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diff --git a/Master/texmf-dist/tex/generic/pgf/math/pgfmathoperations.code.tex b/Master/texmf-dist/tex/generic/pgf/math/pgfmathoperations.code.tex new file mode 100644 index 00000000000..452b9fa0b2c --- /dev/null +++ b/Master/texmf-dist/tex/generic/pgf/math/pgfmathoperations.code.tex @@ -0,0 +1,1488 @@ +% Copyright 2007 by Mark Wibrow +% +% 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. + +% This file defines the mathematical functions and operators. +% +% Version 1.414213 29/9/2007 + +% This file defines the mathematical functions and operators. +% +% Adding/redefining extra operators/functions: +% +% Each operator/function XXX has two forms: +% +% +% \pgfmathXXX#1... a public version which evaluates any +% arguments passed to it and passes the +% results on to... +% +% \pgfmathXXX@#1... a non-public version which performs +% required calculation on arguments which +% must have already been evaluated (i.e. +% *without* dimensions). +% +% If a function XXX is to be included in the parser, it is +% recommended, for consistency, that where possible, the +% pgfmathparser file should define the macro \pgfmath@parseXXX. +% The parser should (ideally) then call \pgfmathXXX@. +% +% It is also recommened that all calculations (where necessary) +% take place within a TeX group. \pgfmath@returnone#1 makes and +% expanded version of #1 global and stores this in \pgfmathresult +% after the group is ended. +% + +\input pgfmathtrig.code.tex% Load the trig. stuff. +\input pgfmathrnd.code.tex% Load the random stuff. + + +% \pgfmathadd +% +% Add #1 and #2. +% +\def\pgfmathadd#1#2{% + \edef\pgfmath@marshal{% + \noexpand\pgfmathparse{#2} + \noexpand\let\noexpand\pgfmath@arg\noexpand\pgfmathresult% + \noexpand\pgfmathparse{#1}% + }% + \pgfmath@marshal% + \pgfmathadd@{\pgfmathresult}{\pgfmath@arg}} +\def\pgfmathadd@#1#2{% + \begingroup% + \pgfmath@x#1pt% + \pgfmath@y#2pt% + \advance\pgfmath@x by\pgfmath@y% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +% \pgfmathsubtract +% +% Subtract #2 from #1. +% +\def\pgfmathsubtract#1#2{% + \edef\pgfmath@marshal{% + \noexpand\pgfmathparse{#2} + \noexpand\let\noexpand\pgfmath@arg\noexpand\pgfmathresult% + \noexpand\pgfmathparse{#1}% + }% + \pgfmath@marshal% + \pgfmathsubtract@{\pgfmathresult}{\pgfmath@arg}} +\def\pgfmathsubtract@#1#2{% + \begingroup% + \pgfmath@x#1pt% + \pgfmath@y#2pt% + \advance\pgfmath@x-\pgfmath@y% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +% \pgfmathmultiply +% +% Multiply #1 by #2. +% +\def\pgfmathmultiply#1#2{% + \edef\pgfmath@marshal{% + \noexpand\pgfmathparse{#2} + \noexpand\let\noexpand\pgfmath@arg\noexpand\pgfmathresult% + \noexpand\pgfmathparse{#1}% + }% + \pgfmath@marshal% + \pgfmathmultiply@{\pgfmathresult}{\pgfmath@arg}} +\def\pgfmathmultiply@#1#2{% + \begingroup% + \pgfmath@x#1pt% + \pgfmath@y#2pt% + \pgfmath@x\pgfmath@tonumber{\pgfmath@y}\pgfmath@x% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +% \pgfmathdivide +% +% Divide #1 by #2. +% +\def\pgfmathdivide#1#2{% + \edef\pgfmath@marshal{% + \noexpand\pgfmathparse{#2} + \noexpand\let\noexpand\pgfmath@arg\noexpand\pgfmathresult% + \noexpand\pgfmathparse{#1}% + }% + \pgfmath@marshal% + \pgfmathdivide@{\pgfmathresult}{\pgfmath@arg}} +\def\pgfmathdivide@#1#2{% + \begingroup% + \pgfmath@x#1pt% + \pgfmath@y#2pt% + \def\pgfmath@sign{}% + \ifdim\pgfmath@y=0pt\relax% + \pgfmath@error{You've asked me to divide `#1' by `#2', % + but I cannot divide any number by `#2'}% + \fi% + \afterassignment\pgfmath@xa% + \expandafter\c@pgfmath@counta\the\pgfmath@y\relax% + % + % If y is an integer, use TeX arithmatic. + % + \ifdim\pgfmath@xa=0pt\relax% + \divide\pgfmath@x\c@pgfmath@counta\relax% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@x}}% + \let\pgfmath@next\pgfmathdivide@@@% + \else% + % + % Simple long division. + % + \ifdim\pgfmath@x<0pt\relax% + \def\pgfmath@sign{-}% + \pgfmath@x-\pgfmath@x% + \fi% + \ifdim\pgfmath@y<0pt\relax% + \edef\pgfmath@sign{\pgfmath@sign-}% + \pgfmath@y-\pgfmath@y% + \fi% + \pgfmath@ya\pgfmath@y% + \c@pgfmath@counta0\relax% + \ifdim\pgfmath@x>\pgfmath@ya% + \ifdim\pgfmath@ya<1638.4pt\relax% + \pgfmathmultiply@dimenbyten\pgfmath@ya% + \ifdim\pgfmath@ya>\pgfmath@x% + \pgfmathdivide@dimenbyten\pgfmath@ya% + \c@pgfmath@counta0\relax% + \else% + \ifdim\pgfmath@ya<1638.4pt\relax% + \pgfmathmultiply@dimenbyten\pgfmath@ya% + \ifdim\pgfmath@ya>\pgfmath@x% + \pgfmathdivide@dimenbyten\pgfmath@ya% + \c@pgfmath@counta1\relax% + \else% + \ifdim\pgfmath@ya<1638.4pt\relax% + \pgfmathmultiply@dimenbyten\pgfmath@ya% + \ifdim\pgfmath@ya>\pgfmath@x% + \pgfmathdivide@dimenbyten\pgfmath@ya% + \c@pgfmath@counta2\relax% + \else% + \ifdim\pgfmath@ya<1638.4pt\relax% + \pgfmathmultiply@dimenbyten\pgfmath@ya% + \ifdim\pgfmath@ya>\pgfmath@x% + \pgfmathdivide@dimenbyten\pgfmath@ya% + \c@pgfmath@counta3\relax% + \else% + \pgfmath@error{The