A set of uniform meshes over a unit hemisphere has been generated using 3 different uniform control grids. The control grids 1, 2, and 3 contain 408, 1680, and 6744, elements, respectively. For each control grid, seven meshes have been generated with target element size 0.448, 0.224, 0.112, 0.056, 0.028, 0.014, and 0.007, respectively. Two sets of results have been produced -- the first one (Fig. 7) with ``exact'' projection (EP) technique to the limit surface and the second (Fig. 8) with ``approximate'' projection (AP) to Bezier triangular patch (with ``exact'' projection in last cycle of the mesh smoothing). The convergence criterion has been set to 0.1 % of the local element size (for ``exact'' as well as ``approximate'' projections). The dependence of the computational time on the number of generated elements for individual control grids is depicted in Figures 7 and 8. Note that instead of using the total computational time rather the part corresponding strictly to the mesh generation process and part related to one cycle of the smoothing phase (either with ``exact'' or ``approximate'' projection) are used.

It is quite clear from Figure 7 that, when using only the ``exact'' projection, each time the size of control triangles is halved the total computing time is reduced by a constant value (specific for a particular element size). This can be explained by the reduction of the level of subdivision required by the convergence criterion by one whenever the control grid element size is halved. It is also apparent that this results in computational complexity of the overall algorithm. On the other hand, when using the ``approximate'' projection, the computational complexity of the mesh generation and smoothing based on the ``approximate'' projection approaches (the logarithmic contribution of the octree data structure is negligible).

An interesting issue to be addressed is, how much the limit surface deviates from the correct shape, or, in other words, how well geometrical information is preserved by the adopted modified Butterfly subdivision scheme. The maximum inside and outside deviation of the nodes from the ideal unit hemisphere exhibited by the initial control grids are in all cases around . The actual quality of geometry representation in the generated meshes is presented again in terms of maximum inside and outside deviation from the ideal unit hemisphere in Figure 9. It reveals that the accuracy of the geometrical representation deteriorates (but not more than one order of magnitude) with decreasing number of elements in the initial control grid and initially also with the decreasing target element size. However, starting with a threshold element size (in this particular case 0.056) the deviations remain almost constant with further decreasing target element size. Similar observations can be made when recursively using the generated mesh as initial control grid for the next discretization.

*Daniel Rypl
2005-12-03*