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## Mesh Density Control

Two levels of mesh density specification on the surface are considered - the required element size specification and the curvature-based mesh density control. The former concept is used to explicitly prescribe the mesh density along the surface. The element size specification is stored at each point of the control polygon of the surface. These values are used to extract the required element spacing at any location on the surface. Moreover, a lower bound limit on the mesh density is used which effectively prescribes the maximum element spacing that is not allowed to be exceeded.

The expression used to interpolate the local element size specification at control points over the surface is similar to the expression describing the geometry of the rational Bezier surface and has the form

 (36)

where is the required element size at point on the surface and are the element size specifications at Bezier control points. The remaining variables have the the same meaning as in Eq. (1).

The curvature-based element size control is employed to enable an accurate representation of a surface by its discretization even if no particular element size is required. The criterion is based on the ratio between the appropriate element spacing and the radius of curvature at a given location on the surface. The principal radii of curvature and on a parameterized surface are given by the solution of the quadratic equation

 (37)

Only the smaller radius of curvature is relevant for the curvature-based element size control. The symbols , , and , , represent coefficients of the first and second fundamental form of the surface and are given by

 (38)

 (39)

where

 (40)

The unit normal vector at the surface can be evaluated as

 (41)

In the actual implementation, it is more convenient to express , , and as

 (42)

The second derivatives of are, in case of rational Bezier surface, given by these expressions

 (43)

 (44)

 (45)

 (46)

where

 (47)

 (48)

 (49)

 (50)

Next: Octree Control Space Up: Direct Triangulation of 3D Surfaces Previous: Direct Triangulation of 3D Surfaces

Daniel Rypl
2005-12-07