result of `#1/#2' is too big for me}{}% + \fi% + \fi% + \fi% + \fi% + \fi% + \fi% + \fi% + \fi% + \fi% + % + % If y < 1pt use reciprocal operation. + % + \ifdim\pgfmath@y<1pt\relax% + \ifdim\pgfmath@y<.00007pt\relax% + \pgfmath@error{The result of `#1/#2' is too big for me}{}% + \fi% + \pgfmathreciprocal@{\pgfmath@tonumber{\pgfmath@y}}% + \pgfmath@x\pgfmathresult\pgfmath@x% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@x}}% + \let\pgfmath@next\pgfmathdivide@@@% + \else% + \pgfmath@y\pgfmath@ya% + \def\pgfmathresult{}% + \let\pgfmath@next\pgfmathdivide@@% + \fi% + \fi% + \pgfmath@next% +} + +\def\pgfmathdivide@dimenbyten#1{% + \edef\pgfmath@temp{\pgfmath@tonumber{#1}}% + \expandafter\pgfmathdivide@@dimenbyten\pgfmath@temp\@@#1\@@} +\def\pgfmathdivide@@dimenbyten#1.#2\@@#3\@@{% + \pgfmath@tempcnta#1\relax% + \divide\pgfmath@tempcnta10\relax% + \pgfmath@tempcntb\pgfmath@tempcnta% + \multiply\pgfmath@tempcnta-10\relax% + \advance\pgfmath@tempcnta#1\relax% + #3\the\pgfmath@tempcntb.\the\pgfmath@tempcnta#2pt\relax% +} + +\def\pgfmathmultiply@dimenbyten#1{% + \edef\pgfmath@temp{\pgfmath@tonumber{#1}}% + \expandafter\pgfmathmultiply@@dimenbyten\pgfmath@temp\@@#1\@@} +\def\pgfmathmultiply@@dimenbyten#1.#2#3\@@#4\@@{#4#1#2.#3pt\relax} + +\def\pgfmathdivide@@{% + \ifdim\pgfmath@x>\pgfmath@y% + \pgfmath@yb6.0\pgfmath@y% + \ifdim\pgfmath@yb>\pgfmath@x% 6*y + \advance\pgfmath@yb-\pgfmath@y% + \advance\pgfmath@yb-\pgfmath@y% + \ifdim\pgfmath@yb>\pgfmath@x% 4*y + \advance\pgfmath@yb-\pgfmath@y% + \advance\pgfmath@yb-\pgfmath@y% + \ifdim\pgfmath@yb>\pgfmath@x% 2*y + \advance\pgfmath@yb-\pgfmath@y% + \ifdim\pgfmath@yb>\pgfmath@x% 1*y + \c@pgfmath@countb0\relax% + \else% + \c@pgfmath@countb1\relax% + \fi% + \else% + \advance\pgfmath@yb\pgfmath@y% 3*y + \ifdim\pgfmath@yb>\pgfmath@x% + \c@pgfmath@countb2\relax% + \else% + \c@pgfmath@countb3\relax% + \fi% + \fi% + \else% + \advance\pgfmath@yb\pgfmath@y% 5*y + \ifdim\pgfmath@yb>\pgfmath@x% + \c@pgfmath@countb4\relax% + \else% + \c@pgfmath@countb5\relax% + \fi% + \fi% + \else% + \advance\pgfmath@yb\pgfmath@y% + \advance\pgfmath@yb\pgfmath@y% + \ifdim\pgfmath@yb>\pgfmath@x% 8*y + \advance\pgfmath@yb-\pgfmath@y% + \ifdim\pgfmath@yb>\pgfmath@x% 7*y + \c@pgfmath@countb6\relax% + \else% + \c@pgfmath@countb7\relax% + \fi% + \else% + \advance\pgfmath@yb\pgfmath@y% + \ifdim\pgfmath@yb>\pgfmath@x% 9*y + \c@pgfmath@countb8\relax% + \else% + \c@pgfmath@countb9\relax% + \fi% + \fi% + \fi% + \else% + \c@pgfmath@countb=0\relax% + \fi% + \ifnum\c@pgfmath@countb=0\relax% + \edef\pgfmathresult{\pgfmathresult0}% + \else% + \pgfmath@yb\c@pgfmath@countb\pgfmath@y% + \advance\pgfmath@x-\pgfmath@yb% + \edef\pgfmathresult{\pgfmathresult\the\c@pgfmath@countb}% + \fi% + \pgfmathdivide@dimenbyten\pgfmath@y% + \ifnum\c@pgfmath@counta=0\relax% + \edef\pgfmathresult{\pgfmathresult.}% + \fi% + \advance\c@pgfmath@counta-1\relax% + \ifdim\pgfmath@y=0pt\relax% + \let\pgfmath@next\pgfmathdivide@@@% + \else% + \let\pgfmath@next\pgfmathdivide@@% + \fi% + \pgfmath@next% +} + +\def\pgfmathdivide@@@{% + \pgfmath@x\pgfmath@sign\pgfmathresult pt\relax% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + + +% \pgfmathgreaterthan +% +% 1.0 if #1 > #2. Otherwise 0.0 +% +\def\pgfmathgreaterthan#1#2{% + \edef\pgfmath@marshal{% + \noexpand\pgfmathparse{#2} + \noexpand\let\noexpand\pgfmath@arg\noexpand\pgfmathresult% + \noexpand\pgfmathparse{#1}% + }% + \pgfmath@marshal% + \pgfmathgreaterthan@{\pgfmathresult}{\pgfmath@arg}} +\def\pgfmathgreaterthan@#1#2{% + \begingroup% + \pgfmath@x#1pt% + \pgfmath@y#2pt% + \advance\pgfmath@x-\pgfmath@y% + \ifdim\pgfmath@x>0pt\relax% + \pgfmath@x1pt\relax% + \else% + \pgfmath@x0pt\relax% + \fi% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +% \pgfmathlessthan +% +% 1.0 if #1< #2. Otherwise 0.0 +% +\def\pgfmathlessthan#1#2{% + \edef\pgfmath@marshal{% + \noexpand\pgfmathparse{#2} + \noexpand\let\noexpand\pgfmath@arg\noexpand\pgfmathresult% + \noexpand\pgfmathparse{#1}% + }% + \pgfmath@marshal% + \pgfmathlessthan@{\pgfmathresult}{\pgfmath@arg}} +\def\pgfmathlessthan@#1#2{% + \begingroup% + \pgfmath@x#1pt% + \pgfmath@y#2pt% + \advance\pgfmath@x-\pgfmath@y\relax% + \ifdim\pgfmath@x<0pt\relax% + \pgfmath@x1pt\relax% + \else% + \pgfmath@x0pt\relax% + \fi% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +% \pgfmathequalto +% +% 1.0 if #1 = #2. Otherwise 0.0 +% +\def\pgfmathequalto#1#2{% + \edef\pgfmath@marshal{% + \noexpand\pgfmathparse{#2} + \noexpand\let\noexpand\pgfmath@arg\noexpand\pgfmathresult% + \noexpand\pgfmathparse{#1}% + }% + \pgfmath@marshal% + \pgfmathequalto@{\pgfmathresult}{\pgfmath@arg}} +\def\pgfmathequalto@#1#2{% + \begingroup% + \pgfmath@x#1pt% + \pgfmath@y#2pt% + \advance\pgfmath@x-\pgfmath@y% + \ifdim\pgfmath@x=0pt\relax% + \pgfmath@x1pt\relax% + \else% + \pgfmath@x0pt\relax% + \fi% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +% \pgfmathreciprocal +% +% 1 / #1 +% +\def\pgfmathreciprocal#1{% + \pgfmathparse{#1}% + \pgfmathreciprocal@{\pgfmathresult}} +\def\pgfmathreciprocal@#1{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \ifdim\pgfmath@x=0pt\relax% + \pgfmath@error{You asked me to calculate `1/#1', but I cannot divide any number by zero}{}% + \fi% + \edef\pgfmath@reciprocaltemp{\pgfmath@tonumber{\pgfmath@x}}% + \expandafter\pgfmathreciprocal@@\pgfmath@reciprocaltemp0000000\pgfmath@} +\def\pgfmathreciprocal@@#1.#2#3#4#5#6#7\pgfmath@{% + \c@pgfmath@counta#2#3#4#5#6\relax% + % If the number is an integer, use TeX arithmatic. + \ifnum\c@pgfmath@counta=0\relax% + \pgfmath@x1pt\relax% + \divide\pgfmath@x#1\relax% + \else% + \ifnum#1>100\relax% + \c@pgfmath@counta#1#2#3#4\relax% + \c@pgfmath@countb1000000000\relax% + \divide\c@pgfmath@countb\c@pgfmath@counta% + \c@pgfmath@counta\c@pgfmath@countb% + \divide\c@pgfmath@counta10000\relax% + \pgfmath@x\c@pgfmath@counta pt\relax% + \multiply\c@pgfmath@counta-10000\relax% + \advance\c@pgfmath@countb\c@pgfmath@counta% + \pgfmath@y\c@pgfmath@countb pt\relax% + \divide\pgfmath@y1000000\relax% + \advance\pgfmath@x\pgfmath@y% + \else% + \c@pgfmath@counta#1#2#3#4#5#6\relax% + \c@pgfmath@countb1000000000\relax% + \divide\c@pgfmath@countb\c@pgfmath@counta% + \c@pgfmath@counta\c@pgfmath@countb% + \divide\c@pgfmath@counta10000\relax% + \pgfmath@x\c@pgfmath@counta pt\relax% + \multiply\c@pgfmath@counta-10000\relax% + \advance\c@pgfmath@countb\c@pgfmath@counta% + \pgfmath@y\c@pgfmath@countb pt\relax% + \pgfmath@y.1\pgfmath@y% Yes! This way is more accurate. Go figure... + \pgfmath@y.1\pgfmath@y% + \pgfmath@y.1\pgfmath@y% + \pgfmath@y.1\pgfmath@y% + \advance\pgfmath@x\pgfmath@y% + \fi% + \fi% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +% \pgfmathln +% +% Calculate ln(#1} +% +% This is based on an algorithm due to Rouben Rostamian and +% uses coefficients contributed by Alain Matthes. +% +\def\pgfmathln#1{\pgfmathparse{#1}\pgfmathln@{\pgfmathresult}} +\def\pgfmathln@#1{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \ifdim\pgfmath@x>0pt\else% + \pgfmath@error{I cannot calculate the logarithm of `#1'}{}% + \fi% + \c@pgfmath@counta0\relax% + \ifdim\pgfmath@x>2pt\relax% + \ifdim\pgfmath@x<128pt\relax% + \ifdim\pgfmath@x<8pt\relax% + \ifdim\pgfmath@x<4pt\relax% + \divide\pgfmath@x2\relax% + \c@pgfmath@counta1\relax% + \else% + \divide\pgfmath@x4\relax% + \c@pgfmath@counta2\relax% + \fi% + \else% + \ifdim\pgfmath@x<32pt\relax% + \ifdim\pgfmath@x<16pt\relax% + \divide\pgfmath@x8\relax% + \c@pgfmath@counta3\relax% + \else% + \divide\pgfmath@x16\relax% + \c@pgfmath@counta4\relax% + \fi% + \else% + \ifdim\pgfmath@x<64pt\relax% + \divide\pgfmath@x32\relax% + \c@pgfmath@counta5\relax% + \else% + \divide\pgfmath@x64\relax% + \c@pgfmath@counta6\relax% + \fi% + \fi% + \fi% + \else% + \ifdim\pgfmath@x<2048pt\relax% + \ifdim\pgfmath@x<512pt\relax% + \ifdim\pgfmath@x<256pt\relax% + \divide\pgfmath@x128\relax% + \c@pgfmath@counta7\relax% + \else% + \divide\pgfmath@x256\relax% + \c@pgfmath@counta8\relax% + \fi% + \else% + \ifdim\pgfmath@x<1024pt\relax% + \divide\pgfmath@x512\relax% + \c@pgfmath@counta9\relax% + \else% + \divide\pgfmath@x1024\relax% + \c@pgfmath@counta10\relax% + \fi% + \fi% + \else% + \ifdim\pgfmath@x<8192pt\relax% + \ifdim\pgfmath@x<4096pt\relax% + \divide\pgfmath@x2048\relax% + \c@pgfmath@counta11\relax% + \else% + \divide\pgfmath@x4096\relax% + \c@pgfmath@counta12\relax% + \fi% + \else% + \divide\pgfmath@x8192\relax% + \c@pgfmath@counta13\relax% + \fi% + \fi% + \fi% + \else% + \ifdim\pgfmath@x<1pt\relax% + \ifdim\pgfmath@x>0.0078125pt\relax% 2^-7 + \ifdim\pgfmath@x>0.125pt\relax% 2^-3 + \ifdim\pgfmath@x>0.5pt\relax% 2^-1 + \multiply\pgfmath@x2\relax% + \c@pgfmath@counta-1\relax% + \else% + \multiply\pgfmath@x4\relax% + \c@pgfmath@counta-2\relax% + \fi% + \else% + \ifdim\pgfmath@x>0.03125pt\relax% 2^-5 + \ifdim\pgfmath@x>0.0625pt\relax% + \multiply\pgfmath@x8\relax% + \c@pgfmath@counta-3\relax% + \else% + \multiply\pgfmath@x16\relax% + \c@pgfmath@counta-4\relax% + \fi% + \else% + \ifdim\pgfmath@x>0.015625pt\relax% 2^-6 + \multiply\pgfmath@x32\relax% + \c@pgfmath@counta-5\relax% + \else% + \multiply\pgfmath@x64\relax% + \c@pgfmath@counta-6\relax% + \fi% + \fi% + \fi% + \else% + \ifdim\pgfmath@x>0.000244140625pt\relax% 2^-11 + \ifdim\pgfmath@x>0.001953125pt\relax% 2^-9 + \ifdim\pgfmath@x>0.00390625pt\relax% 2^-8 + \multiply\pgfmath@x128\relax% + \c@pgfmath@counta-7\relax% + \else% + \multiply\pgfmath@x256\relax% + \c@pgfmath@counta-8\relax% + \fi% + \else% + \ifdim\pgfmath@x>0.0009765625pt\relax% 