## Fachbereich Informatik

This paper describes some new algorithms for the accurate calculation of surface properties. In the first part an arithmetic on Bézier surfaces is introduced. Formulas are given, which determine the Bézier points and weights of the resulting surface from the points and weights of the operand surfaces. An application of the arithmetic operations to the surface interrogation methods are described in the second part. It turns out, that the quality analysis can be reduced to a few numerical stable operations. Finally the advantages and disadvantages of this method are discussed.

Optimal degree reductions, i.e. best approximations of \(n\)-th degree Bezier curves
by Bezier curves of degree \(n\) - 1, with respect to different norms are studied. It
is shown that for any \(L_p\)-norm the euclidean degree reduction where the norm is applied to the euclidean distance function of two curves is identical to componentwise degree reduction. The Bezier points of the degree reductions are found to lie on parallel lines through the Bezier points of any Taylor expansion of degree \(n\) - 1 of the original curve. This geometric situation is shown to hold also in the case of constrained degree reduction. The Bezier points of the degree reduction are explicitly given in the unconstrained case for \(p\) = 1 and \(p\) = 2 and in the constrained case for \(p\) = 2.

The problem of constructing a geometric model of an existing object from a set of boundary points arises in many areas of industry. In this paper we present a new solution to this problem which is an extension of Boissonnat's method [2]. Our approach uses the well known Delaunay triangulation of the data points as an intermediate step. Starting with this structure, we eliminate tetrahedra until we get an appropriate approximation of the desired shape. The method proposed in this paper is capable of reconstructing objects with arbitrary genus and can cope with different point densities in different regions of the object. The
problems which arise during the elimination process, i.e. which tetrahedra can be eliminated, which order has to be used to control the process and finally, how to stop the elimination procedure at the right time, are discussed in detail. Several examples are given to show the validity of the method.