M. Technical Notes
Aspect Ratio, Triangulation Step, and Area Limitation
The aspect ratio (AR) is defined as:
Of the two angles shown the diagram, a3q and monotonic decrease of function 1/tan are used here. In addition, since angle q is the minimum, therefore 2cos(q)31
It means that to prove the above estimation it is necessary to demonstrate that
Since 90o3a3q, then
It remains to note, that .
Since area of a triangle is equal to (L1+L2)h/2, then it leads to, that is the above estimation for area, when angle q is constrained from below, results in to the above estimation of triangulation step.
Moreover, instead of estimation using the more accurate estimation we obtain
Matrix A (referred to as the Jacobian matrix) transfers points of the coordinate triangle to the real triangle. Matrix W transforms coordinate triangle into equilateral (ideal) triangle. Therefore matrix S=AW-1, which transforms ideal element into real one, defines deviation of real element from ideal one.
For any triangle, the free vertex which is located at the point x, and two other vertices – xi, xj, the Jacobian is equal to
det (xi – x, xj – x)
A - triangulation - is a classical Delaunay triangulation (see, for example, ). For other cases a version of J.Rupert  algorithm is used.
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