2^-10 + \multiply\pgfmath@x512\relax% + \c@pgfmath@counta-9\relax% + \else% + \multiply\pgfmath@x1024\relax% + \c@pgfmath@counta-10\relax% + \fi% + \fi% + \else% + \ifdim\pgfmath@x>0.0001220703125pt\relax% 2^13 + \ifdim\pgfmath@x>0.0002441406256pt\relax% 2^12 + \multiply\pgfmath@x2048\relax% + \c@pgfmath@counta-11\relax% + \else% + \multiply\pgfmath@x4096\relax% + \c@pgfmath@counta-12\relax% + \fi% + \else% + \multiply\pgfmath@x8192\relax% + \c@pgfmath@counta-13\relax% + \fi% + \fi% + \fi% + \fi% + \fi% + % + % Use A+(B+(C+(D+(E+F*x)*x)*x)*x)*x + % + % where: + % A = -2.787927935 + % B = 5.047861502 + % C = -3.489886985 + % D = 1.589480044 + % E = -.4025153233 + % F = 0.04300521312 + % + \edef\pgfmath@temp{\pgfmath@tonumber{\pgfmath@x}}% + \pgfmath@x0.04300521312\pgfmath@x% + \advance\pgfmath@x-.4025153233pt\relax% + \pgfmath@x\pgfmath@temp\pgfmath@x% + \advance\pgfmath@x1.589480044pt\relax% + \pgfmath@x\pgfmath@temp\pgfmath@x% + \advance\pgfmath@x-3.489886985pt\relax% + \pgfmath@x\pgfmath@temp\pgfmath@x% + \advance\pgfmath@x5.047861502pt\relax% + \pgfmath@x\pgfmath@temp\pgfmath@x% + \advance\pgfmath@x-2.787927935pt\relax% + \advance\pgfmath@x\c@pgfmath@counta pt\relax% + \pgfmath@x0.6931471806\pgfmath@x% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +% \pgfmathabs +% +% Calculate |#1| +% +\def\pgfmathabs#1{% + \pgfmathparse{#1}% + \pgfmathabsolute@{\pgfmathresult}} +\def\pgfmathabs@#1{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \ifdim\pgfmath@x<0pt\relax% + \pgfmath@x-\pgfmath@x% + \fi% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +% \pgfmathmod +% +% Calculate #1 mod #2. +% +\def\pgfmathmod#1#2{% + \edef\pgfmath@marshal{% + \noexpand\pgfmathparse{#2} + \noexpand\let\noexpand\pgfmath@arg\noexpand\pgfmathresult% + \noexpand\pgfmathparse{#1}% + }% + \pgfmath@marshal% + \pgfmathmod@{\pgfmathresult}{\pgfmath@arg}% +} +\def\pgfmathmod@#1#2{% + \begingroup% + \pgfmath@xa#1pt% + \pgfmath@xb#2pt% + \pgfmath@x\pgfmath@xa% + \c@pgfmath@counta\pgfmath@xa% + \c@pgfmath@countb\pgfmath@xb% + \divide\c@pgfmath@counta\c@pgfmath@countb% + \multiply\pgfmath@xb\c@pgfmath@counta% + \advance\pgfmath@x-\pgfmath@xb% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +% \pgfmathsqrt +% +% Square-root of #1. +% +% +\def\pgfmathsqrt#1{\pgfmathparse{#1}\pgfmathsqrt@{\pgfmathresult}} +\def\pgfmathsqrt@#1{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \ifdim\pgfmath@x<0pt\relax% + \pgfmath@error{I cannot calculate the square-root of the negative number `#1'}{}% + \else% + \ifdim\pgfmath@x<10pt\relax% + \def\pgfmath@zeros{0}% + \def\pgfmath@targetiterations{1}% + \else% + \ifdim\pgfmath@x<100pt\relax% + \def\pgfmath@zeros{}% + \def\pgfmath@targetiterations{1}% + \else% + \ifdim\pgfmath@x<1000pt\relax% + \def\pgfmath@zeros{0}% + \def\pgfmath@targetiterations{2}% + \else% + \ifdim\pgfmath@x<10000pt\relax% + \def\pgfmath@zeros{}% + \def\pgfmath@targetiterations{2}% + \else% + \def\pgfmath@zeros{0}% + \def\pgfmath@targetiterations{3}% + \fi\fi\fi\fi\fi% + \edef\pgfmath@temp{\pgfmath@zeros\pgfmath@tonumber{\pgfmath@x}}% + \expandafter\pgfmath@sqrt@\pgfmath@temp\pgfmath@% +} +\def\pgfmath@sqrt@#1.#2\pgfmath@{\pgfmath@@sqrt@#1#2\pgfmath@empty\pgfmath@empty\pgfmath@} + +\def\pgfmath@@sqrt@#1#2{% + \c@pgfmath@countb#1#2\relax% + \ifnum\c@pgfmath@countb>35\relax% + \ifnum\c@pgfmath@countb>63\relax% + \ifnum\c@pgfmath@countb>80\relax% + \c@pgfmath@counta9\relax% + \else% + \c@pgfmath@counta8\relax% + \fi% + \else% + \ifnum\c@pgfmath@countb>48\relax% + \c@pgfmath@counta7\relax% + \else% + \c@pgfmath@counta6\relax% + \fi% + \fi% + \else% + \ifnum\c@pgfmath@countb>15\relax% + \ifnum\c@pgfmath@countb>24\relax% + \c@pgfmath@counta5\relax% + \else% + \c@pgfmath@counta4\relax% + \fi% + \else% + \ifnum\c@pgfmath@countb>3\relax% + \ifnum\c@pgfmath@countb>8\relax% + \c@pgfmath@counta3\relax% + \else% + \c@pgfmath@counta2\relax% + \fi% + \else% + \ifnum\c@pgfmath@countb>0\relax% + \c@pgfmath@counta1\relax% + \else% + \c@pgfmath@counta0\relax% + \fi% + \fi% + \fi% + \fi% + \edef\pgfmath@root{\the\c@pgfmath@counta}% + \edef\pgfmath@rootspecial{\the\c@pgfmath@counta}% + \multiply\c@pgfmath@counta-\c@pgfmath@counta\relax% + \advance\c@pgfmath@counta#1#2\relax% + \edef\pgfmath@remainder{\the\c@pgfmath@counta}% + \pgfmath@@@sqrt@% +} + +\def\pgfmath@@@sqrt@#1#2{% + \ifx\pgfmath@empty#1% + \edef\pgfmath@remainder{\pgfmath@remainder00}% + \def\pgfmath@tokens{\pgfmath@empty\pgfmath@empty}% + \else% + \ifx\pgfmath@empty#2% + \edef\pgfmath@remainder{\pgfmath@remainder#10}% + \def\pgfmath@tokens{\pgfmath@empty\pgfmath@empty}% + \else% + \edef\pgfmath@remainder{\pgfmath@remainder#1#2}% + \def\pgfmath@tokens{}% + \fi\fi% + \c@pgfmath@counta\pgfmath@rootspecial\relax% + \multiply\c@pgfmath@counta20\relax% + \c@pgfmath@countb\c@pgfmath@counta% + \multiply\c@pgfmath@countb6\relax% + \advance\c@pgfmath@countb36\relax% + \c@pgfmath@countc\c@pgfmath@counta\relax% + \multiply\c@pgfmath@countc2\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \advance\c@pgfmath@countb-\c@pgfmath@countc% + \advance\c@pgfmath@countb-20\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \advance\c@pgfmath@countb-\c@pgfmath@countc% + \advance\c@pgfmath@countb-12\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \advance\c@pgfmath@countb-\c@pgfmath@counta% + \advance\c@pgfmath@countb-3\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \def\pgfmath@digit{0}% + \else% + \def\pgfmath@digit{1}% + \fi% + \else% + \advance\c@pgfmath@countb\c@pgfmath@counta% + \advance\c@pgfmath@countb5\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \def\pgfmath@digit{2}% + \else% + \def\pgfmath@digit{3}% + \fi% + \fi% + \else% + \advance\c@pgfmath@countb\c@pgfmath@counta% + \advance\c@pgfmath@countb9\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \def\pgfmath@digit{4}% + \else% + \def\pgfmath@digit{5}% + \fi% + \fi% + \else% + \advance\c@pgfmath@countb\c@pgfmath@countc% + \advance\c@pgfmath@countb28\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \advance\c@pgfmath@countb-\c@pgfmath@counta% + \advance\c@pgfmath@countb-15\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \def\pgfmath@digit{6}% + \else% + \def\pgfmath@digit{7}% + \fi% + \else% + \advance\c@pgfmath@countb\c@pgfmath@counta% + \advance\c@pgfmath@countb17\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \def\pgfmath@digit{8}% + \else% + \def\pgfmath@digit{9}% + \fi% + \fi% + \fi% + \edef\pgfmath@rootspecial{\pgfmath@rootspecial\pgfmath@digit}% + \advance\c@pgfmath@counta\pgfmath@digit\relax% + \multiply\c@pgfmath@counta-\pgfmath@digit\relax% + \advance\c@pgfmath@counta\pgfmath@remainder\relax% + \edef\pgfmath@remainder{\the\c@pgfmath@counta}% + \c@pgfmath@counta\pgfmath@targetiterations\relax% + \advance\c@pgfmath@counta-1\relax% + \edef\pgfmath@targetiterations{\the\c@pgfmath@counta}% + \ifnum\c@pgfmath@counta=0\relax% + \edef\pgfmath@root{\pgfmath@root.\pgfmath@digit}% + \else% + \edef\pgfmath@root{\pgfmath@root\pgfmath@digit}% + \fi% + \ifnum\c@pgfmath@counta=-4\relax% -n+1 digits of precision. + \let\pgfmath@next\pgfmath@sqrt@end% + \else% + \let\pgfmath@next\pgfmath@@@sqrt@% + \fi% + \expandafter\pgfmath@next\pgfmath@tokens% +} + +\def\pgfmath@sqrt@end#1\pgfmath@{% + \edef\pgfmathresult{\pgfmath@root}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup} + +% \pgfmathpow +% +% Calculates #1 ^ #2 +% +\def\pgfmathpow#1#2{% + \edef\pgfmath@marshal{% + \noexpand\pgfmathparse{#2} + \noexpand\let\noexpand\pgfmath@arg\noexpand\pgfmathresult% + \noexpand\pgfmathparse{#1}% + }% + \pgfmath@marshal% + \pgfmathpow@{\pgfmathresult}{\pgfmath@arg}} +\def\pgfmathpow@#1#2{% + \begingroup% + \pgfmath@xa#1pt% + \pgfmath@xb#2pt% + \afterassignment\pgfmath@x% + \expandafter\c@pgfmath@counta\the\pgfmath@xb\relax% + \ifnum\c@pgfmath@counta<0\relax% + \c@pgfmath@counta-\c@pgfmath@counta% + \pgfmathreciprocal@{#1}% + \pgfmath@xa\pgfmathresult pt\relax% + \fi + \ifdim\pgfmath@x=0pt\relax% + \pgfmath@x1pt\relax% + \pgfmathloop% + \ifnum\c@pgfmath@counta>0\relax% + \ifodd\c@pgfmath@counta% + \pgfmath@x\pgfmath@tonumber{\pgfmath@x}\pgfmath@xa% + \fi + \ifnum\c@pgfmath@counta>1\relax% + \pgfmath@xa\pgfmath@tonumber{\pgfmath@xa}\pgfmath@xa% + \fi% + \divide\c@pgfmath@counta2\relax% + \repeatpgfmathloop% + \else% + \pgfmathln@{#1}% + \pgfmath@x\pgfmathresult pt\relax% + \pgfmath@x\pgfmath@tonumber{\pgfmath@xb}\pgfmath@x% + \pgfmathexp@{\pgfmath@tonumber{\pgfmath@x}}% + \pgfmath@x\pgfmathresult pt\relax% + \fi% + \pgfmath@returnone\pgf@x% + \endgroup% +} + +% \pgfmathround +% +% Asymmetric Half-up rounding. +% +\def\pgfmathround#1{\pgfmathparse{#1}\pgfmathround@{\pgfmathresult}} +\def\pgfmathround@#1{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \afterassignment\pgfmath@xa% + \expandafter\c@pgfmath@counta\the\pgfmath@x\relax% + \pgfmath@xb\pgfmath@x% + \ifdim\pgfmath@xb<0pt\relax% + \ifdim\pgfmath@xa<0.5pt\relax% + \else% + \advance\c@pgfmath@counta-1\relax% + \fi% + \else% + \ifdim\pgfmath@xa<0.5pt\relax% + \else% + \advance\c@pgfmath@counta1\relax% + \fi% + \fi% + \pgfmath@returnone\the\c@pgfmath@counta pt% + \endgroup% +}% + +% \pgfmathfloor +% +% Floor function. +% +\def\pgfmathfloor#1{\pgfmathparse{#1}\pgfmathfloor@{\pgfmathresult}} +\def\pgfmathfloor@#1{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \afterassignment\pgfmath@gobbletilpgfmath@% + \expandafter\c@pgfmath@counta\the\pgfmath@x\relax\pgfmath@% + \pgfmath@x\c@pgfmath@counta