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

/* [wxMaxima: input   start ] */
/* define vertices of a triangle and data given at these vertices */

/* assignment with ":" | array elements via "[.]" */
/* generate output with ";" and suppress with "$" */

v[1] : [-1.0,  0.0];   f[1] :  0.5 $
v[2] : [ 1.0, -0.5];   f[2] : -0.5 $
v[3] : [ 0.8,  0.5];   f[3] :  1.0 $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* generate the list " [ v[1], v[2], v[3] ] " */
poly : makelist( v[i], i,1,3 );
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* plot the vertices in this list by connecting them with lines */
wxplot2d( [ 'discrete, poly ] ) $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* to connect v[3] with v[1], we simply extend the list by adding v[1] */
poly : append( poly, [v[1]] );
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* now the plot shows the complete triangle */
wxplot2d( [ 'discrete, poly ] ) $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* but the "draw" library also offers a simpler way to plot closed polygons */
load( draw ) $
draw2d( polygon( makelist( v[i], i,1,3 ) ) ) $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* let us introduce the general planar point "v" with coordiates (x,y) */
v : [x,y];
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* compute areas of triangles formed by "v" and each of the triangle's edges */
w[1] : determinant( matrix( v[2]-v, v[3]-v ) );
w[2] : determinant( matrix( v[3]-v, v[1]-v ) );
w[3] : determinant( matrix( v[1]-v, v[2]-v ) );
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* compute the sum of these areas */
W : sum( w[i], i,1,3 );
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* this should be equal to the area of the triangle, so let us check */
A : determinant ( matrix ( v[2]-v[1], v[3]-v[1] ) );
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* indeed, if we simplify "W", then we see that this is indeed the case */
/* note: setting "ratprint" to "false" suppresses some additional output */
ratprint : false $
ratsimp( W );
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* finally, we define the barycentric coordinates of "v" w.r.t. the triangle */
for i : 1 thru 3 do (
  b[i] : w[i] / W   
);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* after simplification, we can see the linear nature of the b.c.'s */
ratsimp( b[1] );
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* they clearly add up to one */
ratsimp( sum( b[i], i,1,3 ) );
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* and have the reproduction property */
ratsimp( sum( b[i]*v[i], i,1,3) );
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* this is one way of plotting a bivariate function */
wxplot3d( b[1], [x,-1,1], [y,-1,1], [z,0,1], [gnuplot_pm3d, false] );
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* here is an alternative way, which provides an interactive interface */
poly3d : append( makelist( [ v[i][1], v[i][2], 0 ], i,1,3 ), [[ v[1][1], v[1][2], 0 ]] ) $ 
draw3d( line_type = solid, points_joined = true,
        points( poly3d ), 
        color = green,
        explicit( b[1], x,-1.5,1.5, y,-1.5,1.5 ),
        contour_levels = [0, 0.1, 1.01],
        contour        = both ) $
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* the linear interpolant to the given data is easily defined via b.c.'s */
F : ratsimp( sum( b[i] * f[i], i,1,3 ) );
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* and here is yet another way of plotting bivariate functions */
poly3d : append( makelist( [ v[i][1], v[i][2], f[i] ], i,1,3 ), [[ v[1][1], v[1][2], f[1] ]] ) $ 
draw3d( enhanced3d = true,
        palette = color,
        explicit( F, x,-1.5,1.5, y,-1.5,1.5 ),
        enhanced3d = false,
        color = black,
        line_type = solid, line_width = 3, points_joined = true,
        points( poly3d ) ) $
/* [wxMaxima: input   end   ] */

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