/* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/
/* [ Created with wxMaxima version 0.8.5 ] */

/* [wxMaxima: input   start ] */
/* some initial settings */
load(draw)$
ratprint : false $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* let the polygon have n=5 vertices */
n : 5 $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* define the polygon's vertices */
v[1] : [ 1.0,  0.0] $
v[2] : [ 0.5,  0.8] $
v[3] : [-0.3,  1.0] $
v[4] : [-1.0, -0.2] $
v[5] : [ 0.0, -1.0] $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* plot the polygon */
draw2d( polygon( makelist( v[i], i,1,n ) ) ) $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
v : [x,y];
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* compute relevant areas and distances (as 'functions' of "v") */
for i : 1 thru n do (
  ip : mod( i, n ) + 1,
  im : mod( i+n-2, n ) + 1,
  A[i] : determinant( matrix( v[i ]-v, v[ip]-v ) ),
  B[i] : determinant( matrix( v[im]-v, v[ip]-v ) ),
  r[i] : sqrt( (v[i]-v).(v[i]-v) )
);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* set up some specific choice of functions c[i] */
/* note how the function "f" is define using ":=" */
f(x) := 1;
for i : 1 thru n do (
  c[i] : f(r[i])
);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* general formula for homogeneous barycentric coordinates */
/* note that "ip" and "im" are equal to "i+1" and "i-1", w.r.t. to {1,2,...,n} */
for i : 1 thru n do (
  ip : mod( i, n ) + 1,
  im : mod( i+n-2, n ) + 1,
  w[i] : ( c[ip] * A[im] - c[i] * B[i] + c[im] * A[i] ) / ( A[im] * A[i] )
);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* we sum up all homogeneous coordinates */
W : sum( w[i], i,1,n ) $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* to then define the normalized barycentric coordinates */
for i : 1 thru n do (
  b[i] : ratsimp( w[i] / W )  
);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* check that they add up to one */
ratsimp( sum( b[i], i,1,n ) );
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* and have the reproduction property */
ratsimp( sum( b[i]*v[i], i,1,n ) );
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* note that it can be a rather complicated function */
b[1];
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* plot of one of the barycentric basis functions with contour lines */
poly3d : append( makelist( [ v[i][1], v[i][2], 0 ], i,1,n ), [[ v[1][1], v[1][2], 0 ]] ) $ 
draw3d( line_type = solid, points_joined = true,
        points( poly3d ), 
        color = green,
        explicit( b[1], x,-1.51,1.52, y,-1.53,1.54 ),
        contour_levels = [0, 0.1, 1.01],
        contour        = both ) $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* alternative plot of the j-th basis function, and */
/* the piecewise linear behaviour along the polygon */
j : 1 $
for i : 1 thru n do ( f[i] : 0 ); f[j] : 1 $
poly3d : append( makelist( [ v[i][1], v[i][2], f[i] ], i,1,n ), [[ v[1][1], v[1][2], f[1] ]] ) $ 
draw3d( enhanced3d = true,
        palette = color,
        explicit( b[j], x,-1.51,1.52, y,-1.53,1.54 ),
        enhanced3d = false,
        color = black,
        line_type = solid, line_width = 3, points_joined = true,
        points( poly3d ) ) $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* the non-interactive version of the plot */
wxplot3d( b[j], [x,-1.51,1.52], [y,-1.53,1.54], [z,-1,2], [gnuplot_pm3d, false] );
/* [wxMaxima: input   end   ] */

/* Maxima can't load/batch files which end with a comment! */
"Created with wxMaxima"$