bpolynomial update 07-12-19

git-svn-id: svn://tug.org/texlive/trunk@5806 c570f23f-e606-0410-a88d-b1316a301751

1 files changed, 262 insertions, 45 deletions

diff --git a/Master/texmf-dist/metapost/bpolynomial/bpolynomial.mp b/Master/texmf-dist/metapost/bpolynomial/bpolynomial.mp
index f1c0129bb94..e4f27a434a1 100644
--- a/Master/texmf-dist/metapost/bpolynomial/bpolynomial.mp
+++ b/Master/texmf-dist/metapost/bpolynomial/bpolynomial.mp
@@ -7,73 +7,152 @@
 % license is in http://www.latex-project.org/lppl.txt
 %
 
+%%% Identify yourself.
 if known bpolynomial_fileversion: endinput fi;
 string bpolynomial_fileversion;
-bpolynomial_fileversion := "v0.4 (2007/11/28)";
+bpolynomial_fileversion := "v0.5 (2007/12/12)";
 message "Loading bpolynomial " & bpolynomial_fileversion;
 
 
+%%% Main user macro for defining polynomials.
+%%% Arguments are a suffix and the coefficients
+%%%  of the function a*x^3 + b*x^2 + c*x + d.
 vardef newBPolynomial@#(expr a, b, c, d)=
-  defineBPolynomial.@#(a, b, c, d);
-  defineBPolynomial.@#'(0, 3a, 2b, c);
-  defineBPolynomial.@#''(0, 0, 6a, 2b);
-  defineBPolynomial.@#'''(0, 0, 0, 6a);
+  bpolynomial__defineBPolynomial.@#(a, b, c, d);
+  bpolynomial__defineBPolynomial.@#'(0, 3a, 2b, c);
+  bpolynomial__defineBPolynomial.@#''(0, 0, 6a, 2b);
+  bpolynomial__defineBPolynomial.@#'''(0, 0, 0, 6a);
 enddef;
 
