description: % % This command applies pgf's current transformation matrix to the % given point. Normally, this is done automatically by commands like % lineto or moveto, but sometimes you may wish to access a transformed % point yourself. In the below example, this command is used for a low level % coordinate system shift. % % Example: % % \begin{pgflowleveltransformshiftscope}{\pgfpointtransformed{\pgfpointorigin}} % \pgfbox[center,center]{Hi!} % \end{pgflowleveltransformshiftscope} \def\pgfpointtransformed#1{% \pgf@process{% #1% \pgf@pos@transform@glob% }% } % Return the difference vector of two points. % % #1 = start of vector % #2 = end of vector % % x = x-component of difference % y = y-component of difference % % Example: % % \pgfpathmoveto{\pgfpointdiff{\pgfpointxy{1}{1}}{\pgfpointxy{2}{3}}} \def\pgfpointdiff#1#2{% \pgf@process{#1}% \pgf@xa=\pgf@x% \pgf@ya=\pgf@y% \pgf@process{#2}% \global\advance\pgf@x by-\pgf@xa\relax% \global\advance\pgf@y by-\pgf@ya\relax\ignorespaces} % Add two vectors. % % #1 = first vector % #2 = second vector % % x = x-component of addition % y = y-component of addition % % Example: % % \pgfpathmoveto{\pgfpointadd{\pgfpointxy{0}{1}}{\pgfpointxy{2}{3}}} \def\pgfpointadd#1#2{% \pgf@process{#1}% \pgf@xa=\pgf@x% \pgf@ya=\pgf@y% \pgf@process{#2}% \global\advance\pgf@x by\pgf@xa% \global\advance\pgf@y by\pgf@ya} % Multiply a vector by a factor. % % #1 = factor % #2 = vector % % Example: % % \pgfpointscale{2}{\pgfpointxy{0}{1}} \def\pgfpointscale#1#2{% \pgf@process{#2}% \pgfmathparse{#1}% \global\pgf@x=\pgfmathresult\pgf@x% \global\pgf@y=\pgfmathresult\pgf@y% } % A "quick" variant of \pgfpointscale which doesn't invoke the math parser for '#1'. % #1 must be a number without units, no registers are accepted. \def\pgfqpointscale#1#2{% \pgf@process{#2}% \global\pgf@x=#1\pgf@x% \global\pgf@y=#1\pgf@y% } % The intersection of two lines % % #1 = point on first line % #2 = another point on first line % #3 = point on second line % #4 = another point on second line % % Returns the intersection of the two lines. If there is no % intersection or if the points #1 and #2 or the points #3 and #4 are % identical, the behaviour is not specified. % % Example: % % \pgfpointintersectionoflines{\pgfpointxy{0}{1}}{\pgfpointxy{1}{0}}{\pgfpointxy{2}{2}}{\pgfpointxy{3}{4}} \def\pgfpointintersectionoflines#1#2#3#4{% {% % % Compute orthogonal vector to #1--#2 % \pgf@process{#2}% \pgf@xa=\pgf@x% \pgf@ya=\pgf@y% \pgf@process{#1}% \advance\pgf@xa by-\pgf@x% \advance\pgf@ya by-\pgf@y% \pgf@ya=-\pgf@ya% % Normalise a bit \c@pgf@counta=\pgf@xa% \ifnum\c@pgf@counta<0\relax% \c@pgf@counta=-\c@pgf@counta\relax% \fi% \c@pgf@countb=\pgf@ya% \ifnum\c@pgf@countb<0\relax% \c@pgf@countb=-\c@pgf@countb\relax% \fi% \advance\c@pgf@counta by\c@pgf@countb\relax% \divide\c@pgf@counta by 65536\relax% \ifnum\c@pgf@counta>0\relax% \divide\pgf@xa by\c@pgf@counta\relax% \divide\pgf@ya by\c@pgf@counta\relax% \fi% % % Compute projection % \pgf@xc=\pgf@sys@tonumber{\pgf@ya}\pgf@x% \advance\pgf@xc by\pgf@sys@tonumber{\pgf@xa}\pgf@y% % % The orthogonal vector is (\pgf@ya,\pgf@xa) % % % Compute orthogonal vector to #3--#4 % \pgf@process{#4}% \pgf@xb=\pgf@x% \pgf@yb=\pgf@y% \pgf@process{#3}% \advance\pgf@xb by-\pgf@x% \advance\pgf@yb by-\pgf@y% \pgf@yb=-\pgf@yb% % Normalise a bit \c@pgf@counta=\pgf@xb% \ifnum\c@pgf@counta<0\relax% \c@pgf@counta=-\c@pgf@counta\relax% \fi% \c@pgf@countb=\pgf@yb% \ifnum\c@pgf@countb<0\relax% \c@pgf@countb=-\c@pgf@countb\relax% \fi% \advance\c@pgf@counta by\c@pgf@countb\relax% \divide\c@pgf@counta by 65536\relax% \ifnum\c@pgf@counta>0\relax% \divide\pgf@xb by\c@pgf@counta\relax% \divide\pgf@yb