pt\relax% + \pgfmath@returnone\pgfmath@x% + \endgroup% +}% + +% \pgfmathceil +% +% Ceiling function. +% +\def\pgfmathceil#1{\pgfmathparse{#1}\pgfmathceil@{\pgfmathresult}} +\def\pgfmathceil@#1{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \afterassignment\pgfmath@gobbletilpgfmath@% + \expandafter\c@pgfmath@counta\the\pgfmath@x\relax\pgfmath@% + \pgfmath@y\pgfmath@x% + \advance\pgfmath@y-\c@pgfmath@counta pt\relax% + \pgfmath@x\c@pgfmath@counta pt\relax% + \ifdim\pgfmath@y>0pt\relax% + \advance\pgfmath@x1pt\relax% + \fi% + \pgfmath@returnone\pgfmath@x% + \endgroup% +}% + +% \pgfmathexp +% +% A Maclaurens expansion for e^#1. +% 0 <= #1 < ln(16384). +% +\def\pgfmathexp#1{\pgfmathparse{#1}\pgfmathexp@{\pgfmathresult}} +\def\pgfmathexp@#1{% + \begingroup% + \edef\pgfmath@arg{#1}% + \pgfmath@x1pt\relax% + \pgfmath@xa1pt\relax% + \pgfmath@xb\pgfmath@x% + \pgfmathloop% + \pgfmath@xc\pgfmathcounter pt\relax% + \c@pgfmath@counta\pgfmath@xc% + \divide\c@pgfmath@counta65536\relax% + \pgfmath@xc1pt\relax% + \divide\pgfmath@xc\c@pgfmath@counta% + \pgfmath@xa\pgfmath@tonumber{\pgfmath@xc}\pgfmath@xa% + \expandafter\pgfmath@xa\pgfmath@arg\pgfmath@xa% + \advance\pgfmath@x\pgfmath@xa% + \ifdim\pgfmath@x=\pgfmath@xb% + \else% + \pgfmath@xb\pgfmath@x% + \repeatpgfmathloop% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + + + +% \pgfmathvectorlength +% +% Calcluate the Eulidean length of a 2D vector. +% +% This based on polynomial approximation co-efficents +% contributed by Rouben Rostamian. +% +% #1 - the x component of the vector. +% #2 - the y component of the vector. +% +\def\pgfmathveclen#1#2{% + \edef\pgfmath@marshal{% + \noexpand\pgfmathparse{#2} + \noexpand\let\noexpand\pgfmath@arg\noexpand\pgfmathresult% + \noexpand\pgfmathparse{#1}% + }% + \pgfmath@marshal% + \pgfmathveclen@{\pgfmathresult}{\pgfmath@arg}% +} +\def\pgfmathveclen@#1#2{% + \begingroup% + \pgfmath@x#1pt\relax% + \pgfmath@y#2pt\relax% + \ifdim\pgfmath@x<0pt\relax% + \pgfmath@x-\pgfmath@x% + \fi% + \ifdim\pgfmath@y<0pt\relax% + \pgfmath@y-\pgfmath@y% + \fi% + \ifdim\pgfmath@x=0pt\relax% + \pgfmath@x\pgfmath@y% + \else% + \ifdim\pgfmath@y=0pt\relax% + \else% + \ifdim\pgfmath@x>\pgfmath@y% + \pgfmath@xa\pgfmath@x% + \pgfmath@x\pgfmath@y% + \pgfmath@y\pgfmath@xa% + \fi% + % We use a scaling factor to reduce errors. + \ifdim\pgfmath@y>10000pt\relax% + \c@pgfmath@counta1500\relax% + \else% + \ifdim\pgfmath@y>1000pt\relax% + \c@pgfmath@counta150\relax% + \else% + \ifdim\pgfmath@y>100pt\relax% + \c@pgfmath@counta50\relax% + \else% + \c@pgfmath@counta1\relax% + \fi% + \fi% + \fi% + \divide\pgfmath@x\c@pgfmath@counta\relax% + \divide\pgfmath@y\c@pgfmath@counta\relax% + \pgfmathreciprocal@{\pgfmath@tonumber{\pgfmath@y}}% + \pgfmath@x\pgfmathresult\pgfmath@x% + \pgfmath@xa\pgfmath@tonumber{\pgfmath@x}\pgfmath@x% + \edef\pgfmath@temp{\pgfmath@tonumber{\pgfmath@xa}}% + % + % Use A+x^2*(B+x^2*(C+x^2*(D+E*x^2))) + % where + % A = +1.000012594 + % B = +0.4993615349 + % C = -0.1195159052 + % D = +0.04453994279 + % E = -0.01019210944 + % + \pgfmath@x-0.01019210944\pgfmath@xa% + \advance\pgfmath@x0.04453994279pt\relax% + \pgfmath@x\pgfmath@temp\pgfmath@x% + \advance\pgfmath@x-0.1195159052pt\relax% + \pgfmath@x\pgfmath@temp\pgfmath@x% + \advance\pgfmath@x0.4993615349pt\relax% + \pgfmath@x\pgfmath@temp\pgfmath@x% + \advance\pgfmath@x1.000012594pt\relax% + \ifdim\pgfmath@y<0pt\relax% + \pgfmath@y-\pgfmath@y% + \fi% + \pgfmath@x\pgfmath@tonumber{\pgfmath@y}\pgfmath@x% + % Invert the scaling factor. + \multiply\pgfmath@x\c@pgfmath@counta\relax% + \fi% + \fi% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +% \pgfmathmax +% +% Return the maximum of #1 or #2 +% +\def\pgfmathmax#1#2{% + \edef\pgfmath@marshal{% + \noexpand\pgfmathparse{#2} + \noexpand\let\noexpand\pgfmath@arg\noexpand\pgfmathresult% + \noexpand\pgfmathparse{#1}% + }% + \pgfmath@marshal% + \pgfmathmax@{\pgfmathresult}{\pgfmath@arg}}% +\def\pgfmathmax@#1#2{% + \begingroup + \pgfmath@x#1pt% + \pgfmath@y#2pt\relax% + \ifdim\pgfmath@x>\pgfmath@y% + \pgfmath@returnone\pgfmath@x% + \else% + \pgfmath@returnone\pgfmath@y% + \fi% + \endgroup +} + +% \pgfmathmax +% +% Return the minimim of #1 or #2 +% +\def\pgfmathmin#1#2{% + \edef\pgfmath@marshal{% + \noexpand\pgfmathparse{#2} + \noexpand\let\noexpand\pgfmath@arg\noexpand\pgfmathresult% + \noexpand\pgfmathparse{#1}% + }% + \pgfmath@marshal% + \pgfmathmin@{\pgfmathresult}{\pgfmath@arg}}% +\def\pgfmathmin@#1#2{% + \begingroup + \pgfmath@x#1pt% + \pgfmath@y#2pt% + \relax% + \ifdim\pgfmath@x<\pgfmath@y% + \pgfmath@returnone\pgfmath@x% + \else% + \pgfmath@returnone\pgfmath@y% + \fi% + \endgroup% +} + +% \pgfmathscientific +% +% Return the value of #1e#2 +% +% e.g. \pgfmathscientific{1.23456789123}{4} +% +% defines \pgfmathresult as 12345.67891 +% +% NB These arguments *are not parsed*, as the long mantissa would be lost. +% +\def\pgfmathscientific#1#2{% + \begingroup% + \edef\pgfmath@sci@exponent{#2}% + \pgfmath@x#1pt\relax% + \ifdim\pgfmath@x<0pt\relax% + \pgfmath@x-\pgfmath@x% + \edef\pgfmath@sci@sign{-}% + \edef\pgfmath@temp{\pgfmath@gobbleone#1}% + \else% + \edef\pgfmath@sci@sign{+}% + \edef\pgfmath@temp{#1}% + \fi% + \expandafter\pgfmath@scientific@@\pgfmath@temp00000000000\pgfmath@} + +\def\pgfmath@scientific@@#1.