-%%% This macro defines two macros @#.eval and @#.getPath.
-%%% Parameters are the coefficients of the polynomial a*x^3 + b*x^2 + c*x + d.
-vardef defineBPolynomial@#(expr ca,cb,cc,cd)=
+
+%%% This macro returns the path of a Bezier curve that matches
+%%% a function a*x^3 + b*x^2 + c*x + d between two points A and D.
+%%% This macro is the heart of this package and is used by
+%%% several other macros.
+%%% Arguments are the coefficients of the polynomial and the
+%%% start and end point of the graph/path.
+vardef bpolynomial__getBezierFromPolynomial(expr a, b, c, d, A, D)=
+save xA,xB,xC,xD,yA,yB,yC,yD;
+save xl,yl,xr,yr,dx;
+numeric xA,xB,xC,xD,yA,yB,yC,yD;
+numeric xl,yl,xr,yr,dx;
+  xl := xpart A;
+  yl := ypart A;
+  xr := xpart D;
+  yr := ypart D;
+  dx := xpart D - xpart A;
+  %%% Original equation system for x values.
+%   xA                  = xl;
+%   3(xB - xA)          = dx;
+%   3(xC - 2xB + xA)    = 0;
+%   xD - 3xC + 3xB - xA = 0;
+  %%% Modified equation system.
+  xA := xl;
+  xB := xl + dx/3;
+  xC := xr - dx/3;
+  xD := xr;
+  %%% Original equation system for y values.
+%   yA                  = ((a*xl + b)*xl + c)*xl + d;
+%   3(yB - yA)          = dx*((3a*xl + 2b)*xl + c);
+%   3(yC - 2yB + yA)    = dx*dx*(3a*xl + b);
+%   yD - 3yC + 3yB - yA = a*dx*dx*dx;
+  %%% Modified equation system.
+  yA               := yl;
+  3(yB - yA)       = dx*((3a*xl + 2b)*xl + c);
+  3(yC - 2yB + yA) = dx*dx*(3a*xl + b);
+  yD               := yr;
+  %%% Return path A..controls B and C..D.
+  (xA,yA)..controls (xB,yB) and (xC,yC)..(xD,yD)
+enddef;
+
+
+%%% This macro returns the path of a Bezier curve that matches
+%%% a function a*x^3 + b*x^2 + c*x + d in the range [xl, xr].
+%%% Arguments are the coefficients of the polynomial and the
+%%% range boundaries of the graph/path.
+vardef getBezierFromPolynomial(expr a, b, c, d, xl, xr)=
+  bpolynomial__getBezierFromPolynomial(a, b, c, d,
+    (xl, ((a*xl+b)*xl+c)*xl+d),
+    (xr, ((a*xr+b)*xr+c)*xr+d))
+enddef;
+
+
+%%% This macro returns the path of a Bezier curve that matches
+%%% a function a*x^3 + b*x^2 + c*x + d in the range [xl, xr].
+%%% Arguments are the coefficients of the polynomial and the
+%%% range boundaries of the graph/path.
+vardef getBezierFromSqrRoot(expr u, v, w, xl, xr)=
+save yl, yr;
+numeric yl,yr;
+    if (xl >= -v):
+      yl := xl;
+    else:
+      message "Package bpolynomial warning:  Replacing lower range boundary " & decimal xl & " by " & decimal -v & "!";
+      yl := -v;
+    fi
+    if (xr >= -v):
+      yr := xr;
+    else:
+      message "Package bpolynomial warning:  Replacing upper range boundary " & decimal xr & " by " & decimal -v & "!";
+      yr := -v;
+    fi
+  bpolynomial__getBezierFromPolynomial(0, 1/u/u, -2*w/u/u, (w/u)*(w/u)-v,
+    (u*sqrt(yl+v)+w, yl),
+    (u*sqrt(yr+v)+w, yr)) reflectedabout ((0,0),(1,1))
+enddef;
+
+
+%%% This macro returns the path of a Bezier curve that matches
+%%% a function a*x^3 + b*x^2 + c*x + d in the range [xl, xr].
+%%% Arguments are the coefficients of the polynomial and the
+%%% range boundaries of the graph/path.
+vardef getBezierFromCubRoot(expr u, v, w, xl, xl)=
+save yl, yr;
+numeric yl,yr;
+    if (xl >= -v):
+      yl := xl;
+    else:
+      message "Package bpolynomial warning:  Replacing lower range boundary " & decimal xl & " by " & decimal -v & "!";
+      yl := -v;
+    fi
+    if (xr >= -v):
+      yr := xr;
+    else:
+      message "Package bpolynomial warning:  Replacing upper range boundary " & decimal xr & " by " & decimal -v & "!";
+      yr := -v & "!";
+    fi
+  bpolynomial__getBezierFromPolynomial(1/u/u/u, -3w/u/u/u, 3(w/u)*(w/u)/u, (w/u)*(w/u)*(w/u)-v,
+    (u*((yl+v)**(1/3))+w, yl),
+    (u*((yr+v)**(1/3))+w, yr)) reflectedabout ((0,0),(1,1))
+enddef;
+
+
+%%% This internal macro defines a new polynomial.
+%%% Arguments are a suffix macro and the coefficients
+%%% of the polynomial a*x^3 + b*x^2 + c*x + d.
+vardef bpolynomial__defineBPolynomial@#(expr ca,cb,cc,cd)=
 numeric @#.a, @#.b, @#.c, @#.d;
   %%% Save coefficients for later access.
-  %%% For instance, variable @#.a refers to coefficient a of polynomial @#.
+  %%% Variable @#.a refers to coefficient a of polynomial @#.
   @#.a := ca;
   @#.b := cb;
   @#.c := cc;
   @#.d := cd;
   
-  
-  %%% Define macro that returns values of polynomial @#.
-  %%% Parameter is an x value.
+  %%% This macro returns values of polynomial @#.
+  %%% Argument is an x value.
   vardef @#.eval(expr x)=
     (((@#.a*x + @#.b)*x + @#.c)*x + @#.d)
   enddef;
   
