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[Octave-cvsupdate] SF.net SVN: octave:[10258] trunk/octave-forge/main/plot
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From: <car...@us...> - 2012-04-17 09:29:36
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Revision: 10258
http://octave.svn.sourceforge.net/octave/?rev=10258&view=rev
Author: carandraug
Date: 2012-04-17 09:29:25 +0000 (Tue, 17 Apr 2012)
Log Message:
-----------
implicit: new function submitted by Martin Helm <ma...@mh...>
Modified Paths:
--------------
trunk/octave-forge/main/plot/NEWS
Added Paths:
-----------
trunk/octave-forge/main/plot/inst/implicit.m
Modified: trunk/octave-forge/main/plot/NEWS
===================================================================
--- trunk/octave-forge/main/plot/NEWS 2012-04-17 09:28:37 UTC (rev 10257)
+++ trunk/octave-forge/main/plot/NEWS 2012-04-17 09:29:25 UTC (rev 10258)
@@ -1,4 +1,9 @@
Summary of important user-visible changes for plot 1.1.1:
-------------------------------------------------------------------
+ ** The following functions are new:
+
+ calc_shading
+ implicit
+
** Package is no longer automatically loaded.
Added: trunk/octave-forge/main/plot/inst/implicit.m
===================================================================
--- trunk/octave-forge/main/plot/inst/implicit.m (rev 0)
+++ trunk/octave-forge/main/plot/inst/implicit.m 2012-04-17 09:29:25 UTC (rev 10258)
@@ -0,0 +1,523 @@
+## Copyright (C) 2012 Martin Helm <ma...@mh...>
+##
+## This program is free software; you can redistribute it and/or modify it under
+## the terms of the GNU General Public License as published by the Free Software
+## Foundation; either version 3 of the License, or (at your option) any later
+## version.
+##
+## This program is distributed in the hope that it will be useful, but WITHOUT
+## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
+## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
+## details.
+##
+## You should have received a copy of the GNU General Public License along with
+## this program; if not, see <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {[@var{t}, @var{p}] =} implicit (@var{fun}, @var{iso}, @var{x}, @var{y}, @var{z})
+## @deftypefnx {Function File} {[@var{t}, @var{p}, @var{g}, @var{ng}] =} implicit (@var{fun}, @var{iso}, @var{x}, @var{y}, @var{z})
+##
+## Return the triangulation information @var{t} at points @var{p} for
+## the implicit function @var{fun} and the iso level @var{iso}.
+## The function @var{fun} must accept matrix arguments as given by meshgrid. In most cases this can be
+## accomplished with vectorize(inline("...")).
+##
+## It is considered that the discretisation of the function is
+## given at the points @var{x}, @var{y} and @var{z} which are of type
+## vector as typically given by linspace. The orientation of the triangles
+## is choosen such that the normals point from the lower values to the
+## higher values.
+##
+## The optional output arguments @var{g}, @var{ng} contain an approximation for
+## the normalized gradient vectors and the norm of the gradient vectors.
+##
+## The method is loosely based on the marching cube algorithm, the intersection points with the grids
+## are computed with high accuracy in contrast to the original algorithm which is a linear interpolation.
+## The calculation of the intersection points is numerically intensive and a low resolution for the
+## discretisation should be taken into account (10 to 20 intervals per axis).
+##
+## The triangulation lookup table and the edge table used
+## here are based on Cory Gene Bloyd's implementation and can be found
+## beyond other surface and geometry stuff at Paul Bourke's website
+## @uref{http://local.wasp.uwa.edu.au/~pbourke/geometry/polygonise}.
+##
+## For example:
+##
+## @example
+## @group
+## graphics_toolkit fltk
+## N = 10; # N intervals per axis
+## x = y = z = linspace(-4,4,N+1);
+##
+## f = vectorize(inline("x^3+y^3+z^3"));
+## [T P G, nG] = implicit (f, 3, x, y, z);
+##
+## figure ();
+## trimesh (T, P(:,1), P(:,2), P(:,3));
+## axis equal
+## view (115, 30)
+## @end group
+## @end example
+##
+## Instead of the @command{trimesh} function the @command{patch}
+## function can be used to visualize the geometry. For example:
+##
+## @example
+## @group
+## figure ();
+## cdat = calc_shading (0.1, 1, .7, 4, [1 .2 .3], ...
+## G, [1.0; 0.5; .8], [.5; 1.5; .1]);
+## p = patch ("Faces", T, "Vertices", P, "FaceVertexCData", ...
