Subdivision is a well-known and established method for generating smooth curves and surfaces from discrete data by repeated refinements. The typical input for such a process is a mesh of vertices. In this work we propose to refine 2D data consisting of vertices of a polygon and a normal at each vertex. Our core refinement procedure is based on a circle average, which is a new non-linear weighted average of two points and their corresponding normals. The ability to locally approximate curves by the circle average is demonstrated. With this ability, the circle average is a candidate for modifying linear subdivision schemes refining points, to schemes refining point-normal pairs. This is done by replacing the weighted binary arithmetic means in a linear subdivision scheme, expressed in terms of repeated binary averages, by circle averages with the same weights. Here we investigate the modified Lane-Riesenfeld algorithm and the 4-point scheme. For the case that the initial data consists of a control polygon only, a naive method for choosing initial normals is proposed. An example demonstrates the superiority of the above two modified schemes, with the naive choice of initial normals over the corresponding linear schemes, when applied to a control polygon with edges of significantly different lengths.

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Abstract. Motivated by the method for the reconstruction of 3D objects from a set of parallel cross sections, based on the binary operation between 2D sets termed "metric average", we developed an algorithm for the computation of the metric average between two intersecting convex polygons in 2D. For two 1D sets there is an algorithm for the computation of the metric average, with linear time in the number of intervals in the two 1D sets. The proposed algorithm has linear computation time in the number of vertices of the two polygons. As an application of this algorithm, a new technique for morphing between two convex polygons is developed. The new algorithm performs morphing in a non-intuitive way. Introduction.In this work an algorithm for the computation of the metric average between two intersecting convex polygons in 2D is developed and studied. The metric average of two compact sets is a union of the weighted averages between any point from any set of the two, and the subset of all the closest points to it from the other set. The original application of the metric average is for "piecewise linear" approximation of set-valued functions [2]. It is applied in spline subdivision schemes for compact sets, a procedure which is motivated by the problem of the reconstruction of a 3D smooth object from its parallel cross-sections [4]. An algorithm for the computation of the metric average of 1D compact sets with computation time which is linear in the number of connected subsets in the two 1D sets is introduced in [3].The metric average of two sets is a subset of the Minkowski average of the sets and generally is a non-convex set (even for two convex sets). The Minkowski average is much bigger than the metric average. For example the Minkowski average of a nonconvex set with itself is a larger set containing it, while the metric average is the set itself. The metric property of the metric average is that its Hausdorff distance from any one of the averaged sets changes linearly with the weight parameter in the average.For the reconstruction problem of a smooth 3D object from its parallel cross-sections, the assumption that the projections of two consecutive cross-sections into a parallel plane intersect significantly, is natural. The choice of convex polygons was made although for such polygons the reconstruction from parallel cross-sections can be done by using Minkowski sums [5]. Yet the algorithm developed in this work is regarded as a first step in developing an efficient algorithm for the computation of the metric average of two general polygons. The authors have recently developed a general algorithm for the computation of the metric average of two intersecting regular polygons.An additional outcome of this work is a new morphing technique between two convex polygons, based on the metric average. The morphing of shapes or objects is a common task in producing animations and visual effects. The goal of a morphing process is to

The metric average is a binary operation between sets in Rn which is used in the approximation of set-valued functions. We introduce an algorithm that applies tools of computational geometry to the computation of the metric average of 2D sets with piecewise linear boundaries

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