diff options
Diffstat (limited to 'Master/texmf-dist/source/generic/mfpic/grafbase.dtx')
-rw-r--r-- | Master/texmf-dist/source/generic/mfpic/grafbase.dtx | 193 |
1 files changed, 163 insertions, 30 deletions
diff --git a/Master/texmf-dist/source/generic/mfpic/grafbase.dtx b/Master/texmf-dist/source/generic/mfpic/grafbase.dtx index 32d14a5b676..a309f242f6e 100644 --- a/Master/texmf-dist/source/generic/mfpic/grafbase.dtx +++ b/Master/texmf-dist/source/generic/mfpic/grafbase.dtx @@ -1,10 +1,10 @@ % \iffalse -%%% File: grafbase.dtx -%%% A part of mfpic 1.05 2010/06/10 -%%% +% File: grafbase.dtx +% A part of mfpic 1.06 2011/03/08 +% % ------------------------------------------------------------------- % -% Copyright 2002--2010, Daniel H. Luecking +% Copyright 2002--2011, Daniel H. Luecking % % Mfpic may be distributed and/or modified under the conditions of the % LaTeX Project Public License, either version 1.3b of this license or (at @@ -19,7 +19,7 @@ % %<*driver> \ProvidesFile{grafbase.dtx} - [2010/06/10 v1.05. Metafont/post macros to interface with mfpic.]% + [2011/03/08 v1.06. Metafont/post macros to interface with mfpic.]% \documentclass{ltxdoc} \usepackage{docmfp} @@ -113,7 +113,7 @@ %</driver> %\fi % -% \CheckSum{1453} +% \CheckSum{1473} % \CharacterTable % {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z % Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z @@ -187,7 +187,7 @@ fi boolean grafbase; grafbase := true; string fileversion, filedate; -fileversion := "1.05"; filedate := "2010/06/10"; +fileversion := "1.06"; filedate := "2011/02/25"; message " Loading grafbase macros, version " & fileversion & ", " & filedate & "."; @@ -236,7 +236,7 @@ def checkversions (expr g)= fi enddef; -checkversions (105); +checkversions (106); % \end{macrocode} % @@ -672,11 +672,11 @@ vardef cmykgray (expr g) = cmyk(0,0,0,1 - snapto g) enddef; % with three-way booleans (often containing nested booleans). % % For all three engines we require a definition of the color functions -% \gbc{gray(g)}, \gbc{rgb(r,g,b)}, and \gbc{cmyk(c,m,y,k)}, conversion -% functions \gbc{makegray(x)}, \gbc{makergb(x)}, amd \gbc{makecmyk(x)}, -% and the boolean \gbc{iscolor clr}. The first three have to return -% numerics for \MF{}, colors for early \MP{}, and the associated color -% type for recent \MP{}. +% \gbc{gray(g)}, \gbc{rgb(r,g,b)}, and \gbc{cmyk(c,m,y,k)}, as well as +% conversion functions \gbc{makegray(x)}, \gbc{makergb(x)}, and +% \gbc{makecmyk(x)}, and the boolean \gbc{iscolor clr}. The first three +% have to return numerics for \MF{}, colors for early \MP{}, and the +% associated color type for recent \MP{}. % \begin{macrocode} %<*MP> if has_cmyk : @@ -729,7 +729,8 @@ if has_cmyk : enddef; vardef makergb primary clr = colorchoice (clr)(rgbblack)(rgbgray(clr))(clr) - (cmyktorgb(cyanpart clr, magentapart clr, yellowpart clr, blackpart clr)) + (cmyktorgb(cyanpart clr, magentapart clr, yellowpart clr, + blackpart clr)) enddef; vardef makegray primary clr = colorchoice (clr)(grayscaleblack)(grayscalegray(clr)) @@ -4513,12 +4514,14 @@ enddef; % Once we know what segment we are in, we determine the time by linear % interpolation between the times corresponding to \gbc{ct} and % \gbc{ct+1}. Note: in the \mfc{forever} loop, the exit must come before -% the increment. +% the increment. The function \gbc{inrange} is defined in +% section~\ref{axes}. It checks if the third argument is between the +% first two, or equal to one of them. % \begin{macrocode} vardef gettime (suffix arr, ct) (expr lngth) = setnumeric (_gtl) emax (arr[ct], emin (arr[arr], lngth)); setsplit (_s) segment_split; - forever: exitif ( (arr[ct] <= _gtl) and (_gtl <= arr[ct+1]) ); + forever: exitif inrange (arr[ct], arr[ct+1]) (_gtl); next ct; endfor if arr[ct] = arr[ct+1]: ct @@ -4987,17 +4990,48 @@ vardef axisline.t = axisline.x shifted (0, yhigh) enddef; vardef axis@# (expr len) = headpath (len, 0, 0) axisline@# enddef; +% \end{macrocode} +% +% \DescribeRoutine{borderrect} +% These are mostly for the simplification of \mfpic{} and readability of +% code. The command \gbc{borderrect} produces the border of the picture +% in graph coordinates, taking into account the four margins. +% +% \DescribeRoutine{between} +% The boolean \gbc{between} checks if the last argument