+## cdat, "FaceColor", "interp", "EdgeColor", "black");
+## axis equal
+## view (115, 30)
+## @end group
+## @end example
+##
+## @end deftypefn
+
+## Author: Martin Helm <ma...@mh...>
+## Created: 2012-03-28
+
+function [T, p, G, nG] = implicit (fun, iso, x, y, z)
+
+ persistent edge_table=[];
+ persistent tri_table=[];
+
+ lindex = 4;
+
+ if (isempty (tri_table) || isempty (edge_table))
+ [edge_table, tri_table] = init_mc ();
+ endif
+
+ if (nargin != 5)
+ print_usage ();
+ endif
+
+ if (!isvector (x) || !isvector (y) || !isvector (z))
+ error ("implicit: X, Y, Z must be vectors");
+ endif
+
+ if (length (x) < 2 || length (y) < 2 || length (z) < 2)
+ error ("implicit: grid size must be at least 2x2x2");
+ endif
+
+ if (!isscalar (iso))
+ error ("implicit: ISO must be scalar value");
+ endif
+
+ [xx yy zz] = meshgrid (x, y, z);
+ c = fun(xx, yy, zz);
+ n = size (xx) - 1;
+
+ ## phase I: assign information to each voxel which edges are intersected by
+ ## the isosurface
+ cc = zeros (n(1), n(2), n(3), "uint16");
+ cedge = zeros (size (cc), "uint16");
+
+ vertex_idx = {1:n(1), 1:n(2), 1:n(3); ...
+ 2:n(1)+1, 1:n(2), 1:n(3); ...
+ 2:n(1)+1, 2:n(2)+1, 1:n(3); ...
+ 1:n(1), 2:n(2)+1, 1:n(3); ...
+ 1:n(1), 1:n(2), 2:n(3)+1; ...
+ 2:n(1)+1, 1:n(2), 2:n(3)+1; ...
+ 2:n(1)+1, 2:n(2)+1, 2:n(3)+1; ...
+ 1:n(1), 2:n(2)+1, 2:n(3)+1 };
+
+ ## calculate which vertices have values >= iso
+ for ii=1:8
+ idx = c(vertex_idx{ii, :}) >= iso;
+ cc(idx) = bitset (cc(idx), ii);
+ endfor
+
+ cedge = edge_table(cc+1); # assign the info about intersected edges
+ id = find (cedge); # select only voxels which are intersected
+ if (isempty (id))
+ T = p = G = nG = [];
+ return
+ endif
+
+ ## phase II: calculate the list of intersection points
+ xyz_off = [1, 1, 1; 2, 1, 1; 2, 2, 1; 1, 2, 1; 1, 1, 2; 2, 1, 2; 2, 2, 2; 1, 2, 2];
+ edges = [1 2; 2 3; 3 4; 4 1; 5 6; 6 7; 7 8; 8 5; 1 5; 2 6; 3 7; 4 8];
+ offset = sub2ind (size (c), xyz_off(:, 1), xyz_off(:, 2), xyz_off(:, 3)) -1;
+ pp = zeros (length (id), lindex, 12);
+ ccedge = [vec(cedge(id)), id];
+ ix_offset=0;
+ for jj=1:12
+ id__ = bitget (ccedge(:, 1), jj);
+ id_ = ccedge(id__, 2);
+ [ix iy iz] = ind2sub (size (cc), id_);
+ id_c = sub2ind (size (c), ix, iy, iz);
+ id1 = id_c + offset(edges(jj, 1));
+ id2 = id_c + offset(edges(jj, 2));
+ pp(id__, 1:4, jj) = [vertex_interp(fun, iso, xx(id1), yy(id1), zz(id1), ...
+ xx(id2), yy(id2), zz(id2)), ...
+ (1:size (id_, 1))' + ix_offset ];
+ ix_offset += size (id_, 1);
+ endfor
+
+ ## phase III: calculate the triangulation from the point list
+ T = [];
+ tri = tri_table(cc(id)+1, :);
+ for jj=1:3:15
+ id_ = find (tri(:, jj)>0);
+ p = [id_, lindex*ones(size (id_, 1), 1),tri(id_, jj:jj+2)];
+ if (!isempty (p))
+ p1 = sub2ind (size (pp), p(:,1), p(:,2), p(:,3));
+ p2 = sub2ind (size (pp), p(:,1), p(:,2), p(:,4));
+ p3 = sub2ind (size (pp), p(:,1), p(:,2), p(:,5));
+ T = [T; pp(p1), pp(p2), pp(p3)];
+ endif
+ endfor
+
+ p = [];
+ for jj = 1:12
+ idp = pp(:, lindex, jj) > 0;
+ if (any (idp))
+ p(pp(idp, lindex, jj), 1:3) = pp(idp, 1:3, jj);
+ endif
+ endfor
+
+ if (nargout == 3 || nargout == 4)
+ sqeps = sqrt(eps);
+ delta_x = (max(x)-min(x))*sqeps;
+ delta_y = (max(y)-min(y))*sqeps;
+ delta_z = (max(z)-min(z))*sqeps;
+ [dx dy dz] = gradient (fun, p, delta_x, delta_y, delta_z);
+ nG = norm ([dx dy dz], 2, "rows");
+ G = [dx dy dz] ./ repmat (nG, 1, 3);
+ endif
+endfunction
+
+function p = vertex_interp(fun, isolevel,p1x, p1y, p1z, p2x, p2y, p2z)
+
+ if (nargin == 8)
+ p = zeros (length (p1x), 3);
+ else
+ error ("implicit: wrong number of arguments");
+ endif
+ for ii=1:length (p1x)
+ f = @(t) fun ((1-t)*p1x(ii) + t*p2x(ii), (1-t)*p1y(ii) + t*p2y(ii), ...
