A new stationary subdivision scheme is presented which performs slower topological refinement than the usual dyadic split operation. The number of triangles increases in every step by a factor of 3 instead of 4. Applying the subdivision operator twice causes a uniform refinement with tri-section of every original edge (hence the name sqrt(3)-subdivision) while two dyadic splits would quad-sect every original edge. Besides the finer gradation of the hierarchy levels, the new scheme has several important properties: The stencils for the subdivision rules have minimum size and maximum symmetry. The smoothness of the limit surface is C2 everywhere except for the extraordinary points where it is C1. The convergence analysis of the scheme is presented based on a new general technique which also applies to the analysis of other subdivision schemes. The new splitting operation enables locally adaptive refinement under built-in preservation of the mesh consistency without temporary crack-fixing between neighboring faces from different refinement levels. The size of the surrounding mesh area which is affected by selective refinement is smaller than for the dyadic split operation. We further present a simple extension of the new subdivision scheme which makes it applicable to meshes with boundary and allows us to generate sharp feature lines.
A simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented which generates C1 surfaces in the limit. The scheme satisfies important requirements for practical applications in computer graphics and engineering. These requirements include the necessity to generate smooth surfaces with local creases and cusps. The scheme can be applied to open nets in which case it generates boundary curves that allow a C0-join of several subdivision patches. Due to the local support of the scheme, adaptive refinement strategies can be applied. We present a simple device to preserve the consistency of such adaptively refined nets.