-  
-  %%% Define a macro that returns a path of the polynomial
-  %%% on a given intervall [xl, xr].
+  %%% This macro returns the path corresponding to polynomial @#
+  %%% on the intervall [xl, xr].
   vardef @#.getPath(expr xl,xr)=
-  save xA,xB,xC,xD,yA,yB,yC,yD;
-  save dx;
-  numeric xA,xB,xC,xD,yA,yB,yC,yD;
-  numeric dx;
-    dx := xr - xl;
-    %%% Original equation system for x values.
-%     xA                  = xl;
-%     3(xB - xA)          = dx;
-%     3(xC - 2xB + xA)    = 0;
-%     xD - 3xC + 3xB - xA = 0;
-    %%% Modified equation system.
-    xA := xl;
-    xB := xl + dx/3;
-    xC := xr - dx/3;
-    xD := xr;
-    %%% Original equation system for y values.
-%     yA                  = ((@#.a*xl + @#.b)*xl + @#.c)*xl + @#.d;
-%     3(yB - yA)          = dx*((3@#.a*xl + 2@#.b)*xl + @#.c);
-%     3(yC - 2yB + yA)    = dx*dx*(3@#.a*xl + @#.b);
-%     yD - 3yC + 3yB - yA = @#.a*dx*dx*dx;
-    %%% Modified equation system.
-    yA               := @#.eval(xl);
-    3(yB - yA)       = dx*((3@#.a*xl + 2@#.b)*xl + @#.c);
-    3(yC - 2yB + yA) = dx*dx*(3@#.a*xl + @#.b);
-    yD               := @#.eval(xr);
-    %%% Return path A..controls B and C..D.
-    (xA,yA)..controls (xB,yB) and (xC,yC)..(xD,yD)
+    bpolynomial__getBezierFromPolynomial(@#.a, @#.b, @#.c, @#.d, (xl, @#.eval(xl)), (xr, @#.eval(xr)))
   enddef;
   
-
-  %%% Define macro that returns a path tangent to @# at point (x, f(x))
-  %%% covering interval (x+xm, x+xp).
+  %%% This macro returns a path tangent to @# at point (x, f(x))
+  %%% covering the interval [x+xm, x+xp].
   vardef @#.getTangent(expr x, xm, xp)=
   save m, y;
   numeric m, y;
@@ -83,3 +162,141 @@ numeric @#.a, @#.b, @#.c, @#.d;
   enddef;
   