+ (1-t)*p1z(ii) + t*p2z(ii)) -isolevel;
+ mu = fzero (f, [0 1]);
+ p(ii, 1:3) = [p1x(ii) + mu .* (p2x(ii) - p1x(ii)), ...
+ p1y(ii) + mu .* (p2y(ii) - p1y(ii)), ...
+ p1z(ii) + mu .* (p2z(ii) - p1z(ii))];
+ endfor
+endfunction
+
+function [edge_table, tri_table] = init_mc()
+ edge_table = [
+ 0x0 , 0x109, 0x203, 0x30a, 0x406, 0x50f, 0x605, 0x70c, ...
+ 0x80c, 0x905, 0xa0f, 0xb06, 0xc0a, 0xd03, 0xe09, 0xf00, ...
+ 0x190, 0x99 , 0x393, 0x29a, 0x596, 0x49f, 0x795, 0x69c, ...
+ 0x99c, 0x895, 0xb9f, 0xa96, 0xd9a, 0xc93, 0xf99, 0xe90, ...
+ 0x230, 0x339, 0x33 , 0x13a, 0x636, 0x73f, 0x435, 0x53c, ...
+ 0xa3c, 0xb35, 0x83f, 0x936, 0xe3a, 0xf33, 0xc39, 0xd30, ...
+ 0x3a0, 0x2a9, 0x1a3, 0xaa , 0x7a6, 0x6af, 0x5a5, 0x4ac, ...
+ 0xbac, 0xaa5, 0x9af, 0x8a6, 0xfaa, 0xea3, 0xda9, 0xca0, ...
+ 0x460, 0x569, 0x663, 0x76a, 0x66 , 0x16f, 0x265, 0x36c, ...
+ 0xc6c, 0xd65, 0xe6f, 0xf66, 0x86a, 0x963, 0xa69, 0xb60, ...
+ 0x5f0, 0x4f9, 0x7f3, 0x6fa, 0x1f6, 0xff , 0x3f5, 0x2fc, ...
+ 0xdfc, 0xcf5, 0xfff, 0xef6, 0x9fa, 0x8f3, 0xbf9, 0xaf0, ...
+ 0x650, 0x759, 0x453, 0x55a, 0x256, 0x35f, 0x55 , 0x15c, ...
+ 0xe5c, 0xf55, 0xc5f, 0xd56, 0xa5a, 0xb53, 0x859, 0x950, ...
+ 0x7c0, 0x6c9, 0x5c3, 0x4ca, 0x3c6, 0x2cf, 0x1c5, 0xcc , ...
+ 0xfcc, 0xec5, 0xdcf, 0xcc6, 0xbca, 0xac3, 0x9c9, 0x8c0, ...
+ 0x8c0, 0x9c9, 0xac3, 0xbca, 0xcc6, 0xdcf, 0xec5, 0xfcc, ...
+ 0xcc , 0x1c5, 0x2cf, 0x3c6, 0x4ca, 0x5c3, 0x6c9, 0x7c0, ...
+ 0x950, 0x859, 0xb53, 0xa5a, 0xd56, 0xc5f, 0xf55, 0xe5c, ...
+ 0x15c, 0x55 , 0x35f, 0x256, 0x55a, 0x453, 0x759, 0x650, ...
+ 0xaf0, 0xbf9, 0x8f3, 0x9fa, 0xef6, 0xfff, 0xcf5, 0xdfc, ...
+ 0x2fc, 0x3f5, 0xff , 0x1f6, 0x6fa, 0x7f3, 0x4f9, 0x5f0, ...
+ 0xb60, 0xa69, 0x963, 0x86a, 0xf66, 0xe6f, 0xd65, 0xc6c, ...
+ 0x36c, 0x265, 0x16f, 0x66 , 0x76a, 0x663, 0x569, 0x460, ...
+ 0xca0, 0xda9, 0xea3, 0xfaa, 0x8a6, 0x9af, 0xaa5, 0xbac, ...
+ 0x4ac, 0x5a5, 0x6af, 0x7a6, 0xaa , 0x1a3, 0x2a9, 0x3a0, ...
+ 0xd30, 0xc39, 0xf33, 0xe3a, 0x936, 0x83f, 0xb35, 0xa3c, ...
+ 0x53c, 0x435, 0x73f, 0x636, 0x13a, 0x33 , 0x339, 0x230, ...
+ 0xe90, 0xf99, 0xc93, 0xd9a, 0xa96, 0xb9f, 0x895, 0x99c, ...
+ 0x69c, 0x795, 0x49f, 0x596, 0x29a, 0x393, 0x99 , 0x190, ...
+ 0xf00, 0xe09, 0xd03, 0xc0a, 0xb06, 0xa0f, 0x905, 0x80c, ...