by\c@pgf@counta\relax% \fi% % % Compute projection % \pgf@yc=\pgf@sys@tonumber{\pgf@yb}\pgf@x% \advance\pgf@yc by\pgf@sys@tonumber{\pgf@xb}\pgf@y% % % The orthogonal vector is (\pgf@yb,\pgf@xb) % % Setup transformation matrix (this is just to use the matrix % inversion) % \pgfsettransform{{\pgf@sys@tonumber\pgf@ya}{\pgf@sys@tonumber\pgf@yb}{\pgf@sys@tonumber\pgf@xa}{\pgf@sys@tonumber\pgf@xb}{0pt}{0pt}}% \pgftransforminvert% \pgf@process{\pgfpointtransformed{\pgfpoint{\pgf@xc}{\pgf@yc}}}% }% } % The intersection of two circles % % #1 = center of first circle % #2 = center of second circle % #3 = radius of first circle % #4 = radius of second circle % #5 = solution number % % Returns the intersection of the two circles. If #5 is to "1", the % first intersection is returned, otherwise the second. If the circles % do not intersect, an error may occur. % % Example: % % \pgfpointintersectionofcircles{\pgfpointxy{0}{1}}{\pgfpointxy{1}{0}}{1cm}{1cm}{1} \def\pgfpointintersectionofcircles#1#2#3#4#5{% {% % Store first point in (xa,ya) and radius in xc. \pgf@process{#1}% \pgf@xa=\pgf@x% \pgf@ya=\pgf@y% \pgfmathsetlength{\pgf@xc}{#3}% % Store second point in (xb,yb) and radius in yc. \pgf@process{#2}% \pgf@xb=\pgf@x% \pgf@yb=\pgf@y% \pgfmathsetlength{\pgf@yc}{#4}% % % Ok, now make numbers smaller, in case they are too large % \c@pgf@counta=1\relax% \loop% \pgf@scale@downfalse% \ifdim\pgf@xc>50pt\relax% \pgf@scale@downtrue% \fi% \ifdim\pgf@yc>50pt\relax% \pgf@scale@downtrue% \fi% \ifpgf@scale@down% \multiply\c@pgf@counta by2\relax% \divide\pgf@xa by2\relax% \divide\pgf@ya by2\relax% \divide\pgf@xb by2\relax% \divide\pgf@yb by2\relax% \divide\pgf@xc by2\relax% \divide\pgf@yc by2\relax% \repeat% % The following code is taken from the Dr. Math internet forum: % % Let the centers be: (a,b), (c,d) % Let the radii be: r, s % % e = c - a [difference in x coordinates] % f = d - b [difference in y coordinates] % p = sqrt(e^2 + f^2) [distance between centers] % k = (p^2 + r^2 - s^2)/(2p) [distance from center 1 to line % joining points of intersection] % x = a + ek/p + (f/p)sqrt(r^2 - k^2) % y = b + fk/p - (e/p)sqrt(r^2 - k^2) % OR % x = a + ek/p - (f/p)sqrt(r^2 - k^2) % y = b + fk/p + (e/p)sqrt(r^2 - k^2) % % Since we are running low on registers, use % \dimen0 for e % \dimen1 for f % \dimen2 for p % \dimen3 for p^2 % \dimen4 for k % \dimen5 for 1/p % \dimen6 for sqrt(r^2 - k^2) % \dimen7 for k^2 % \dimen8 for k/p % \dimen9 for sqrt(r^2 - k^2)/p % Also note that: % \pgf@xa for a % \pgf@ya for b % \pgf@xb for c % \pgf@yb for d % \pgf@xc for r % \pgf@yc for s % % Now: % e = c - a \dimen0=\pgf@xb% \advance\dimen0 by-\pgf@xa% % f = d - b \dimen1=\pgf@yb% \advance\dimen1 by-\pgf@ya% % p^2 = e^2 + f^2 \pgf@x=\dimen0\relax% \pgf@x=\pgf@sys@tonumber{\pgf@x}\pgf@x% \dimen3=\pgf@x% \pgf@x=\dimen1\relax% \advance\dimen3 by\pgf@sys@tonumber{\pgf@x}\pgf@x% % p = sqrt(p^2) \pgfmathparse{sqrt(\the\dimen3)}% \dimen2=\pgfmathresult pt% % 1/p = 1/p \pgfmathreciprocal@{\pgfmathresult}% \dimen5=\pgfmathresult pt% % k = (p^2 + r^2 - s^2)/(2p) \dimen4=\dimen3\relax% \pgf@x=\pgf@xc% \advance\dimen4 by\pgf@sys@tonumber{\pgf@x}\pgf@x\relax% \pgf@x=\pgf@yc% \advance\dimen4 by-\pgf@sys@tonumber{\pgf@x}\pgf@x\relax% \dimen4=.5\dimen4% \dimen4=\pgf@sys@tonumber{\dimen5}\dimen4% % dimen7 is k^2 \dimen7=\pgf@sys@tonumber{\dimen4}\dimen4\relax% % dimen6 is sqrt(r^2 - k^2) \pgfmathparse{sqrt(\pgf@sys@tonumber{\pgf@xc}\pgf@xc-\the\dimen7)}% \dimen6=\pgfmathresult pt% % dimen8 is k/p \dimen8=\pgf@sys@tonumber{\dimen4}\dimen5\relax% % dimen9 is sqrt(r^2 - k^2)/p \dimen9=\pgf@sys@tonumber{\dimen6}\dimen5\relax% \ifnum#5=1\relax% % x = a + ek/p + (f/p)sqrt(r^2 - k^2) \pgf@x=\pgf@xa% \advance\pgf@x by\pgf@sys@tonumber{\dimen0}\dimen8\relax% \advance\pgf@x by\pgf@sys@tonumber{\dimen1}\dimen9\relax% % y = b + fk/p - (e/p)sqrt(r^2 - k^2) \pgf@y=\pgf@ya% \advance\pgf@y by\pgf@sys@tonumber{\dimen1}\dimen8\relax% %temp \pgf@xb=\pgf@sys@tonumber{\dimen0}\dimen9% \pgf@xb=-\pgf@xb% \advance\pgf@y by\pgf@xb\relax% \else% % x = a + ek/p - (f/p)sqrt(r^2 - k^2) \pgf@x=\pgf@xa% \advance\pgf@x by\pgf@sys@tonumber{\dimen0}\dimen8\relax% %temp \pgf@xb=\pgf@sys@tonumber{\dimen1}\dimen9% \pgf@xb=-\pgf@xb% \advance\pgf@x by\pgf@xb\relax% % y = b + fk/p + (e/p)sqrt(r^2 - k^2) \pgf@y=\pgf@ya% \advance\pgf@y by\pgf@sys@tonumber{\dimen1}\dimen8\relax% \advance\pgf@y by\pgf@sys@tonumber{\dimen0}\dimen9\relax% \fi% \pgf@x=\c@pgf@counta\pgf@x% \pgf@y=\c@pgf@counta\pgf@y% \pgf@process{}% get results outside }% } \newif\ifpgf@scale@down % Returns point on a line from #2 to #3 at time #1. % % #1 = a time, where 0 is the start and 1 is the end % #2 = start point % #3 = end point % % x = x-component of #1*start + (1-#1)*end % y = y-component of #1*start + (1-#1)*end % xa/ya = #1*start + (1-#1)*end % xb/yb = start point % xc/yc = end point % % Example: % % % Middle of (1,1) and (2,3) % \pgfpathmoveto{\pgfpointlineattime{0.5}{\pgfpointxy{0}{1}}{\pgfpointxy{2}{3}}} \def\pgfpointlineattime#1#2#3{% \pgf@process{#3}% \pgf@xa\pgf@x% \pgf@ya\pgf@y% \pgf@xc\pgf@x% \pgf@yc\pgf@y% \pgf@process{#2}% \pgf@xb\pgf@x% \pgf@yb\pgf@y% \pgfmathsetmacro\pgf@temp{#1}% \advance\pgf@xa by-\pgf@x% \advance\pgf@ya by-\pgf@y% \global\advance\pgf@x by\pgf@temp\pgf@xa% \global\advance\pgf@y by\pgf@temp\pgf@ya% } % Move point #2 #1 many units in the direction of #3. % % #1 = a distance % #2 = start point % #3 = end point % % Description: % % Computes % % x/y = start + #1*(normalise(end-start)) % % and additionally % % xa/ya = #1*(normalise(end-start)) % xb/yb = start % xc/yc = end % % Example: % % \pgfpathmoveto{\pgfpointlineatdistance{2pt}{\pgfpointxy{0}{1}}{\pgfpointxy{2}{3}}} % \pgfpathlineto{\pgfpointlineatdistance{3pt}{\pgfpointxy{2}{3}}{\pgfpointxy{0}{1}}} \def\pgfpointlineatdistance#1#2#3{% \pgfmathsetlength\pgf@xa{#1}% \pgf@process{#2}% \pgf@xb\pgf@x% xb/yb = start point \pgf@yb\pgf@y% \pgf@process{#3}% \pgf@xc\pgf@x% \pgf@yc\pgf@y% \global\advance\pgf@x by-\pgf@xb\relax% \global\advance\pgf@y by-\pgf@yb\relax% \pgf@process{\pgfpointnormalised{}}% x/y = normalised vector \pgf@ya=\pgf@xa\relax% \pgf@xa=\pgf@sys@tonumber{\pgf@x}\pgf@xa% \pgf@ya=\pgf@sys@tonumber{\pgf@y}\pgf@ya% \global\pgf@x=\pgf@xb\relax% \global\pgf@y=\pgf@yb\relax% \global\advance\pgf@x by\pgf@xa\relax% \global\advance\pgf@y by\pgf@ya\relax% } % Returns point on a curve from #2 to #5 with controls #3 and #4 at time #1. % % #1 = a time % #2 = start point % #3 = first control point % #4 = second control point % #5 = end point % % x = x-component of place on the curve at time t % y = y-component of place on the curve at time t % % Additionally, (\pgf@xa,\pgf@ya) and (\pgf@xb,\pgf@yb) will be on a % tangent to the point on the curve (this can be useful for computing % a label rotation). (\pgf@xc,\pgf@yc) will be equal to the end % point. \pgf@time@s will equal the value of #1 and \pgf@time@t will % equal 1-#1. % % Example: % % % Middle of (1,1) and (2,3) % \pgfpathmoveto{\pgfpointcurveattime{0.5}{\pgfpointxy{0}{1}}{\pgfpointxy{1}{1}}{\pgfpointxy{1}{1}}{\pgfpointxy{2}{3}}} \def\pgfpointcurveattime#1#2#3#4#5{% \pgfmathparse{#1}% \let\pgf@time@s=\pgfmathresult% \global\pgf@x=\pgfmathresult pt% \global\pgf@x=-\pgf@x% \advance\pgf@x by 