# Convexity preserving interpolation: stationary nonlinear subdivision and splines

Frans Kuijt

Research output: ThesisPhD Thesis - Research UT, graduation UT

## Abstract

This thesis deals with interpolation and approximation while preserving shape conditions. The problem is defined as follows. Given is a set of points, the data, which possesses certain shape properties, such as convexity or monotonicity. The question is to construct a function, for example a curve or a surface, that describes these data well and that possesses the same shape properties. A function describes the given data well when it interpolates the data or, otherwise, when it approximates the data well in a suitable norm. The additional requirement on the function is that it is sufficiently smooth, i.e., it is at least continuously differentiable. The techniques described in this thesis can be split up into two groups. The first, more traditional group deals with the use of splines, which consist of piecewise polynomials. The requirement on the shape is attained by imposing conditions on the coefficients of these splines. The conditions are most convenient if they are linear in the spline coefficients. Therefore, attention is paid to the linearisation of conditions for convexity and monotonicity. Beside the examination of linear conditions several linear objective functions are investigated and compared. The second group of methods are so-called subdivision schemes, of which the linear four-point scheme is a well-known example. In subdivision, new points are inserted between existing data points by calculation from a local group of data points. The density of the data can be increased to arbitrarily high level by repeated application of this process. In the limit of infinitely many data points a function which interpolates the given data is obtained, and, in addition, this function is continuous or even one or more times continuously differentiable. For most applications however, a limited number of iterations is sufficient to arrive at a useful result. As, in addition, subdivision methods are local, above mentioned iteration process requires a relatively small amount of computational effort.
Original language English University of Twente Traas, C.R., Supervisorvan Damme, Rudolf Martinus Josephus, Co-Supervisor 9 Oct 1998 Enschede Universiteit Twente 90-3651201-8 Published - 9 Oct 1998

## Keywords

• IR-29655
• METIS-140289

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