function vs = sphere_triangle_vertices_to_centroid ( r, v1, v2, v3 )
%*****************************************************************************80
%
%% sphere_triangle_vertices_to_centroid() gets a spherical triangle centroid in 3D.
%
% Discussion:
%
% A sphere centered at 0 in 3D satisfies the equation:
%
% X*X + Y*Y + Z*Z = R*R
%
% A spherical triangle is specified by three points on the sphere.
%
% The (true) centroid of a spherical triangle is the point
%
% VT = (XT,YT,ZT) = Integral ( X, Y, Z ) dArea / Integral 1 dArea
%
% Note that the true centroid does NOT, in general, lie on the sphere.
%
% The "flat" centroid VF is the centroid of the planar triangle defined by
% the vertices of the spherical triangle.
%
% The "spherical" centroid VS of a spherical triangle is computed by
% the intersection of the geodesic bisectors of the triangle angles.
% The spherical centroid lies on the sphere.
%
% VF, VT and VS lie on a line through the center of the sphere. We can
% easily calculate VF by averaging the vertices, and from this determine
% VS by normalizing.
%
% Of course, we still will not have actually computed VT, which lies
% somewhere between VF and VS!
%
% Licensing:
%
% This code is distributed under the MIT license.
%
% Modified:
%
% 21 February 2005
%
% Author:
%
% John Burkardt
%
% Input:
%
% real R, the radius of the sphere.
%
% real V1(3), V2(3), V3(3), the vertices of the triangle.
%
% Output:
%
% real VS(3), the coordinates of the "spherical
% centroid" of the spherical triangle.
%
dim_num = 3;
vs(1:dim_num) = ( v1(1:dim_num) + v2(1:dim_num) + v3(1:dim_num) ) / 3.0;
norm = sqrt ( sum ( vs(1:dim_num).^2 ) );
vs(1:dim_num) = r * vs(1:dim_num) / norm;
return
end