1pt% \edef\pgf@time@t{\pgf@sys@tonumber{\pgf@x}}% \pgf@process{#5}% \pgf@xc=\pgf@x% \pgf@yc=\pgf@y% \pgf@process{#4}% \pgf@xb=\pgf@x% \pgf@yb=\pgf@y% \pgf@process{#3}% \pgf@xa=\pgf@x% \pgf@ya=\pgf@y% \pgf@process{#2}% % First iteration: \global\pgf@x=\pgf@time@t\pgf@x\global\advance\pgf@x by\pgf@time@s\pgf@xa% \global\pgf@y=\pgf@time@t\pgf@y\global\advance\pgf@y by\pgf@time@s\pgf@ya% \pgf@xa=\pgf@time@t\pgf@xa\advance\pgf@xa by\pgf@time@s\pgf@xb% \pgf@ya=\pgf@time@t\pgf@ya\advance\pgf@ya by\pgf@time@s\pgf@yb% \pgf@xb=\pgf@time@t\pgf@xb\advance\pgf@xb by\pgf@time@s\pgf@xc% \pgf@yb=\pgf@time@t\pgf@yb\advance\pgf@yb by\pgf@time@s\pgf@yc% % Second iteration: \global\pgf@x=\pgf@time@t\pgf@x\global\advance\pgf@x by\pgf@time@s\pgf@xa% \global\pgf@y=\pgf@time@t\pgf@y\global\advance\pgf@y by\pgf@time@s\pgf@ya% \pgf@xa=\pgf@time@t\pgf@xa\advance\pgf@xa by\pgf@time@s\pgf@xb% \pgf@ya=\pgf@time@t\pgf@ya\advance\pgf@ya by\pgf@time@s\pgf@yb% % Save x/y \pgf@xb=\pgf@x% \pgf@yb=\pgf@y% % Third iteration: \global\pgf@x=\pgf@time@t\pgf@x\global\advance\pgf@x by\pgf@time@s\pgf@xa% \global\pgf@y=\pgf@time@t\pgf@y\global\advance\pgf@y by\pgf@time@s\pgf@ya% } % Returns point on an arc at a certain "time" % % #1 = a time % #2 = center of a ellipse % #3 = 0-degree axis of the ellipse % #4 = 90-degree axis of the ellipse % #5 = start angle of an arc on the ellipse % #6 = end angle of an arc on the ellipse % % Result: % % \pgf@x = x-component of place on the arc at time t % \pgf@y = y-component of place on the arc at time t % % Additionally, (\pgf@xa,\pgf@ya) will be a tangent to the % point on the arc (this can be useful for computing a label rotation). % % Example: % % \pgfpointarcaxesattime{0.25}{\pgfpoint{1cm}{1cm}}{\pgfpoint{1cm}{0cm}}{\pgfpoint{0cm}{1cm}}{1cm}{30}{40} \def\pgfpointarcaxesattime#1#2#3#4#5#6{% \pgfmathsetmacro\pgf@angle@start{#5}% \pgfmathsetmacro\pgf@angle@end{#6}% \pgfmathparse{#1}% \global\pgf@x=\pgfmathresult pt% \global\pgf@x=-\pgf@x% \advance\pgf@x by 1pt% \pgfmathsetmacro\pgf@angle@mid{\pgf@angle@end*\pgfmathresult+\pgf@angle@start*\pgf@sys@tonumber{\pgf@x}}% \pgfmathcos@{\pgf@angle@mid}% \let\pgf@angle@cos\pgfmathresult% \pgfmathsin@{\pgf@angle@mid}% \let\pgf@angle@sin\pgfmathresult% \pgf@process{#3}% \edef\pgf@angle@zero@axis{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}} \pgf@process{#4}% \edef\pgf@angle@ninety@axis{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}} % Compute tangent \pgf@process{\pgfpointadd{\pgfpointscale{\pgf@angle@sin}{\pgf@angle@zero@axis}}% {\pgfpointscale{-\pgf@angle@cos}{\pgf@angle@ninety@axis}}}% \ifdim\pgf@angle@start pt>\pgf@angle@end pt% \pgf@xa=\pgf@x% \pgf@ya=\pgf@y% \else% \pgf@xa=-\pgf@x% \pgf@ya=-\pgf@y% \fi% % Compute position \pgf@process{\pgfpointadd{#2}{% \pgfpointadd{\pgfpointscale{\pgf@angle@cos}{\pgf@angle@zero@axis}}% {\pgfpointscale{\pgf@angle@sin}{\pgf@angle@ninety@axis}}}}% } % Internal registers \newdimen\pgf@xx \newdimen\pgf@xy \newdimen\pgf@yx \newdimen\pgf@yy \newdimen\pgf@zx \newdimen\pgf@zy % A polar coordinate % % #1 = a degree % #2 = a radius -- either a dimension or two dimensions separated by % " and ". % % x = (first dimension in #2) * cos(#1) % y = (second dimension in #2) * sin(#2) % % Example: % % \pgfpathmoveto{\pgfpointpolar{30}{1cm}} % \pgfpathlineto{\pgfpointpolar{30}{1cm and 2cm}} \def\pgfpointpolar#1#2{% \pgfutil@in@{and }{#2}% \ifpgfutil@in@% \pgf@polar@#2\@@% \else% \pgf@polar@#2 and #2\@@% \fi% \pgfmathparse{#1}% \let\pgfpoint@angle=\pgfmathresult% \pgfmathcos@{\pgfpoint@angle}% \global\pgf@x=\pgfmathresult\pgf@x% \pgfmathsin@{\pgfpoint@angle}% \global\pgf@y=\pgfmathresult\pgf@y% } \def\pgf@polar@#1and #2\@@{% \pgfmathsetlength{\pgf@y}{#2}% \pgfmathsetlength{\pgf@x}{#1}% } % Quick