Wendelin L. F. Degen

Explicit continuity conditions for adjacent Be´zier surface patches

Abstract Liu and Hoschek recently gave necessary and sufficient conditions for G1-continuity of two adjacent Bezier surface patches. In this paper, not only the common edge but also the position of the tangent planes at its points are considered to be given. Then one obtains by algebraic methods explicit representations of the first order cross-boundary tangent vectors of the two patches. These methods can be extended to the G2 case. In combination with new results of J. Hahn, a complete and explicit solution is derived for G2-continuity too.

Best approximations of parametric curves by splines

A normal distance d N (C,C’) for pairs of plane parametric curves C, C’ (with given C and some restrictions for C’) is introduced. For certain classes of Bezier curves.ℬ’ we get a differentiable manifold of deviation functions and can apply the non-linear approximation theory. As main result we obtain that the error of the best approximant C’0∈ℬ’ has the alternant property like in the case of Chebyshev approximation. An algorithm to calculate C’0 explicitly and some examples are included.

Exploiting curvatures to compute the medial axis for domains with smooth boundary

In the paper, strong proofs of some basics facts about the medial axis transform of a planar region Ω with smooth boundary curve(s) are given. In particular the decomposition of Choi et al. (1997) is derived from the set of regular disks. Two special algorithms to compute the medial axis are presented. Due to the decomposition, they can be applied to each part of Ω separately. Emphasis is given to the local analysis of the boundarys curvatures. This leads to an interesting connection to the theory of Dupin cyclides. With this mean, a predictor/corrector algorithm (as in the numerical analysis of ODEs) is developed. The examples show that the predictor yields already an excellent approximation; so only a few refining steps of the corrector are necessary.

Geometric Hermite interpolation: in memoriam Josef Hoschek

In this paper we present an overview over the more recent developments of Geometric Hermite Approximation Theory for planar curves. A general method to solve those problems is presented. Emphasis is put on the relations to differential geometry and to invariance against parameter transformations and the motion group of the underlying geometry. However, besides a few elementary cases, this leads to nonlinear systems of algebraic equations.Furthermore we give some geometric interpretations, a couple of examples and a detailed discussion of the case degree n = 4 with one contact point.

Construction of Be´zier rectangles and triangles on the symmetric Dupin horn cyclide by means of inversion

Abstract Dupin cyclides may be obtained as offsets of a special Dupin cyclide, the so-called symmetric Dupin horn cyclide. A novel approach based on the concept of inversion is presented for generating rational Bezier patches on the symmetric Dupin horn cyclide. This leads to a new formulation for rational rectangular biquadratic cyclide Bezier patches, and to a rational Bezier representation of triangular patches of degree 4 on the symmetric Dupin horn cyclide.

Die zweifachen Blutelschen Kegelschnittflächen

A surface Φ in projective space generated by a one parameter family of conics is called a conic surface of Blutel if the tangent planes of Φ taken along a generating conic, envelop a quadratic cone. If the conjugate curves (with respect to the generating conics) are conics, too, we call Φ a two-fold Blutels conic surface. In an earlier paper [4] it was shown that the planes of both conic families, the generating and the conjugate one, belong to a pencil, each. The present paper completes these investigations by integrating the derivative equations (3), (8), (9), (10). As a final result, a complete classification of all these surfaces is given. They are all algebraic of at most fourth order and furthermore—besides the quadrics and certain degenerate cases—they are complex projectively equivalent to the cyclides of Dupin.

The geometric meaning of Nielson's affine invariant norm

Abstract This paper presents a geometric interpretation of the affine invariant metric introduced in (Nielson, 1987) and (Nielson and Foley, 1989). The norm allows the modification of several methods of scattered data interpolation to achieve affine invariance of the interpolating surfaces.

Sharp error bounds for piecewise linear interpolation of planar curves

Curves are commonly drawn by piecewise linear interpolation, but to worry about the error is rather seldom. In the present paper we give a strong mathematical error analysis for curve segments with bounded curvature and length. Though the result seems very clear, the proof turned out to be unexpectedly hard, comparable to that of the famous four vertex theorem.

The types of rational (2,1)-Bézier surfaces

Abstract Bezier surfaces with degrees n =2 and m =1 are ruled surfaces joining two conics. They are algebraic surfaces of orders 4, 3 and 2. Characteristic properties distinguishing these main types are derived by the “method of projection”. In particular the last case where cones, cylinders and hyperboloids occur, deserves interest.

The Shape of the Overhauser Spline

Among the well-known class of tangent-continuous parametric spline curves with cubic Bezier segments, interpolating a sequence of data points, the construction of A. Overhauser is one of the earliest examples after Ferguson [8]. Besides its extreme simplicity and robustness, this spline (in the cardinal case) deserves some interest because of its affine invariance. This allows a complete analysis of its shape-preserving properties, which is given in the present paper.