function boundary_num = triangulation_order6_boundary_edge_count_euler ( ...
node_num, triangle_num, hole_num, boundary_num )
%*****************************************************************************80
%
%% triangulation_order6_boundary_edge_count_euler() counts boundary edges.
%
% Discussion:
%
% We assume we are given information about an order 6 triangulation
% of a set of points in the plane.
%
% By ignoring the midside nodes, we can determine the corresponding
% information for an order 3 triangulation, and apply Euler's formula
% to determine the number of edges that lie on the boundary of the
% set of points.
%
% Thus, if we have TRIANGLE_NUM triangles, and NODE_NUM nodes, we
% imagine that each triangle is replaced by 4 triangles, created
% by adding the edges created by joining the midside nodes.
%
% Thus, for 4 * TRIANGLE_NUM triangles, we can apply Euler's formula
% to compute the number of boundary edges.
%
% Now, to adjust the data to our order 6 triangles, we divide the
% number of boundary edges by 2.
%
% Licensing:
%
% This code is distributed under the MIT license.
%
% Modified:
%
% 11 June 2005
%
% Author:
%
% John Burkardt
%
% Reference:
%
% de Berg, Krevald, Overmars and Schwarzkopf,
% Computational Geometry, Section 9.1,
% Springer, 2000.
%
% Input:
%
% integer NODE_NUM, the number of nodes.
%
% integer TRIANGLE_NUM, the number of triangles.
%
% integer HOLE_NUM, the number of internal nodes.
%
% Output:
%
% integer BOUNDARY_NUM, the number of edges that lie on the
% boundary of the triangulation.
%
boundary_num = ( 2 * node_num + 2 * hole_num - 4 * triangle_num - 2 ) / 2;
return
end