+ 0x70c, 0x605, 0x50f, 0x406, 0x30a, 0x203, 0x109, 0x0 ];
+
+ tri_table =[
+ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 0, 8, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 0, 1, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 1, 8, 3, 9, 8, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 1, 2, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 0, 8, 3, 1, 2, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 9, 2, 10, 0, 2, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 2, 8, 3, 2, 10, 8, 10, 9, 8, -1, -1, -1, -1, -1, -1, -1;
+ 3, 11, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 0, 11, 2, 8, 11, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 1, 9, 0, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 1, 11, 2, 1, 9, 11, 9, 8, 11, -1, -1, -1, -1, -1, -1, -1;
+ 3, 10, 1, 11, 10, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 0, 10, 1, 0, 8, 10, 8, 11, 10, -1, -1, -1, -1, -1, -1, -1;
+ 3, 9, 0, 3, 11, 9, 11, 10, 9, -1, -1, -1, -1, -1, -1, -1;
+ 9, 8, 10, 10, 8, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 4, 7, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 4, 3, 0, 7, 3, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 0, 1, 9, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 4, 1, 9, 4, 7, 1, 7, 3, 1, -1, -1, -1, -1, -1, -1, -1;
+ 1, 2, 10, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 3, 4, 7, 3, 0, 4, 1, 2, 10, -1, -1, -1, -1, -1, -1, -1;
+ 9, 2, 10, 9, 0, 2, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1;
+ 2, 10, 9, 2, 9, 7, 2, 7, 3, 7, 9, 4, -1, -1, -1, -1;
+ 8, 4, 7, 3, 11, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 11, 4, 7, 11, 2, 4, 2, 0, 4, -1, -1, -1, -1, -1, -1, -1;
+ 9, 0, 1, 8, 4, 7, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1;
+ 4, 7, 11, 9, 4, 11, 9, 11, 2, 9, 2, 1, -1, -1, -1, -1;
+ 3, 10, 1, 3, 11, 10, 7, 8, 4, -1, -1, -1, -1, -1, -1, -1;
+ 1, 11, 10, 1, 4, 11, 1, 0, 4, 7, 11, 4, -1, -1, -1, -1;
+ 4, 7, 8, 9, 0, 11, 9, 11, 10, 11, 0, 3, -1, -1, -1, -1;
+ 4, 7, 11, 4, 11, 9, 9, 11, 10, -1, -1, -1, -1, -1, -1, -1;
+ 9, 5, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 9, 5, 4, 0, 8, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 0, 5, 4, 1, 5, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 8, 5, 4, 8, 3, 5, 3, 1, 5, -1, -1, -1, -1, -1, -1, -1;
+ 1, 2, 10, 9, 5, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 3, 0, 8, 1, 2, 10, 4, 9, 5, -1, -1, -1, -1, -1, -1, -1;
+ 5, 2, 10, 5, 4, 2, 4, 0, 2, -1, -1, -1, -1, -1, -1, -1;
+ 2, 10, 5, 3, 2, 5, 3, 5, 4, 3, 4, 8, -1, -1, -1, -1;
+ 9, 5, 4, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 0, 11, 2, 0, 8, 11, 4, 9, 5, -1, -1, -1, -1, -1, -1, -1;
+ 0, 5, 4, 0, 1, 5, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1;
+ 2, 1, 5, 2, 5, 8, 2, 8, 11, 4, 8, 5, -1, -1, -1, -1;
+ 10, 3, 11, 10, 1, 3, 9, 5, 4, -1, -1, -1, -1, -1, -1, -1;
+ 4, 9, 5, 0, 8, 1, 8, 10, 1, 8, 11, 10, -1, -1, -1, -1;
+ 5, 4, 0, 5, 0, 11, 5, 11, 10, 11, 0, 3, -1, -1, -1, -1;
+ 5, 4, 8, 5, 8, 10, 10, 8, 11, -1, -1, -1, -1, -1, -1, -1;
+ 9, 7, 8, 5, 7, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 9, 3, 0, 9, 5, 3, 5, 7, 3, -1, -1, -1, -1, -1, -1, -1;
