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Interpolating subdivision with triangle aspect ratio control

The triangle meshes are often used to approximate smooth or piecewise smooth surfaces. Improving the uniformity of the approximation error distribution is crucial for the mesh quality. In order to achieve such improvement it is often necessary to increase the mesh resolution (by introducing new mesh vertices and triangles) where needed depending on a local estimator for the error distribution. Usually such estimators work per triangle producing an estimation for the local mesh curvature or the local deviation from the target surface.Many methods for subdivision of triangle meshes have been developed in the recent decades. The major flaw in most of them is the tendency to produce poor quality subdivided meshes when working on (starting from) irregular meshes. Even starting from a regular mesh which unevenly approximates the target surface, most methods tend to produce poor quality result (bad triangles aspect ratios and/or highly irregular mesh connectivity in some regions). Our research shows that the main problem with most of the methods is that they don't control the aspect ratios of the newly produced triangles and their close vicinity during the subdivision. Another flaw in most of the existing approaches is that the interpolant they construct doesn't behave well on highly irregular meshes (highly constrained and/or low quality) which often produces bumpy meshes as a result. Here we take advantage of the 2.5D type of the meshes that we process which allows us to simplify the interpolant and make it more stable in the irregular cases.In this paper we present a smooth interpolating subdivision approach for refinement of irregular 2.5D triangle meshes. It combines an edge-splitting subdivision with a Hermite-based smooth interpolation of the triangle mesh. Working on 2.5D meshes (defined in 2D domains) makes the task simpler so that the only thing required from the interpolant is to calculate the elevations on the target interpolation surface for the new mesh vertices introduced by the subdivision. The interpolant provides the data for the estimation of local approximation error at each subdivision step used to decide whether the target surface is locally well approximated (which is the stop criteria for the subdivision process). At the same time, the method estimates how well the local mesh quality is being maintained after each subdivision step, analyses the possibilities to improve the triangles aspect ratios and applies them whenever possible which also speeds up the convergence of the overall process.The described approach produces multiresolution triangle mesh which uniformly approximates the smooth interpolation surface while maintaining the mesh quality.