New approaches in the modeling of subdivision surfaces

Abstract

Recursive subdivision surfaces allow considerable freedom in designing surfaces of arbitrary topology. Tools to manipulate them, however, are still not as powerful as existing tools for longer established modeling paradigms, such as B-spline surfaces. Furthermore, the most well-behaved and most widely used schemes are approximating schemes, which do not interpolate their initial control points. This dissertation describes a new modeling paradigm, providing the possibility of locally choosing an interpolating variant of the conventionally approximating subdivision scheme. Our approach combines the advantages of approximating schemes with the precise control of interpolating schemes. Unlike other solutions that mostly focus on locally changing the weighting factors of the subdivision scheme, we keep the underlying uniform scheme intact. Our method is based upon introducing additional control points on well-chosen locations, with optional interactive user control over the tangent plane (or surface normal) and the tension of the surface near the interpolating control points. The same techniques used for surface modeling and editing are also adapted to implement a versatile free-form deformation tool, especially designed for 2D textured objects. Based on subdivision surfaces applied in 2D, our method successfully combines the following features: fluid good-looking movement, both general global and precise local control and explicit discontinuities. As a different item of interest, we noticed a lack in the current range of subdivision surface schemes. Quadrilateral schemes are organized logically as primal and dual schemes, but for triangular subdivision, only primal schemes are described in the literature. This is a pity, as recently research papers have been published showing that primal and dual schemes can be successfully combined to create surfaces with an arbitrarily high degree of continuity. Therefore, we introduce a new hexagonal scheme, opening a fascinating range of possibilities.