+ 0, 7, 8, 0, 1, 7, 1, 5, 7, -1, -1, -1, -1, -1, -1, -1;
+ 1, 5, 3, 3, 5, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 9, 7, 8, 9, 5, 7, 10, 1, 2, -1, -1, -1, -1, -1, -1, -1;
+ 10, 1, 2, 9, 5, 0, 5, 3, 0, 5, 7, 3, -1, -1, -1, -1;
+ 8, 0, 2, 8, 2, 5, 8, 5, 7, 10, 5, 2, -1, -1, -1, -1;
+ 2, 10, 5, 2, 5, 3, 3, 5, 7, -1, -1, -1, -1, -1, -1, -1;
+ 7, 9, 5, 7, 8, 9, 3, 11, 2, -1, -1, -1, -1, -1, -1, -1;
+ 9, 5, 7, 9, 7, 2, 9, 2, 0, 2, 7, 11, -1, -1, -1, -1;
+ 2, 3, 11, 0, 1, 8, 1, 7, 8, 1, 5, 7, -1, -1, -1, -1;
+ 11, 2, 1, 11, 1, 7, 7, 1, 5, -1, -1, -1, -1, -1, -1, -1;
+ 9, 5, 8, 8, 5, 7, 10, 1, 3, 10, 3, 11, -1, -1, -1, -1;
+ 5, 7, 0, 5, 0, 9, 7, 11, 0, 1, 0, 10, 11, 10, 0, -1;
+ 11, 10, 0, 11, 0, 3, 10, 5, 0, 8, 0, 7, 5, 7, 0, -1;
+ 11, 10, 5, 7, 11, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 10, 6, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 0, 8, 3, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 9, 0, 1, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 1, 8, 3, 1, 9, 8, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1;
+ 1, 6, 5, 2, 6, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 1, 6, 5, 1, 2, 6, 3, 0, 8, -1, -1, -1, -1, -1, -1, -1;
+ 9, 6, 5, 9, 0, 6, 0, 2, 6, -1, -1, -1, -1, -1, -1, -1;
+ 5, 9, 8, 5, 8, 2, 5, 2, 6, 3, 2, 8, -1, -1, -1, -1;
+ 2, 3, 11, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 11, 0, 8, 11, 2, 0, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1;
+ 0, 1, 9, 2, 3, 11, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1;
+ 5, 10, 6, 1, 9, 2, 9, 11, 2, 9, 8, 11, -1, -1, -1, -1;
+ 6, 3, 11, 6, 5, 3, 5, 1, 3, -1, -1, -1, -1, -1, -1, -1;
+ 0, 8, 11, 0, 11, 5, 0, 5, 1, 5, 11, 6, -1, -1, -1, -1;
+ 3, 11, 6, 0, 3, 6, 0, 6, 5, 0, 5, 9, -1, -1, -1, -1;
+ 6, 5, 9, 6, 9, 11, 11, 9, 8, -1, -1, -1, -1, -1, -1, -1;
+ 5, 10, 6, 4, 7, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 4, 3, 0, 4, 7, 3, 6, 5, 10, -1, -1, -1, -1, -1, -1, -1;
+ 1, 9, 0, 5, 10, 6, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1;
+ 10, 6, 5, 1, 9, 7, 1, 7, 3, 7, 9, 4, -1, -1, -1, -1;
+ 6, 1, 2, 6, 5, 1, 4, 7, 8, -1, -1, -1, -1, -1, -1, -1;
+ 1, 2, 5, 5, 2, 6, 3, 0, 4, 3, 4, 7, -1, -1, -1, -1;
+ 8, 4, 7, 9, 0, 5, 0, 6, 5, 0, 2, 6, -1, -1, -1, -1;
+ 7, 3, 9, 7, 9, 4, 3, 2, 9, 5, 9, 6, 2, 6, 9, -1;
+ 3, 11, 2, 7, 8, 4, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1;
+ 5, 10, 6, 4, 7, 2, 4, 2, 0, 2, 7, 11, -1, -1, -1, -1;
+ 0, 1, 9, 4, 7, 8, 2, 3, 11, 5, 10, 6, -1, -1, -1, -1;
+ 9, 2, 1, 9, 11, 2, 9, 4, 11, 7, 11, 4, 5, 10, 6, -1;
+ 8, 4, 7, 3, 11, 5, 3, 5, 1, 5, 11, 6, -1, -1, -1, -1;
+ 5, 1, 11, 5, 11, 6, 1, 0, 11, 7, 11, 4, 0, 4, 11, -1;
+ 0, 5, 9, 0, 6, 5, 0, 3, 6, 11, 6, 3, 8, 4, 7, -1;
+ 6, 5, 9, 6, 9, 11, 4, 7, 9, 7, 11, 9, -1, -1, -1, -1;
+ 10, 4, 9, 6, 4, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 4, 10, 6, 4, 9, 10, 0, 8, 3, -1, -1, -1, -1, -1, -1, -1;
+ 10, 0, 1, 10, 6, 0, 6, 4, 0, -1, -1, -1, -1, -1, -1, -1;
+ 8, 3, 1, 8, 1, 6, 8, 6, 4, 6, 1, 10, -1, -1, -1, -1;
+ 1, 4, 9, 1, 2, 4, 2, 6, 4, -1, -1, -1, -1, -1, -1, -1;
+ 3, 0, 8, 1, 2, 9, 2, 4, 9, 2, 6, 4, -1, -1, -1, -1;
+ 0, 2, 4, 4, 2, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 8, 3, 2, 8, 2, 4, 4, 2, 6, -1, -1, -1, -1, -1, -1, -1;