version of the polar coordinate method \def\pgfqpointpolar#1#2{% \global\pgf@x=#2% \global\pgf@y=\pgf@x% \pgfmathcos@{#1}% \global\pgf@x=\pgfmathresult\pgf@x% \pgfmathsin@{#1}% \global\pgf@y=\pgfmathresult\pgf@y\relax% } % A polar coordinate in the xy plane. % % #1 = a degree % #2 = a radius given as a number or two radii % % result = (first dim in #2) * x-vector * cos(#1) + % (second dim in #2) * y-vector * sin(#1) % % Example: % % \pgfpathmoveto{\pgfpointpolarxy{30}{2}} \def\pgfpointpolarxy#1#2{% \pgfutil@in@{and }{#2}% \ifpgfutil@in@% \pgf@polarxy@#2\@@% \else% \pgf@polarxy@#2and #2\@@% \fi% \pgfmathparse{#1}% \let\pgfpoint@angle=\pgfmathresult% \pgfmathcos@{\pgfpoint@angle}% \pgf@xa=\pgfmathresult\pgf@xa% \pgfmathsin@{\pgfpoint@angle}% \pgf@ya=\pgfmathresult\pgf@ya% \global\pgf@x=\pgf@sys@tonumber{\pgf@xa}\pgf@xx% \global\advance\pgf@x by \pgf@sys@tonumber{\pgf@ya}\pgf@yx% \global\pgf@y=\pgf@sys@tonumber{\pgf@xa}\pgf@xy% \global\advance\pgf@y by \pgf@sys@tonumber{\pgf@ya}\pgf@yy} \def\pgf@polarxy@#1and #2\@@{% \pgfmathsetlength{\pgf@xa}{#1}% \pgfmathsetlength{\pgf@ya}{#2}% } % A cylindrical coordinate. % % #1 = a degree % #2 = a radius given as a number % #3 = a height given as a number % % result = #2*(x-vector * cos(#1) + y-vector * sin(#1)) + #3*z-vector % % Example: % % \pgfpathmoveto{\pgfpointcylindrical{30}{2}{1}} \def\pgfpointcylindrical#1#2#3{% \pgfpointpolarxy{#1}{#2}% \pgfmathparse{#3}% \global\advance\pgf@x by \pgfmathresult\pgf@zx% \global\advance\pgf@y by \pgfmathresult\pgf@zy} % A spherical coordinate. % % #1 = a longitude % #2 = a latitude % #3 = a radius % % result = #3*(cos(#2)*(x-vector * cos(#1) + y-vector * sin(#1)) + sin(#2)*z-vector) % % Example: % % \pgfpathmoveto{\pgfpointspherical{30}{30}{2}} \def\pgfpointspherical#1#2#3{% \pgfmathparse{#1}% \let\pgfpoint@angle=\pgfmathresult% \pgfmathsin@{\pgfpoint@angle}% \pgf@xb=\pgfmathresult\pgf@xx% \pgf@yb=\pgfmathresult\pgf@xy% \pgfmathcos@{\pgfpoint@angle}% \advance\pgf@xb by \pgfmathresult\pgf@yx% \advance\pgf@yb by \pgfmathresult\pgf@yy% % \pgfmathparse{#2}% \let\pgfpoint@angle=\pgfmathresult% \pgfmathcos@{\pgfpoint@angle}% \pgf@xc=\pgfmathresult\pgf@xb% \pgf@yc=\pgfmathresult\pgf@yb% \pgfmathsin@{\pgfpoint@angle}% \advance\pgf@xc by \pgfmathresult\pgf@zx% \advance\pgf@yc by \pgfmathresult\pgf@zy% \pgfmathparse{#3}% \global\pgf@x=\pgfmathresult\pgf@xc% \global\pgf@y=\pgfmathresult\pgf@yc\relax% } % Store the vector #1 * x-vec + #2 * y-vec % % #1 = a factor for the x-vector % #2 = a factor for the y-vector % % x = x-component of result vector % y = y-component of result vector % % Description: % % This command can be used to create a new coordinate system % without using the rotate/translate/scale commands. This % may be useful, if you do not want arrows and line width to % be scaled/transformed together with the coordinate system. % % Example: % % % Create a slanted rectangle % % \pgfsetxvec{\pgfpoint{1cm}{1cm}} % \pgfsetyvec{\pgfpoint{0cm}{1cm}} % % \pgfpathmoveto{\pgfpointxy{0}{0}} % \pgfpathlineto{\pgfpointxy{1}{0}} % \pgfpathlineto{\pgfpointxy{1}{1}} % \pgfpathlineto{\pgfpointxy{0}{1}} % \pgfclosestroke \def\pgfpointxy#1#2{% \pgfmathparse{#1}% \let\pgftemp@x=\pgfmathresult% \pgfmathparse{#2}% \let\pgftemp@y=\pgfmathresult% \global\pgf@x=\pgftemp@x\pgf@xx% \global\advance\pgf@x by \pgftemp@y\pgf@yx% \global\pgf@y=\pgftemp@x\pgf@xy% \global\advance\pgf@y by \pgftemp@y\pgf@yy} % "Quick" variant for \pgfpointxy. % % Only numbers without unit are allowed here. \def\pgfqpointxy#1#2{% \global\pgf@x=#1\pgf@xx% \global\advance\pgf@x by #2\pgf@yx% \global\pgf@y=#1\pgf@xy% \global\advance\pgf@y by #2\pgf@yy} % Store