Geometry Near Singularities

Subdivision surfaces have to be analyzed in the terms of differential geometry. This chapter summarizes well-known concepts, such as the Gauss map, the principal curvatures and the fundamental forms, but also develops material that is not found in standard text books, such as the embedded Weingarten map, that is crucial to understanding subdivision surfaces.

Parametric singularities in the form of isolated ‘extraordinary points’ are a key feature of subdivision surfaces. The analysis of such singularities requires a separate assessment of parametric and geometric continuity. Accordingly, we will define function spaces C ^k _r where k indicates the smoothness of the parametrization, except at isolated points, and r measures the smoothness of the resulting surface in the geometric sense.

After providing special notations for dot and cross products in Sect. 2.1/16, we consider basic concepts from the differential geometry of regularly parametrized surfaces in Sect. 2.2/17. In particular, the embedded Weingarten map, which is given by a (3 × 3)-matrix, is introduced as a geometric invariant for the study of curvature properties. Unlike the principal directions, it is uniquely defined and continuous even at umbillic points. This property is crucial for our subsequent considerations of limit properties of subdivision surfaces at singular points.

In Sect. 2.3/23, the standard requirement on the regularity of the parametrization is suspended at an isolated point to allow for the structural conditions of subdivision surfaces. To establish geometric continuity, we first introduce the concept of ‘normal continuity’. That is, we require that the normal map can be continuously extended from the regular neighborhood to the singular point. This unique normal is used to define a differential-geometric notion of smoothness. If and only if the projection of the surface to the tangent plane is injective, the surface is single-sheeted and meets the requirements of a two-dimensional manifold. Then, the surface can be viewed as the graph of a scalar-valued function in a local coordinate system: the parameters are associated with the tangent plane, and function values are measured in the normal direction. To capture both analytic and geometric smoothness, we call a single-sheeted surface C ^k _r if its parametrization is C^k and the local height function is C^r. In case of single-sheetedness, we can use continuity of the Gauss map and the embedded Weingarten map to decide membership in C ^k₁ and C ^k₂ , respectively. This approach circumvents an explicit construction of the local height function. Using the embedded Weingarten map avoids having to select consistent coordinate systems in the set of tangent planes, as is necessary when working with the standard Weingarten map.