+ 10, 4, 9, 10, 6, 4, 11, 2, 3, -1, -1, -1, -1, -1, -1, -1;
+ 0, 8, 2, 2, 8, 11, 4, 9, 10, 4, 10, 6, -1, -1, -1, -1;
+ 3, 11, 2, 0, 1, 6, 0, 6, 4, 6, 1, 10, -1, -1, -1, -1;
+ 6, 4, 1, 6, 1, 10, 4, 8, 1, 2, 1, 11, 8, 11, 1, -1;
+ 9, 6, 4, 9, 3, 6, 9, 1, 3, 11, 6, 3, -1, -1, -1, -1;
+ 8, 11, 1, 8, 1, 0, 11, 6, 1, 9, 1, 4, 6, 4, 1, -1;
+ 3, 11, 6, 3, 6, 0, 0, 6, 4, -1, -1, -1, -1, -1, -1, -1;
+ 6, 4, 8, 11, 6, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 7, 10, 6, 7, 8, 10, 8, 9, 10, -1, -1, -1, -1, -1, -1, -1;
+ 0, 7, 3, 0, 10, 7, 0, 9, 10, 6, 7, 10, -1, -1, -1, -1;
+ 10, 6, 7, 1, 10, 7, 1, 7, 8, 1, 8, 0, -1, -1, -1, -1;
+ 10, 6, 7, 10, 7, 1, 1, 7, 3, -1, -1, -1, -1, -1, -1, -1;
+ 1, 2, 6, 1, 6, 8, 1, 8, 9, 8, 6, 7, -1, -1, -1, -1;
+ 2, 6, 9, 2, 9, 1, 6, 7, 9, 0, 9, 3, 7, 3, 9, -1;
+ 7, 8, 0, 7, 0, 6, 6, 0, 2, -1, -1, -1, -1, -1, -1, -1;
+ 7, 3, 2, 6, 7, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 2, 3, 11, 10, 6, 8, 10, 8, 9, 8, 6, 7, -1, -1, -1, -1;
+ 2, 0, 7, 2, 7, 11, 0, 9, 7, 6, 7, 10, 9, 10, 7, -1;
+ 1, 8, 0, 1, 7, 8, 1, 10, 7, 6, 7, 10, 2, 3, 11, -1;
+ 11, 2, 1, 11, 1, 7, 10, 6, 1, 6, 7, 1, -1, -1, -1, -1;
+ 8, 9, 6, 8, 6, 7, 9, 1, 6, 11, 6, 3, 1, 3, 6, -1;
+ 0, 9, 1, 11, 6, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 7, 8, 0, 7, 0, 6, 3, 11, 0, 11, 6, 0, -1, -1, -1, -1;
+ 7, 11, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 7, 6, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 3, 0, 8, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 0, 1, 9, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 8, 1, 9, 8, 3, 1, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1;
+ 10, 1, 2, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 1, 2, 10, 3, 0, 8, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1;
+ 2, 9, 0, 2, 10, 9, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1;
+ 6, 11, 7, 2, 10, 3, 10, 8, 3, 10, 9, 8, -1, -1, -1, -1;
+ 7, 2, 3, 6, 2, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 7, 0, 8, 7, 6, 0, 6, 2, 0, -1, -1, -1, -1, -1, -1, -1;
+ 2, 7, 6, 2, 3, 7, 0, 1, 9, -1, -1, -1, -1, -1, -1, -1;
+ 1, 6, 2, 1, 8, 6, 1, 9, 8, 8, 7, 6, -1, -1, -1, -1;
+ 10, 7, 6, 10, 1, 7, 1, 3, 7, -1, -1, -1, -1, -1, -1, -1;
+ 10, 7, 6, 1, 7, 10, 1, 8, 7, 1, 0, 8, -1, -1, -1, -1;
+ 0, 3, 7, 0, 7, 10, 0, 10, 9, 6, 10, 7, -1, -1, -1, -1;
+ 7, 6, 10, 7, 10, 8, 8, 10, 9, -1, -1, -1, -1, -1, -1, -1;
+ 6, 8, 4, 11, 8, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 3, 6, 11, 3, 0, 6, 0, 4, 6, -1, -1, -1, -1, -1, -1, -1;
+ 8, 6, 11, 8, 4, 6, 9, 0, 1, -1, -1, -1, -1, -1, -1, -1;
+ 9, 4, 6, 9, 6, 3, 9, 3, 1, 11, 3, 6, -1, -1, -1, -1;
+ 6, 8, 4, 6, 11, 8, 2, 10, 1, -1, -1, -1, -1, -1, -1, -1;
+ 1, 2, 10, 3, 0, 11, 0, 6, 11, 0, 4, 6, -1, -1, -1, -1;
+ 4, 11, 8, 4, 6, 11, 0, 2, 9, 2, 10, 9, -1, -1, -1, -1;
+ 10, 9, 3, 10, 3, 2, 9, 4, 3, 11, 3, 6, 4, 6, 3, -1;
+ 8, 2, 3, 8, 4, 2, 4, 6, 2, -1, -1, -1, -1, -1, -1, -1;
+ 0, 4, 2, 4, 6, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 1, 9, 0, 2, 3, 4, 2, 4, 6, 4, 3, 8, -1, -1, -1, -1;
+ 1, 9, 4, 1, 4, 2, 2, 4, 6, -1, -1, -1, -1, -1, -1, -1;
+ 8, 1, 3, 8, 6, 1, 8, 4, 6, 6, 10, 1, -1, -1, -1, -1;
+ 10, 1, 0, 10, 0, 6, 6, 0, 4, -1, -1, -1, -1, -1, -1, -1;