the vector #1 * x-vec + #2 * y-vec + #3 * z-vec % % #1 = a factor for the x-vector % #2 = a factor for the y-vector % #3 = a factor for the z-vector % % x = x-component of result vector % y = y-component of result vector % % % Description: % % This command allows you to use a 3d coordinate system. % % % Example: % % % Draw a cubus % % \pgfline{\pgfpointxyz{0}{0}{0}}{\pgfpointxyz{0}{0}{1}} % \pgfline{\pgfpointxyz{0}{1}{0}}{\pgfpointxyz{0}{1}{1}} % \pgfline{\pgfpointxyz{1}{0}{0}}{\pgfpointxyz{1}{0}{1}} % \pgfline{\pgfpointxyz{1}{1}{0}}{\pgfpointxyz{1}{1}{1}} % \pgfline{\pgfpointxyz{0}{0}{0}}{\pgfpointxyz{0}{1}{0}} % \pgfline{\pgfpointxyz{0}{0}{1}}{\pgfpointxyz{0}{1}{1}} % \pgfline{\pgfpointxyz{1}{0}{0}}{\pgfpointxyz{1}{1}{0}} % \pgfline{\pgfpointxyz{1}{0}{1}}{\pgfpointxyz{1}{1}{1}} % \pgfline{\pgfpointxyz{0}{0}{0}}{\pgfpointxyz{1}{0}{0}} % \pgfline{\pgfpointxyz{0}{0}{1}}{\pgfpointxyz{1}{0}{1}} % \pgfline{\pgfpointxyz{0}{1}{0}}{\pgfpointxyz{1}{1}{0}} % \pgfline{\pgfpointxyz{0}{1}{1}}{\pgfpointxyz{1}{1}{1}} \def\pgfpointxyz#1#2#3{% \pgfmathparse{#1}% \let\pgftemp@x=\pgfmathresult% \pgfmathparse{#2}% \let\pgftemp@y=\pgfmathresult% \pgfmathparse{#3}% \let\pgftemp@z=\pgfmathresult% \global\pgf@x=\pgftemp@x\pgf@xx% \global\advance\pgf@x by \pgftemp@y\pgf@yx% \global\advance\pgf@x by \pgftemp@z\pgf@zx% \global\pgf@y=\pgftemp@x\pgf@xy% \global\advance\pgf@y by \pgftemp@y\pgf@yy% \global\advance\pgf@y by \pgftemp@z\pgf@zy} % "Quick" variant for \pgfpointxyz. % % Only numbers without unit are allowed. \def\pgfqpointxyz#1#2#3{% \global\pgf@x=#1\pgf@xx% \global\advance\pgf@x by #2\pgf@yx% \global\advance\pgf@x by #3\pgf@zx% \global\pgf@y=#1\pgf@xy% \global\advance\pgf@y by #2\pgf@yy% \global\advance\pgf@y by #3\pgf@zy} % Set the x-vector % % #1 = a point the is the new x-vector % % Example: % % \pgfsetxvec{\pgfpoint{1cm}{0cm}} \def\pgfsetxvec#1{% \pgf@process{#1}% \pgf@xx=\pgf@x% \pgf@xy=\pgf@y% \ignorespaces} % Set the y-vector % % #1 = a point the is the new y-vector % % Example: % % \pgfsetyvec{\pgfpoint{0cm}{1cm}} \def\pgfsetyvec#1{% \pgf@process{#1}% \pgf@yx=\pgf@x% \pgf@yy=\pgf@y% \ignorespaces} % Set the z-vector % % #1 = a point the is the new z-vector % % Example: % % \pgfsetzvec{\pgfpoint{-0.385cm}{-0.385cm}} \def\pgfsetzvec#1{% \pgf@process{#1}% \pgf@zx=\pgf@x% \pgf@zy=\pgf@y% \ignorespaces} % Default values \pgfsetxvec{\pgfpoint{1cm}{0cm}} \pgfsetyvec{\pgfpoint{0cm}{1cm}} \pgfsetzvec{\pgfpoint{-0.385cm}{-0.385cm}} % Normalise a point. % % #1 = point with coordinates (a,b) % % x = a/\sqrt(a*a+b*b) % y = b/\sqrt(a*a+b*b) % % Example: % % \pgfpointnormalised{\pgfpointxy{2}{1}} \def\pgfpointnormalised#1{% \pgf@process{#1}% \pgfmathatantwo{\the\pgf@y}{\the\pgf@x}% \let\pgf@tmp=\pgfmathresult% \pgfmathcos@{\pgf@tmp}% \pgf@x=\pgfmathresult pt\relax% \pgfmathsin@{\pgf@tmp}% \pgf@y=\pgfmathresult pt\relax% } % A point on a rectangle in a certain direction. % % #1 = a point pointing in some direction (length should be about 1pt, % but need not be exact) % #2 = upper right corner of a rectangle centered at the origin % % Returns the intersection of a line starting at the origin going in % the given direction and the rectangle's border. % % Example: % % \pgfpointborderrectangle{\pgfpointnormalised{\pgfpointxy{2}{1}} % {\pgfpoint{1cm}{2cm}} \def\pgfpointborderrectangle#1#2{% \pgf@process{#2}% \pgf@xb=\pgf@x% \pgf@yb=\pgf@y% \pgf@process{#1}% % Ok, let's find out about the direction: \pgf@xa=\pgf@x% \pgf@ya=\pgf@y% \ifnum\pgf@xa<0\relax% move into first quadrant \global\pgf@x=-\pgf@x% \fi% \ifnum\pgf@ya<0\relax% \global\pgf@y=-\pgf@y% \fi% \pgf@xc=.125\pgf@x% \pgf@yc=.125\pgf@y% \c@pgf@counta=\pgf@xc% \c@pgf@countb=\pgf@yc% \ifnum\c@pgf@countb<\c@pgf@counta% \ifnum\c@pgf@counta<255\relax% \global\pgf@y=\pgf@yb\relax% \global\pgf@x=0pt\relax% \else% \pgf@xc=8192pt% \divide\pgf@xc by\c@pgf@counta% \pgf@xc = 1/\pgf@x \global\pgf@y=\pgf@sys@tonumber{\pgf@xc}\pgf@y% \global\pgf@y=\pgf@sys@tonumber{\pgf@xb}\pgf@y% \ifnum\pgf@y<\pgf@yb% \global\pgf@x=\pgf@xb% \else% rats, calculate intersection on upper side \ifnum\c@pgf@countb<255\relax% \global\pgf@x=\pgf@xb\relax% \global\pgf@y=0pt\relax% \else% \pgf@yc=8192pt% \divide\pgf@yc by\c@pgf@countb% \pgf@xc = 1/\pgf@x \global\pgf@x=\pgf@sys@tonumber{\pgf@yc}\pgf@x% \global\pgf@x=\pgf@sys@tonumber{\pgf@yb}\pgf@x% \global\pgf@y=\pgf@yb% \fi% \fi% \fi% \else% \ifnum\c@pgf@countb<255\relax% \global\pgf@x=\pgf@xb\relax% \global\pgf@y=0pt\relax% \else% \pgf@yc=8192pt% \divide\pgf@yc by\c@pgf@countb% \pgf@xc = 1/\pgf@x \global\pgf@x=\pgf@sys@tonumber{\pgf@yc}\pgf@x% \global\pgf@x=\pgf@sys@tonumber{\pgf@yb}\pgf@x% \ifnum\pgf@x<\pgf@xb% \global\pgf@y=\pgf@yb% \else% \ifnum\c@pgf@counta<255\relax% \global\pgf@y=\pgf@yb\relax% \global\pgf@x=0pt\relax% \else% \pgf@xc=8192pt% \divide\pgf@xc by\c@pgf@counta% \pgf@xc = 1/\pgf@x \global\pgf@y=\pgf@sys@tonumber{\pgf@xc}\pgf@y% \global\pgf@y=\pgf@sys@tonumber{\pgf@xb}\pgf@y% \global\pgf@x=\pgf@xb% \fi% \fi% \fi% \fi% \ifnum\pgf@xa<0\relax\global\pgf@x=-\pgf@x\fi% \ifnum\pgf@ya<0\relax\global\pgf@y=-\pgf@y\fi% } % An approximation to a point on an ellipse in a certain % direction. Will be exact only if the ellipse is a circle. % % #1 = a point pointing in some direction % #2 = upper right corner of a bounding box for the ellipse % % Returns the intersection of a line starting at the origin going in % the given direction and the ellipses border. % % Example: % % \pgfpointborderellipse{\pgfpointnormalised{\pgfpointxy{2}{1}} % {\pgfpoint{1cm}{2cm}} \def\pgfpointborderellipse#1#2{% \pgf@process{#2}% \pgf@xa=\pgf@x% \pgf@ya=\pgf@y% \ifdim\pgf@xa=\pgf@ya% circle. that's easy! \pgf@process{\pgfpointnormalised{#1}}% \global\pgf@x=\pgf@sys@tonumber{\pgf@xa}\pgf@x% \global\pgf@y=\pgf@sys@tonumber{\pgf@xa}\pgf@y% \else% \ifdim\pgf@xa<\pgf@ya% % Ok, first, let's compute x/y: \c@pgf@countb=\pgf@ya% \divide\c@pgf@countb by65536\relax% \global\divide\pgf@x by\c@pgf@countb% \global\divide\pgf@y by\c@pgf@countb% \pgf@xc=\pgf@x% \pgf@yc=8192pt% \global\pgf@y=.125\pgf@y% \c@pgf@countb=\pgf@y% \divide\pgf@yc by\c@pgf@countb% \pgf@process{#1}% \global\pgf@y=\pgf@sys@tonumber{\pgf@yc}\pgf@y% \global\pgf@y=\pgf@sys@tonumber{\pgf@xc}\pgf@y% \pgf@process{\pgfpointnormalised{}}% \global\pgf@x=\pgf@sys@tonumber{\pgf@xa}\pgf@x% \global\pgf@y=\pgf@sys@tonumber{\pgf@ya}\pgf@y% \else% % Ok, now let's compute y/x: \c@pgf@countb=\pgf@xa% \divide\c@pgf@countb by65536\relax% \global\divide\pgf@x by\c@pgf@countb% \global\divide\pgf@y by\c@pgf@countb% \pgf@yc=\pgf@y% \pgf@xc=8192pt% \global\pgf@x=.125\pgf@x% \c@pgf@countb=\pgf@x% \divide\pgf@xc by\c@pgf@countb% \pgf@process{#1}% \global\pgf@x=\pgf@sys@tonumber{\pgf@yc}\pgf@x% \global\pgf@x=\pgf@sys@tonumber{\pgf@xc}\pgf@x% \pgf@process{\pgfpointnormalised{}}% \global\pgf@x=\pgf@sys@tonumber{\pgf@xa}\pgf@x% \global\pgf@y=\pgf@sys@tonumber{\pgf@ya}\pgf@y% \fi% \fi% } % Extract the x-coordinate of a point to a dimensions % % #1 = a TeX dimension % #2 = a point % % Example: % % \newdimen\mydim % \pgfextractx{\mydim}{\pgfpoint{2cm}{4pt}} % % \mydim is now 2cm \def\pgfextractx#1#2{% \pgf@process{#2}% #1=\pgf@x\relax} % Extract the y-coordinate of a point to a dimensions % % #1 = a TeX dimension % #2 = a point % % Example: % % \newdimen\mydim % \pgfextracty{\mydim}{\pgfpoint{2cm}{4pt}} % % \mydim is now 