triangulation_delaunay_discrepancy, a MATLAB code which computes the local Delaunay discrepancy of a triangulation.
A (maximal) triangulation is Delaunay if and only if it is locally Delaunay.
A triangulation is Delaunay if the minimum angle over all triangles in the triangulation is maximized. That is, there is no other triangulation of the points which has a larger minimum angle.
A triangulation is locally Delaunay if, for every pair of triangles that share an edge, the minimum angle in the two triangles is larger than the minimum angle in the two triangles formed by removing the common edge and joining the two opposing vertices.
This function examines the question of whether a given triangulation is locally Delaunay. It does this by looking at every pair of neighboring triangles and comparing the minimum angle attained for the current triangle pair and the alternative triangle pair.
Let A(i,j) be the minimum angle formed by triangles T(i) and T(j), which are two triangles in the triangulation which share a common edge. Let B(I,J) be the minimum angle formed by triangles S(i) and S(j), where S(i) and S(j) are formed by removing the common edge of T(i) and T(j), and joining the opposing vertices.
Then the triangulation is Delaunay if B(i,j) <= A(i,j) for every pair of neighbors T(i) and T(j).
If A(i,j) < B(i,j) for at least one pair of neighbors, the triangulation is not a Delaunay triangulation.
This program returns VALUE = min ( A(i,j) - B(i,j) ) over all triangle neighbors. VALUE is scaled to be in degrees, for comprehensibility. If VALUE is negative, then at least one pair of triangles violates the Delaunay condition, and so the entire triangulation is not a Delaunay triangulation. If VALUE is nonnegative, then the triangulation is a Delaunay triangulation.
It is useful to return VALUE, rather than a simple True/False value, because there can be cases where the Delaunay condition is only "slightly" violated. A simple example is a triangulation formed by starting with a mesh of squares and dividing each square into two triangles by choosing one of the diagonals of the square. The Delaunay discrepancy for this mesh, if computed exactly, is 0, but roundoff could easily result in discrepancies that were very slightly negative.
If VALUE is positive, and not very small in magnitude, then every pair of triangles in the triangulation satisfies the local Delaunay condition, and so the triangulation is a Delaunay triangulation.
If VALUE is negative, and not very small in magnitude, then at least one pair of triangles violates the Delaunay condition, and to a significant degree. The triangulation is not a Delaunay triangulation.
If the magnitude of VALUE is very close to zero, then the triangulation is numerically ambiguous. At least one pair of triangles violates or almost violates the condition, but no triangle violates the condition to a great extent. The user must judge whether the violation is significant or not.
triangulation_delaunay_discrepancy ( 'prefix' )where 'prefix' is the common prefix for the node and element files
The computer code and data files described and made available on this web page are distributed under the MIT license
triangulation_delaunay_discrepancy is available in a C++ version and a FORTRAN90 version and a MATLAB version.
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