#2#3#4#5#6{% + \edef\pgfmath@sci@int{#1}% + \edef\pgfmath@sci@mantissaA{#2#3#4#5#6}% + \pgfmath@scientific@@@} + +\def\pgfmath@scientific@@@#1#2#3#4#5#6\pgfmath@{% + \edef\pgfmath@sci@mantissaB{#1#2#3#4#5}% + \c@pgfmath@counta\pgfmath@sci@exponent\relax% + \c@pgfmath@countb\c@pgfmath@counta% + \ifnum\c@pgfmath@counta<0\relax% + \c@pgfmath@counta-\c@pgfmath@counta% + \fi% + \pgfmathpow@{10}{\the\c@pgfmath@counta}% + \afterassignment\pgfmath@gobbletilpgfmath@ + \c@pgfmath@countc\pgfmathresult\relax\pgfmath@ + \edef\pgfmath@sci@factor{\the\c@pgfmath@countc}% + \ifnum\c@pgfmath@countb<0\relax% + % xE-y: easy... + \pgfmath@x\pgfmath@sci@int.\pgfmath@sci@mantissaA pt\relax% + \divide\pgfmath@x\pgfmath@sci@factor\relax% + \else% + % xE+y: + % Must do this way so as not lose digits in a long mantissa. Sigh... + \c@pgfmath@counta\pgfmath@sci@int% + \c@pgfmath@countb\pgfmath@sci@mantissaA% + \multiply\c@pgfmath@counta\pgfmath@sci@factor\relax% + \multiply\c@pgfmath@countb\pgfmath@sci@factor\relax% + \c@pgfmath@countc\c@pgfmath@countb% + \divide\c@pgfmath@countb100000\relax% + \advance\c@pgfmath@counta\c@pgfmath@countb% + \multiply\c@pgfmath@countb100000\relax% + \advance\c@pgfmath@countc-\c@pgfmath@countb% + \c@pgfmath@countb\pgfmath@sci@mantissaB\relax% + \multiply\c@pgfmath@countb\pgfmath@sci@factor\relax% + \divide\c@pgfmath@countb100000\relax% + \advance\c@pgfmath@countc\c@pgfmath@countb% + \advance\c@pgfmath@countc100000\relax% + \edef\pgfmath@sci@result{\pgfmath@sci@sign\the\c@pgfmath@counta.\expandafter\pgfmath@gobbleone\the\c@pgfmath@countc pt}% + \pgfmath@x\pgfmath@sci@result\relax% + \fi% + \pgfmath@returnone\pgfmath@x% + \endgroup} + + + + + +% *** The following commands DO NOT WORK without the rest of PGF *** +% +% (a dumping ground for stuff that doesn't really belong anywhere else) + +% \pgfmathanglebetweenpoints +% +% Define \pgfmathresult as the angle between points #1 and #2 +% Should get the quadrants right as well. +% +\def\pgfmathanglebetweenpoints#1#2{% + \begingroup% + \pgf@process{\pgfpointdiff{#1}{#2}}% + % + % First approximate the angle of the external point... + % + \pgf@xa\pgf@x% + \pgf@ya\pgf@y% + \pgf@xb\pgf@x% + \pgf@yb\pgf@y% + \ifdim\pgf@xa<0pt\relax% + \pgf@xa-\pgf@xa% + \fi% + \ifdim\pgf@ya<0pt\relax% + \pgf@ya-\pgf@ya% + \fi% + \ifdim\pgf@ya>\pgf@xa% + \pgf@x\pgf@xa% + \pgf@y\pgf@ya% + \else% + \pgf@x\pgf@ya% + \pgf@y\pgf@xa% + \fi% + \ifdim\pgf@y=0pt\relax% + \pgf@x0pt% + \else% + \pgfmathreciprocal@{\pgfmath@tonumber{\pgf@y}}% + \pgf@x\pgfmathresult\pgf@x% + \fi% + \multiply\pgf@x1000\relax% + \afterassignment\pgfmath@gobbletilpgfmath@% + \expandafter\c@pgf@counta\the\pgf@x\relax\pgfmath@% + \expandafter\pgf@x\csname pgfmath@atan@\the\c@pgf@counta\endcsname pt\relax% + \ifdim\pgfmath@ya>\pgfmath@xa\relax% + \pgf@x-\pgf@x% + \advance\pgf@x90pt% + \fi% + \ifdim\pgf@xb<0pt% + \ifdim\pgf@yb>0pt% + \pgf@x-\pgf@x% + \fi% + \advance\pgf@x180pt\relax% + \else% + \ifdim\pgf@yb<0pt% + \pgf@x-\pgf@x% + \advance\pgf@x360pt\relax% + \fi\fi% + \pgfmath@returnone\pgf@x% + \endgroup% +} + +% +% \pgfmathrotatepointaround +% +% Rotate point #1 about point #2 by #3 degrees. +% +\def\pgfmathrotatepointaround#1#2#3{% + \pgf@process{% + \pgf@process{#2}% + \pgf@xa\pgf@x% + \pgf@ya\pgf@y% + \pgf@xb\pgf@x% + \pgf@yb\pgf@y% + \pgf@process{#1}% + \advance\pgf@x-\pgf@xa% + \advance\pgf@y-\pgf@ya% + \pgfmathsetmacro\angle{#3}% + \pgfmathsin@{\angle}% + \let\sineangle\pgfmathresult% + \pgfmathcos@{\angle}% + \let\cosineangle\pgfmathresult% + \pgf@xa\cosineangle\pgf@x% + \advance\pgf@xa-\sineangle\pgf@y% + \pgf@ya\sineangle\pgf@x% + \advance\pgf@ya\cosineangle\pgf@y% + \pgf@x\pgf@xb% + \pgf@y\pgf@yb% + \advance\pgf@x\pgf@xa% + \advance\pgf@y\pgf@ya% + }% +} + +% \pgfmathanglebetweenlines +% +% Calculate the clockwise angle between a line from point #1 +% to point #2 and a line from #3 to point #4. +% +\def\pgfmathanglebetweenlines#1#2#3#4{% + \begingroup% + \pgfmathanglebetweenpoints{#1}{#2}% + \let\firstangle\pgfmathresult% + \pgfmathanglebetweenpoints{#3}{#4}% + \let\secondangle\pgfmathresult% + \ifdim\firstangle pt>\secondangle pt\relax% + \pgfmathadd@{\secondangle}{360}% + \let\secondangle\pgfmathresult% + \fi% + \pgfmathsubtract@{\secondangle}{\firstangle}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + +% \pgfmathpointreflectalongaxis +% +% Reflects point #2 around an axis centered on #2 at an angle #3. +% +\def\pgfmathreflectpointalongaxis#1#2#3{% + \pgf@process{% + \pgfmathanglebetweenpoints{#2}{#1}% + \pgfmath@tempdima\pgfmathresult pt\relax% + \pgfmathparse{#3}% + \advance\pgfmath@tempdima-\pgfmathresult pt\relax% + \pgfmath@tempdima-2.0\pgfmath@tempdima% + \pgfmathrotatepointaround{#1}{#2}{\pgfmath@tonumber{\pgfmath@tempdima}}% + }% +} + + +% \pgfmathpointintersectionoflineandarc +% +% A bit experimental at the moment: +% +% Locates the point where a line crosses an eliptical arc. If the line +% does not cross the arc, a meaningless point will result. +% +% #1 the point of the line on the "convex" side of the arc. +% #2 the point of the line on the "concave" side of the arc. +% #3 the center of the eliptical arc. +% #4 start angle of the arc. +% #5 end angle of the arc. +% #6 radii of the arc. +% +\def\pgfmathpointintersectionoflineandarc#1#2#3#4#5#6{% + \pgf@process{% + % + % Get the required angle. + % + \pgfmathanglebetweenpoints{#2}{#1}% + \let\x\pgfmathresult% + % + % Get the radii of the arc. + % + \pgfmath@in@{and }{#6}% + \ifpgfmath@in@% + \pgf@polar@#6\@@% + \else% + \pgf@polar@#6 and #6\@@% + \fi% + \edef\xarcradius{\the\pgf@x}% + \edef\yarcradius{\the\pgf@y}% + % + % Get the start and end angles of the arc... + % + \pgfmathsetmacro\s{#4}% + \pgfmathsetmacro\e{#5}% + % + % ...and also with rounding. + % + \pgfmathmod@{\s}{360}% + \ifdim\pgfmathresult pt<0pt\relax% + \pgfmathadd@{\pgfmathresult}{360}% + \fi% + \let\ss\pgfmathresult% + \pgfmathmod@{\e}{360}% + \ifdim\pgfmathresult pt<0pt\relax% + \pgfmathadd@{\pgfmathresult}{360}% + \fi% + \let\ee\pgfmathresult% + % + % Hackery for when arc straddles zero. + % + \ifdim\ee pt<\ss pt\relax% + \pgfmathadd@{\x}{180}% + \pgfmathmod@{\pgfmathresult}{360}% + \let\x\pgfmathresult% + \fi% + \def\m{360}% Measure of nearness. + \pgfmathadd@{\s}{\e}% + \pgfmathdivide@{\pgfmathresult}{2}% + \let\n\pgfmathresult% The best estimate (default to middle of arc). + \pgfmathloop% + \pgfmathadd@{\s}{\e}% + \pgfmathdivide@{\pgfmathresult}{2}% + \let\p\pgfmathresult% + \ifdim\p pt=\s pt\relax% + \else% + \pgfmathanglebetweenpoints{#2}{% + \pgfpointadd{#3}{% + \pgf@x\xarcradius\relax% + \pgfmathcos@{\p}% + \pgf@x\pgfmathresult\pgf@x% + \pgf@y\yarcradius\relax% + \pgfmathsin@{\p}% + \pgf@y\pgfmathresult\pgf@y% + }% + }% + % + % Hackery for when arc straddles zero. + % + \ifdim\ee pt<\ss pt\relax% + \pgfmathadd@{\pgfmathresult}{180}% + \pgfmathmod@{\pgfmathresult}{360}% + \fi% + \let\q\pgfmathresult% + % + % More hackery... + % + \ifdim\x pt>335pt\relax% + \ifdim\q pt<45pt\relax% + \pgfmathadd@{\q}{360}% + \let\q\pgfmathresult% + \fi% + \fi% + \ifdim\x pt=\q pt% Found it! + \pgfmathbreakloop% Breaks after current iteration is complete. + \else + \ifdim\x pt<\q pt\relax% + \let\e\p% + \else% + \let\s\p% + \fi% + \fi% + \pgfmathsubtract@{\x}{\q}% + \pgfmathabs@{\pgfmathresult}% + % + % Save the estimate if it is better than any previous estimate. + % + \ifdim\pgfmathresult pt<\m pt\relax% + \let\m\pgfmathresult% + \let\n\p% + \fi% + \repeatpgfmathloop% + \pgfpointadd{#3}{\pgfpointpolar{\n}{\xarcradius and \yarcradius}}% + }% +} + +% \pgfmathangleonellipse +% +% Find the angle corresponding to a point on the border of an ellispe. +% +% #1 - the point on the border. +% #2 - the radii of the ellipse. +% +\def\pgfmathangleonellipse#1#2{% + \begingroup% + \pgfmath@in@{and }{#2}% + \ifpgfmath@in@% + \pgf@polar@#2\@@% + \else% + \pgf@polar@#2 and #2\@@% + \fi% + \pgf@xa\pgf@x% + \pgf@ya\pgf@y% + \pgf@process{#1}% + \ifdim\pgf@x=0pt\relax% + \pgfutil@tempdima1pt\relax% + \else% + \pgfutil@tempdima\pgf@x% + \pgfmathdivide@{\pgfmath@tonumber{\pgf@xa}}{\pgfmath@tonumber{\pgfutil@tempdima}}% + \pgfutil@tempdima\pgfmathresult pt\relax% + \fi% + \ifdim\pgf@y=0pt\relax% + \pgfutil@tempdima1pt\relax% + \else% + \pgfmathdivide@{\pgfmath@tonumber{\pgf@y}}{\pgfmath@tonumber{\pgf@ya}}% + \pgfutil@tempdima\pgfmathresult\pgfutil@tempdima% + \pgfmathatan@{\pgfmath@tonumber{\pgfutil@tempdima}}% + \fi% + % + \pgfutil@tempdima\pgfmathresult pt\relax% + \ifdim\pgfutil@tempdima<0pt\relax% + \advance\pgfutil@tempdima360pt\relax% + \fi% + \ifdim\pgf@x<0pt\relax% + \ifdim\pgf@y=0pt\relax% + \pgfutil@tempdima180pt\relax% + \else% + \ifdim\pgf@y<0pt\relax% + \advance\pgfutil@tempdima180pt\relax% + \else% + \advance\pgfutil@tempdima-180pt\relax% + \fi% + \fi% + \else% + \ifdim\pgf@x=0pt\relax% + \ifdim\pgf@y<0pt\relax% + \pgfutil@tempdima270pt\relax% + \else% + \pgfutil@tempdima90pt\relax% + \fi% + \else% + \ifdim\pgf@y=0pt\relax% + \pgfutil@tempdima0pt\relax% + \fi% + \fi% + \fi% + \pgfmath@returnone\pgfutil@tempdima% + \endgroup% +}
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