+ 4, 6, 3, 4, 3, 8, 6, 10, 3, 0, 3, 9, 10, 9, 3, -1;
+ 10, 9, 4, 6, 10, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 4, 9, 5, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 0, 8, 3, 4, 9, 5, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1;
+ 5, 0, 1, 5, 4, 0, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1;
+ 11, 7, 6, 8, 3, 4, 3, 5, 4, 3, 1, 5, -1, -1, -1, -1;
+ 9, 5, 4, 10, 1, 2, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1;
+ 6, 11, 7, 1, 2, 10, 0, 8, 3, 4, 9, 5, -1, -1, -1, -1;
+ 7, 6, 11, 5, 4, 10, 4, 2, 10, 4, 0, 2, -1, -1, -1, -1;
+ 3, 4, 8, 3, 5, 4, 3, 2, 5, 10, 5, 2, 11, 7, 6, -1;
+ 7, 2, 3, 7, 6, 2, 5, 4, 9, -1, -1, -1, -1, -1, -1, -1;
+ 9, 5, 4, 0, 8, 6, 0, 6, 2, 6, 8, 7, -1, -1, -1, -1;
+ 3, 6, 2, 3, 7, 6, 1, 5, 0, 5, 4, 0, -1, -1, -1, -1;
+ 6, 2, 8, 6, 8, 7, 2, 1, 8, 4, 8, 5, 1, 5, 8, -1;
+ 9, 5, 4, 10, 1, 6, 1, 7, 6, 1, 3, 7, -1, -1, -1, -1;
+ 1, 6, 10, 1, 7, 6, 1, 0, 7, 8, 7, 0, 9, 5, 4, -1;
+ 4, 0, 10, 4, 10, 5, 0, 3, 10, 6, 10, 7, 3, 7, 10, -1;
+ 7, 6, 10, 7, 10, 8, 5, 4, 10, 4, 8, 10, -1, -1, -1, -1;
+ 6, 9, 5, 6, 11, 9, 11, 8, 9, -1, -1, -1, -1, -1, -1, -1;
+ 3, 6, 11, 0, 6, 3, 0, 5, 6, 0, 9, 5, -1, -1, -1, -1;
+ 0, 11, 8, 0, 5, 11, 0, 1, 5, 5, 6, 11, -1, -1, -1, -1;
+ 6, 11, 3, 6, 3, 5, 5, 3, 1, -1, -1, -1, -1, -1, -1, -1;
+ 1, 2, 10, 9, 5, 11, 9, 11, 8, 11, 5, 6, -1, -1, -1, -1;
+ 0, 11, 3, 0, 6, 11, 0, 9, 6, 5, 6, 9, 1, 2, 10, -1;
+ 11, 8, 5, 11, 5, 6, 8, 0, 5, 10, 5, 2, 0, 2, 5, -1;
+ 6, 11, 3, 6, 3, 5, 2, 10, 3, 10, 5, 3, -1, -1, -1, -1;
+ 5, 8, 9, 5, 2, 8, 5, 6, 2, 3, 8, 2, -1, -1, -1, -1;
+ 9, 5, 6, 9, 6, 0, 0, 6, 2, -1, -1, -1, -1, -1, -1, -1;
+ 1, 5, 8, 1, 8, 0, 5, 6, 8, 3, 8, 2, 6, 2, 8, -1;
+ 1, 5, 6, 2, 1, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 1, 3, 6, 1, 6, 10, 3, 8, 6, 5, 6, 9, 8, 9, 6, -1;
+ 10, 1, 0, 10, 0, 6, 9, 5, 0, 5, 6, 0, -1, -1, -1, -1;
+ 0, 3, 8, 5, 6, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 10, 5, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 11, 5, 10, 7, 5, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 11, 5, 10, 11, 7, 5, 8, 3, 0, -1, -1, -1, -1, -1, -1, -1;
+ 5, 11, 7, 5, 10, 11, 1, 9, 0, -1, -1, -1, -1, -1, -1, -1;
+ 10, 7, 5, 10, 11, 7, 9, 8, 1, 8, 3, 1, -1, -1, -1, -1;
+ 11, 1, 2, 11, 7, 1, 7, 5, 1, -1, -1, -1, -1, -1, -1, -1;
+ 0, 8, 3, 1, 2, 7, 1, 7, 5, 7, 2, 11, -1, -1, -1, -1;
+ 9, 7, 5, 9, 2, 7, 9, 0, 2, 2, 11, 7, -1, -1, -1, -1;
+ 7, 5, 2, 7, 2, 11, 5, 9, 2, 3, 2, 8, 9, 8, 2, -1;
+ 2, 5, 10, 2, 3, 5, 3, 7, 5, -1, -1, -1, -1, -1, -1, -1;
+ 8, 2, 0, 8, 5, 2, 8, 7, 5, 10, 2, 5, -1, -1, -1, -1;
+ 9, 0, 1, 5, 10, 3, 5, 3, 7, 3, 10, 2, -1, -1, -1, -1;
+ 9, 8, 2, 9, 2, 1, 8, 7, 2, 10, 2, 5, 7, 5, 2, -1;
+ 1, 3, 5, 3, 7, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 0, 8, 7, 0, 7, 1, 1, 7, 5, -1, -1, -1, -1, -1, -1, -1;
+ 9, 0, 3, 9, 3, 5, 5, 3, 7, -1, -1, -1, -1, -1, -1, -1;
+ 9, 8, 7, 5, 9, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 5, 8, 4, 5, 10, 8, 10, 11, 8, -1, -1, -1, -1, -1, -1, -1;
+ 5, 0, 4, 5, 11, 0, 5, 10, 11, 11, 3, 0, -1, -1, -1, -1;
+ 0, 1, 9, 8, 4, 10, 8, 10, 11, 10, 4, 5, -1, -1, -1, -1;
+ 10, 11, 4, 10, 4, 5, 11, 3, 4, 9, 4, 1, 3, 1, 4, -1;
+ 2, 5, 1, 2, 8, 5, 2, 11, 8, 4, 5, 8, -1, -1, -1, -1;
