function o = sphere_triangle_vertices_to_orientation ( a, b, c )
%*****************************************************************************80
%
%% SPHERE_TRIANGLE_VERTICES_TO_ORIENTATION seeks the orientation of a spherical triangle.
%
% Discussion:
%
% Three points on a sphere actually compute two triangles; typically
% we are interested in the smaller of the two.
%
% As long as our triangle is "small", we can define an orientation
% by comparing the direction of the centroid against the normal
% vector (C-B) x (A-B). If the dot product of these vectors
% is positive, we say the triangle has positive orientation.
%
% By using information from the triangle orientation, we can correctly
% determine the area of a Voronoi polygon by summing up the pieces
% of Delaunay triangles, even in the case when the Voronoi vertex
% lies outside the Delaunay triangle. In that case, the areas of
% some of the Delaunay triangle pieces must be formally negative.
%
% Licensing:
%
% This code is distributed under the GNU LGPL license.
%
% Modified:
%
% 11 May 2010
%
% Author:
%
% John Burkardt
%
% Input:
%
% real A(3), B(3), C(3), three points on a sphere.
%
% Output:
%
% integer O, is +1 if the spherical triangle is judged to
% have positive orientation, and -1 otherwise.
%
%
% Please, column vectors only!
%
a = a( : );
b = b( : );
c = c( : );
%
% Centroid.
%
cd = ( a + b + c ) / 3.0;
%
% Cross product ( C - B ) x ( A - B );
%
v1 = c - b;
v2 = a - b;
cp = zeros ( 3, 1 );
cp(1) = v1(2) * v2(3) - v1(3) * v2(2);
cp(2) = v1(3) * v2(1) - v1(1) * v2(3);
cp(3) = v1(1) * v2(2) - v1(2) * v2(1);
%
% Compare the directions.
%
if ( cp' * cd < 0 )
o = - 1;
else
o = + 1;
end
return
end