4pt \def\pgfextracty#1#2{% \pgf@process{#2}% #1=\pgf@y\relax} % Stores the most recently used (x,y) coordinates into two macros, #1 and #2. \def\pgfgetlastxy#1#2{% \edef#1{\the\pgf@x}% \edef#2{\the\pgf@y}% }% \def\pgf@def#1#2#3{\expandafter\def\csname pgf@#1#2\endcsname{#3}} \pgf@def{cosfrac}{0}{1} \pgf@def{cosfrac}{1}{0.99995} \pgf@def{cosfrac}{2}{0.9998} \pgf@def{cosfrac}{3}{0.99955} \pgf@def{cosfrac}{4}{0.999201} \pgf@def{cosfrac}{5}{0.998752} \pgf@def{cosfrac}{6}{0.998205} \pgf@def{cosfrac}{7}{0.997559} \pgf@def{cosfrac}{8}{0.996815} \pgf@def{cosfrac}{9}{0.995974} \pgf@def{cosfrac}{10}{0.995037} \pgf@def{cosfrac}{11}{0.994004} \pgf@def{cosfrac}{12}{0.992877} \pgf@def{cosfrac}{13}{0.991656} \pgf@def{cosfrac}{14}{0.990342} \pgf@def{cosfrac}{15}{0.988936} \pgf@def{cosfrac}{16}{0.987441} \pgf@def{cosfrac}{17}{0.985856} \pgf@def{cosfrac}{18}{0.984183} \pgf@def{cosfrac}{19}{0.982424} \pgf@def{cosfrac}{20}{0.980581} \pgf@def{cosfrac}{21}{0.978653} \pgf@def{cosfrac}{22}{0.976644} \pgf@def{cosfrac}{23}{0.974555} \pgf@def{cosfrac}{24}{0.972387} \pgf@def{cosfrac}{25}{0.970143} \pgf@def{cosfrac}{26}{0.967823} \pgf@def{cosfrac}{27}{0.965429} \pgf@def{cosfrac}{28}{0.962964} \pgf@def{cosfrac}{29}{0.960429} \pgf@def{cosfrac}{30}{0.957826} \pgf@def{cosfrac}{31}{0.955157} \pgf@def{cosfrac}{32}{0.952424} \pgf@def{cosfrac}{33}{0.949629} \pgf@def{cosfrac}{34}{0.946773} \pgf@def{cosfrac}{35}{0.943858} \pgf@def{cosfrac}{36}{0.940887} \pgf@def{cosfrac}{37}{0.937862} \pgf@def{cosfrac}{38}{0.934784} \pgf@def{cosfrac}{39}{0.931655} \pgf@def{cosfrac}{40}{0.928477} \pgf@def{cosfrac}{41}{0.925252} \pgf@def{cosfrac}{42}{0.921982} \pgf@def{cosfrac}{43}{0.918669} \pgf@def{cosfrac}{44}{0.915315} \pgf@def{cosfrac}{45}{0.911922} \pgf@def{cosfrac}{46}{0.90849} \pgf@def{cosfrac}{47}{0.905024} \pgf@def{cosfrac}{48}{0.901523} \pgf@def{cosfrac}{49}{0.89799} \pgf@def{cosfrac}{50}{0.894427} \pgf@def{cosfrac}{51}{0.890835} \pgf@def{cosfrac}{52}{0.887217} \pgf@def{cosfrac}{53}{0.883573} \pgf@def{cosfrac}{54}{0.879905} \pgf@def{cosfrac}{55}{0.876216} \pgf@def{cosfrac}{56}{0.872506} \pgf@def{cosfrac}{57}{0.868777} \pgf@def{cosfrac}{58}{0.865031} \pgf@def{cosfrac}{59}{0.861269} \pgf@def{cosfrac}{60}{0.857493} \pgf@def{cosfrac}{61}{0.853704} \pgf@def{cosfrac}{62}{0.849903} \pgf@def{cosfrac}{63}{0.846092} \pgf@def{cosfrac}{64}{0.842271} \pgf@def{cosfrac}{65}{0.838444} \pgf@def{cosfrac}{66}{0.834609} \pgf@def{cosfrac}{67}{0.83077} \pgf@def{cosfrac}{68}{0.826927} \pgf@def{cosfrac}{69}{0.82308} \pgf@def{cosfrac}{70}{0.819232} \pgf@def{cosfrac}{71}{0.815383} \pgf@def{cosfrac}{72}{0.811534} \pgf@def{cosfrac}{73}{0.807687} \pgf@def{cosfrac}{74}{0.803842} \pgf@def{cosfrac}{75}{0.8} \pgf@def{cosfrac}{76}{0.796162} \pgf@def{cosfrac}{77}{0.792329} \pgf@def{cosfrac}{78}{0.788502} \pgf@def{cosfrac}{79}{0.784682} \pgf@def{cosfrac}{80}{0.780869} \pgf@def{cosfrac}{81}{0.777064} \pgf@def{cosfrac}{82}{0.773268} \pgf@def{cosfrac}{83}{0.769481} \pgf@def{cosfrac}{84}{0.765705} \pgf@def{cosfrac}{85}{0.761939} \pgf@def{cosfrac}{86}{0.758185} \pgf@def{cosfrac}{87}{0.754443} \pgf@def{cosfrac}{88}{0.750714} \pgf@def{cosfrac}{89}{0.746997} \pgf@def{cosfrac}{90}{0.743294} \pgf@def{cosfrac}{91}{0.739605} \pgf@def{cosfrac}{92}{0.735931} \pgf@def{cosfrac}{93}{0.732272} \pgf@def{cosfrac}{94}{0.728628} \pgf@def{cosfrac}{95}{0.724999} \pgf@def{cosfrac}{96}{0.721387} \pgf@def{cosfrac}{97}{0.717792} \pgf@def{cosfrac}{98}{0.714213} \pgf@def{cosfrac}{99}{0.710651} \pgf@def{cosfrac}{100}{0.707107} % Forward declarations for nonlinear stuff (have no effect till module % nonlineartransformations is loaded) \let\pgfpointtransformednonlinear\pgfpointtransformed \endinput