+ 0, 4, 11, 0, 11, 3, 4, 5, 11, 2, 11, 1, 5, 1, 11, -1;
+ 0, 2, 5, 0, 5, 9, 2, 11, 5, 4, 5, 8, 11, 8, 5, -1;
+ 9, 4, 5, 2, 11, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 2, 5, 10, 3, 5, 2, 3, 4, 5, 3, 8, 4, -1, -1, -1, -1;
+ 5, 10, 2, 5, 2, 4, 4, 2, 0, -1, -1, -1, -1, -1, -1, -1;
+ 3, 10, 2, 3, 5, 10, 3, 8, 5, 4, 5, 8, 0, 1, 9, -1;
+ 5, 10, 2, 5, 2, 4, 1, 9, 2, 9, 4, 2, -1, -1, -1, -1;
+ 8, 4, 5, 8, 5, 3, 3, 5, 1, -1, -1, -1, -1, -1, -1, -1;
+ 0, 4, 5, 1, 0, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 8, 4, 5, 8, 5, 3, 9, 0, 5, 0, 3, 5, -1, -1, -1, -1;
+ 9, 4, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 4, 11, 7, 4, 9, 11, 9, 10, 11, -1, -1, -1, -1, -1, -1, -1;
+ 0, 8, 3, 4, 9, 7, 9, 11, 7, 9, 10, 11, -1, -1, -1, -1;
+ 1, 10, 11, 1, 11, 4, 1, 4, 0, 7, 4, 11, -1, -1, -1, -1;
+ 3, 1, 4, 3, 4, 8, 1, 10, 4, 7, 4, 11, 10, 11, 4, -1;
+ 4, 11, 7, 9, 11, 4, 9, 2, 11, 9, 1, 2, -1, -1, -1, -1;
+ 9, 7, 4, 9, 11, 7, 9, 1, 11, 2, 11, 1, 0, 8, 3, -1;
+ 11, 7, 4, 11, 4, 2, 2, 4, 0, -1, -1, -1, -1, -1, -1, -1;
+ 11, 7, 4, 11, 4, 2, 8, 3, 4, 3, 2, 4, -1, -1, -1, -1;
+ 2, 9, 10, 2, 7, 9, 2, 3, 7, 7, 4, 9, -1, -1, -1, -1;
+ 9, 10, 7, 9, 7, 4, 10, 2, 7, 8, 7, 0, 2, 0, 7, -1;
+ 3, 7, 10, 3, 10, 2, 7, 4, 10, 1, 10, 0, 4, 0, 10, -1;
+ 1, 10, 2, 8, 7, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 4, 9, 1, 4, 1, 7, 7, 1, 3, -1, -1, -1, -1, -1, -1, -1;
+ 4, 9, 1, 4, 1, 7, 0, 8, 1, 8, 7, 1, -1, -1, -1, -1;
+ 4, 0, 3, 7, 4, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 4, 8, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 9, 10, 8, 10, 11, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 3, 0, 9, 3, 9, 11, 11, 9, 10, -1, -1, -1, -1, -1, -1, -1;
+ 0, 1, 10, 0, 10, 8, 8, 10, 11, -1, -1, -1, -1, -1, -1, -1;
+ 3, 1, 10, 11, 3, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 1, 2, 11, 1, 11, 9, 9, 11, 8, -1, -1, -1, -1, -1, -1, -1;
+ 3, 0, 9, 3, 9, 11, 1, 2, 9, 2, 11, 9, -1, -1, -1, -1;
+ 0, 2, 11, 8, 0, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 3, 2, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 2, 3, 8, 2, 8, 10, 10, 8, 9, -1, -1, -1, -1, -1, -1, -1;
+ 9, 10, 2, 0, 9, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 2, 3, 8, 2, 8, 10, 0, 1, 8, 1, 10, 8, -1, -1, -1, -1;
+ 1, 10, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 1, 3, 8, 9, 1, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 0, 9, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ 0, 3, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
+ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 ] + 1;
+endfunction
+
+%!demo
+%! graphics_toolkit fltk
+%! N = 10; # N intervals per axis
+%! x = y = z = linspace(-4,4,N+1);
+%!
+%! f = vectorize(inline("x^3+y^3+z^3"));
+%! [T P G, nG] = implicit (f, 3, x, y, z);
+%!
+%! figure ();
+%! trimesh (T, P(:,1), P(:,2), P(:,3));
+%! axis equal
+%! view (115, 30)
+%!
+%! figure ();
+%! cdat = calc_shading (0.1, 1, .7, 4, [1 .2 .3], ...
+%! G, [1.0; 0.5; .8], [.5; 1.5; .1]);
+%! p = patch ("Faces", T, "Vertices", P, "FaceVertexCData", ...
+%! cdat, "FaceColor", "interp", "EdgeColor", "black");
+%! axis equal
+%! view (115, 30)
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