is strictly +% between the first two (which must be in order). +% +% \DescribeRoutine{inrange} +% The boolean \gbc{inrange} checks if the last argument is in the closed +% interval determined by the first two (which must be in order). +% +% \DescribeRoutine{inbounds} +% The boolean \gbc{inbounds} checks if the argument (a pair) is in +% the closed border rectangle (\gbc{borderrect}). It is not yet used in +% \grafbase{}, though it would seem it ought to be useful. +% \begin{macrocode} vardef borderrect = rect((xlow,ylow),(xhigh,yhigh)) enddef; +vardef between (expr A, B, X) = (A < X) and (X < B) enddef; +vardef inrange (expr A, B, X) = (A <= X) and (X <= B) enddef; + vardef inbounds (expr Z) = - (xpart Z >= xlow ) and (ypart Z >= ylow ) and - (xpart Z <= xhigh) and (ypart Z <= yhigh) + inrange (xlow, xhigh) (xpart Z) and inrange (ylow, yhigh) (ypart Z) enddef; % \end{macrocode} % +% Possible binary relation versions. The last is just a reversal of the +% order of the first. These are not yet used in \grafbase{}. +% \begin{macrocode} +tertiarydef X isbetween P = between (xpart P, ypart P, X) enddef; +tertiarydef X isinrange P = inrange (xpart P, ypart P, X) enddef; +tertiarydef P contains X = between (xpart P, ypart P, X) enddef; + +% \end{macrocode} +% % Tick marks can be on the inside or outside of a border axis, % above or below any horizontal axes, left or right of any vertical axis % or centered on any axis. The following numerics are merely used to @@ -5216,7 +5250,7 @@ def plrvectorfield (expr len, rsp, tsp) (text fcn) (text cond) = vardef _vf (expr r,t) = ((0,0)--(fcn)) shifted (r*dir t) enddef; vardef _is_OK (expr r,t) = save _X, _Y; _X := r*cosd t; _Y := r*sind t; - (cond) and (_A < _X) and (_X < _B) and (_C < _Y) and (_Y < _D) + (cond) and between (_A, _B) (_X) and between (_C, _D) (_Y) enddef; mkplrvectorfield (len, rsp, tsp) (_vf, _is_OK); enddef; @@ -5385,7 +5419,7 @@ def getpolarbounds = % \gbc{tmin}${} < \theta < {}$\gbc{tmax} includes the graph. % \begin{macrocode} rmin := 0; - if (xneg < 0) and (xpos > 0) and (yneg < 0) and (ypos > 0): + if between (xneg, xpos) (0) and between (yneg, ypos) (0): tmin := 0; tmax := 360; elseif (p0 = origin): tmin := 0; tmax := 90; elseif (p1 = origin): tmin := -90; tmax := 0; @@ -5406,9 +5440,9 @@ def getpolarbounds = % the following order: (1)~above or below, (2)~left or right, and (3)~one % of the four corner regions. % \begin{macrocode} - if (xneg < 0) and (0 < xpos): + if between (xneg, xpos) (0): rmin := emin (abs(yneg), abs(ypos)); - elseif (yneg < 0) and (0 < ypos): + elseif between (yneg, ypos) (0): rmin := emin (abs(xneg), abs(xpos)); else: rmin := min (r0, r1, r2, r3); @@ -5596,8 +5630,10 @@ vardef anglefromto (expr u, v) = enddef; vardef cornerangle (expr A, B, C) = - if (A = B) and (B = C) : 60 - elseif (A=B) or (A=C) : 90 + if (A = B) or (A = C) : + if (B = C) : 60 + else: 90 + fi else: anglefromto (B - A, C - A) fi enddef; @@ -6834,14 +6870,14 @@ vardef pshcircle (expr disk, ctr, rad) = if disk: if rad >= 1 : if rad > 1: - GBerrmsg ("Impossible pseudohyperbolic circle.") + GBerrmsg ("Impossible radius of pseudohyperbolic circle.") "The radius of a pseudohyperbolic circle can be at most 1."; fi circle ((0,0),1) elseif abs(ctr) >= 1 : if abs(ctr) > 1: - GBerrmsg ("Impossible pseudohyperbolic circle.") - "The center of a pseudohyperbolic circle must be in" + GBerrmsg ("Impossible center of pseudohyperbolic circle.") + "The center of a pseudohyperbolic circle must be in " & "the unit disk."; fi onepointpath (true,ctr) @@ -6861,7 +6897,7 @@ vardef pshcircle (expr disk, ctr, rad) = elseif ypart ctr <= 0: if ypart ctr < 0: GBerrmsg ("Impossible pseudohyperbolic circle.") - "The center of a pseudohyperbolic circle must be in" + "The center of a pseudohyperbolic circle must be in " & "the upper half-plane."; fi onepointpath (true,ctr) @@ -7112,8 +7148,7 @@ tolerancefactor := .02; vardef mklevelset (expr sm, tens, X, Y, t, a, b, c, d) = save _inside_; vardef _inside_ (expr U, V) = - inside_levelset (U, V) and (a < U) and (U < b) - and (c < V) and (V < d) + inside_levelset (U, V) and between (a, b) (U) and between (c, d) (V) enddef; if not _inside_ (X, Y): GBwarn "Invalid seed point for levelset."; @@ -7175,6 +7210,104 @@ enddef; % \end{macrocode} % +% Our next set of macros produce approximations to the solutions of +% differential equations. While we could have several different macros +% each using a different method (Euler, two-step Runge-Kutta, four-step +% Runge-Kutta, etc.), our point of view is that we just want to draw a +% reasonably accurate solution, so we only utilize one method: four-step +% Runge-Kutta. The variations we allow are the following: +% \begin{enumerate} +% \item Drawing the graph of a one-dimensional differential +% equation, +% \[ \frac{dy}{dx} = g(x,y)\,.\] +% +% \item Drawing the trajectory of a two-dimensional differential +% equation, +% \[ \left( \frac{dx}{dt},\frac{dy}{dt} \right) = +% (f(x,y,t), g(x,y,t))\,.\] +% \end{enumerate} +% The first of these is implemented using the second with $f(x,y,t) \equiv +% 1$ and $g(x,y,t)$ not depending on $t$. The parameters passed include +% the starting point, the step size, the number of steps and an expression +% representing the right side of the equation. +% +% We do not use exactly the traditional Runge-Kutta method: we use the +% Runge-Kutta algorithm, but with a variable step size. The time step +% $\Delta t$ is chosen so that $|\mathbf{F}(x,y,t)|\Delta t$ equals the +% given step size parameter, and thus the parameter passed is actually a +% distance step. This makes drawing more stable, especially if the DE is +% one that produces an infinite path in finite time. +% +% This modification is itself unstable if $|\mathbf{F}|$ is very +% small (and impossible if it is zero), so we never use a $\Delta t$ +% larger than the given step size parameter $\Delta s$. That is, we +% actually use $\Delta t = \Delta s/\max(1,|\mathbf{F}|)$. +% +% As with our other function-like paths, we offer two variants. The basic +% version has a final text parameter which is the name of a pair-valued +% function of a numeric (representing $t$) and a pair variable +% (representing $x$ and $y$). The other version takes a text +% parameter, which must be a pair-valued expression in \mfc{x}, \mfc{y} +% and \mfc{t}. This parameter is copied into the definition text of a +% function and then the first form is called with that function's name. +% +% Also like other function-like paths, we offer polygonal or smooth +% versions controlled by a boolean argument, and the smooth versions make +% use of a tension parameter. +% \begin{macrocode} +def RKIV (expr sm) = tRKIV (sm, default_tension) +enddef; +vardef tRKIV (expr sm, tens, zstart, ds, N) (text _RHS_) = + save _trj, _ztr, _dz, _ztmp, _ctm; + pair _trj[], % The trajectory + _ztr, % current point + _dz[], % array[4] of displacements + _ztmp; % current point for calculating velocity +% + _trj := N+1; % ultimate size of _trj array + _trj1 := _ztr := zstart; + save _tt, % current time + _dt, % current time step + _th; % current time plus half a step + _tt := 0; + for _idx := 2 upto _trj: + _dt := ds/emax(1,abs(_RHS_(_tt,_ztr))); + _th := _tt + .5_dt; + _dz1 := _dt*_RHS_(_tt, _ztr); % displacement based on current point + _ztmp := _ztr + .5_dz1; % 1st midpoint + % use _th instead of twice calculating (_tt + .5_dt) + _dz2 := _dt*_RHS_(_th, _ztmp); % displacement based on 1st midpoint + _ztmp := _ztr + .5_dz2; % 2nd midpoint + _dz3 := _dt*_RHS_(_th, _ztmp); % displacement based on 2nd midpoint + _ztmp := _ztr + _dz3; % temporary end point + % get time for next loop now since we need it right away in next line: + _tt := _tt + _dt; + _dz4 := _dt*_RHS_(_tt, _ztmp); % displacement based on end point + % get next point + _ztr := _ztr + (_dz1 + 2_dz2 + 2_dz3 + _dz4)/6; + _trj[_idx] := _ztr; + endfor + mkpath (sm, tens, false, _trj) +enddef; + +def xyRKIV (expr sm) = txyRKIV (sm, default_tension) +enddef; +vardef txyRKIV (expr sm, tens, zstart, ds, N) (text _RHS_) = + save _fgxy, __fgxy; + vardef __fgxy (expr t, x, y) = _RHS_ enddef; + vardef _fgxy (expr t, Z) = __fgxy(t, xpart Z, ypart Z) enddef; + tRKIV (sm, tens, zstart, ds, N) (_fgxy) +enddef; + +def odeRKIV (expr sm) = todeRKIV (sm, default_tension) +enddef; +vardef todeRKIV (expr sm, tens, xstart, ystart, ds, N) +(text _fxy) = + txyRKIV (sm, tens, (xstart, ystart), ds, N) ((1, _fxy)) +enddef; + +% \end{macrocode} +% % \section{Modification of Paths}\label{modification} % % \subsection{Closing a path}\label{closing} |