 enddef;
+
+
+%%% This macro defines a new square root.
+%%% Arguments are a suffix macro and the parameters
+%%% of the function u*(x + v)^(1/2) + w.
+vardef newBSqrRoot@#(expr cu,cv,cw)=
+numeric @#.a, @#.b, @#.c, @#.d;
+numeric @#.u, @#.v, @#.w;
+  %%% Save parameters for later access.
+  %%% Variable @#.v refers to parameters of square root @#.
+  %%% Variables @#.a to @#.d store the coefficients of the
+  %%% corresponding polynomial.
+  @#.u := cu;
+  @#.v := cv;
+  @#.w := cw;
+  @#.a := 0;
+  @#.b := 1/cu/cu;
+  @#.c := -2*cw/cu/cu;
+  @#.d := (cw/cu)*(cw/cu)-cv;
+  
+  %%% This macro returns values of polynomial @#.
+  %%% Argument is an x value.
+  vardef @#.eval(expr x)=
+    if (x >= -@#.v):
+      @#.u*sqrt(x + @#.v) + @#.w
+    else:
+      message "Package bpolynomial warning:  Cannot evaluate function at x = " & decimal x & "!";
+      @#.w
+    fi
+  enddef;
+  
+  %%% This macro returns the path corresponding to square root @#
+  %%% on the intervall [yl, yr].  The path of the corresponing
+  %%% polynomial is computed and then transformed.
+  vardef @#.getPath(expr xl,xr)=
+  save yl, yr;
+  numeric yl, yr;
+    if (xl >= -@#.v):
+      yl := xl;
+    else:
+      message "Package bpolynomial warning:  Replacing lower range boundary " & decimal xl & " by " & decimal -@#.v & "!";
+      yl := -@#.v;
+    fi
+    if (xr >= -@#.v):
+      yr := xr;
+    else:
+      message "Package bpolynomial warning:  Replacing upper range boundary " & decimal xr & " by " & decimal -@#.v & "!";
+      yr := -@#.v;
+    fi
+    bpolynomial__getBezierFromPolynomial(@#.a, @#.b, @#.c, @#.d, (@#.eval(yl), yl), (@#.eval(yr), yr))
+    reflectedabout ((0,0),(1,1))
+  enddef;
+  
+  %%% This macro returns a path tangent to square root @#
+  %%% at point (x, f(x)) covering the interval [x+xm, x+xp].
+  vardef @#.getTangent(expr x, epsl, epsr)=
+  save m, y;
+  numeric m, y;
+    if (x >= -@#.v):
+      m := @#.u/(2sqrt(x + @#.v));
+      y := @#.eval(x);
+      (x+epsl, y + m*epsl) -- (x+epsr, y + m*epsr)
+    else:
+      message "Package bpolynomial warning:  Cannot draw tangent at x = " & decimal x & "!";
+      (-@#.v, @#.w)--(-@#.v, @#.w+1)
+    fi
+  enddef;
+  
+enddef;
+
+  
+%%% This macro defines a new cubic root.
+%%% Arguments are a suffix macro and the parameters
+%%% of the function u*(x + v)^(1/3) + w.
+vardef newBCubRoot@#(expr cu,cv,cw)=
+numeric @#.a, @#.b, @#.c, @#.d;
+numeric @#.u, @#.v, @#.w;
+  %%% Save parameters for later access.
+  %%% Variable @#.v refers to parameters of cubic root @#.
+  %%% Variables @#.a to @#.d store the coefficients of the
+  %%% corresponding polynomial.
+  @#.u := cu;
+  @#.v := cv;
+  @#.w := cw;
+  @#.a := 1/cu/cu/cu;
+  @#.b := -3cw/cu/cu/cu;
+  @#.c := 3(cw/cu)*(cw/cu)/cu;
+  @#.d := (cw/cu)*(cw/cu)*(cw/cu)-cv;
+  
+  %%% This macro returns values of polynomial @#.
+  %%% Argument is an x value.
+  vardef @#.eval(expr x)=
+    if (x >= -@#.v):
+      @#.u*((x+@#.v)**(1/3)) + @#.w
+    else:
+      message "Package bpolynomial warning:  Cannot evaluate function at x = " & decimal x & "!";
+      @#.w
+    fi
+  enddef;
+  
+  %%% This macro returns the path corresponding to cubic root @#
+  %%% on the intervall [yl, yr].  The path of the corresponing
+  %%% polynomial is computed and then transformed.
+  vardef @#.getPath(expr xl,xr)=
+  save yl, yr;
+  numeric yl, yr;
+    if (xl >= -@#.v):
+      yl := xl;
+    else:
+      message "Package bpolynomial warning:  Replacing lower range boundary " & decimal xl & " by " & decimal -@#.v & "!";
+      yl := -@#.v;
+    fi
+    if (xr >= -@#.v):
+      yr := xr;
+    else:
+      message "Package bpolynomial warning:  Replacing upper range boundary " & decimal xr & " by " & decimal -@#.v & "!";
+      yr := -@#.v;
+    fi
+    bpolynomial__getBezierFromPolynomial(@#.a, @#.b, @#.c, @#.d, (@#.eval(yl), yl), (@#.eval(yr), yr))
+      reflectedabout ((0,0),(1,1))
+  enddef;
+  
+  %%% This macro returns a path tangent to cubic root @#
+  %%% at point (x, f(x)) covering the interval [x+xm, x+xp].
+  vardef @#.getTangent(expr x, epsl, epsr)=
+  save m, y;
+  numeric m, y;
+    if (x >= -@#.v):
+      m := @#.u/3/((x + @#.v)**(2/3));
+      y := @#.eval(x);
+      (x+epsl, y + m*epsl) -- (x+epsr, y + m*epsr)
+    else:
+      message "Package bpolynomial warning:  Cannot draw tangent at x = " & decimal x & "!";
+      (-@#.v, @#.w)--(-@#.v, @#.w+1)
+    fi
+  enddef;
+  
+enddef;