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Convergence and normal continuity analysis of a bivariate nonstationary (level-dependent) subdivision scheme for 2-manifold meshes with arbitrary topology is still an open issue. Exploiting ideas from the theory of asymptotically equivalent subdivision schemes, in this paper we derive new sufficient conditions for establishing convergence and normal continuity of any rotationally symmetric, nonstationary subdivision scheme near an extraordinary vertex/face. [ABSTRACT FROM AUTHOR]

We present a framework for analyzing nonlinear $$\mathbb {R}^d$$ -valued subdivision schemes which are geometric in the sense that they commute with similarities in $$\mathbb {R}^d$$ . It allows us to establish $$C^{1,\alpha }$$ -regularity for arbitrary schemes of this type, and $$C^{2,\alpha }$$ -regularity for an important subset thereof, which includes all real-valued schemes. Safe bounds on the domain of convergence of the scheme and on the Hölder exponent of the limit curves can be found by determining the range of certain real-valued functions. This task can be executed automatically and rigorously by a computer when using interval arithmetics. [ABSTRACT FROM AUTHOR]

Five points in general position in R always lie on a unique conic, and three points plus two tangents also have a unique interpolating conic, the type of which depends on the data. These well-known facts from projective geometry are generalized: an odd number 2 n+1≥5 of points in R, if they can be interpolated at all by a smooth curve with nonvanishing curvature, will have a unique GC interpolant consisting of pieces of conics of varying type. This interpolation process reproduces conics of arbitrary type and preserves strict convexity. Under weak additional assumptions its approximation order is ϑ( h), where h is the maximal distance of adjacent data points f(t) sampled from a smooth and regular planar curve f with nonvanishing curvature. Two algorithms for the construction of the interpolant are suggested, and some examples are presented. [ABSTRACT FROM AUTHOR]