A non-linear circle-preserving subdivision scheme

Abstract

We describe a new method for constructing a sequence of refined polygons, which starts with a sequence of points and associated normals. The newly generated points are sampled from circles which approximate adjacent points and the corresponding normals.By iterating the refinement procedure, we get a limit curve interpolating the data. We show that the limit curve is G 1 , and that it reproduces circles. The method is invariant with respect to group of Euclidean similarities (including rigid transformations and scaling). We also discuss an experimental setup for a G 2 construction and various possible extensions of the method.

Clothoid fitting and geometric Hermite subdivision

“…In this paper, we present a family of geometric Hermite subdivision schemes for the generation of planar curves where the data to be refined are point-vector pairs, the latter serving as information on tangents or normals. Schemes refining pointvector pairs of that type were already suggested in [14,28]. The basic idea of these two approaches is to locally fit circles to the data and then to sample new points from them, but the specific methods are different.…”

“…1 shows the Fig. 1 Midpoint refinement using the proposed clothoid average (upper row), the circle average of [14] (middle row), and the circle average of [28] (lower row). We always display the points p j together with the normals n j = i exp iα j at level refinement of initial data consisting of two point-normal pairs by the so-called clothoid average, as described in this paper.…”

“…The resulting curve is S-shaped and interpolates the initial data. Also the scheme of Chalmoviansky and Jüttler [14], as shown in the middle row, interpolates the initial data, but has three turning points, leading to a less natural shape and a less optimal distribution of curvature. The scheme of Dyn and Lipovetsky [28], shown in the bottom row, produces a straight line (as it would for any choice of parallel normals).…”

Curvature of Approximating Curve Subdivision Schemes

“…A main task of curve modeling in product design is to reproduce segments of a variety of basic shapes, such as conics, spirals and clothoids exactly, and to transition smoothly between them. Since the standard uniform, polynomial subdivision algorithms cannot reproduce these basic shapes, a number of non-stationary curve subdivision algorithms have recently been devised to reproduce, in particular, circles and ellipses [15,20,3,6,18,4,7,2,19]. However, the introduction of parameter-dependent subdivision means that explicit basis functions for the control points are no longer easily available, removing a reliable technique to compute curvature.…”

Shape controlled interpolatory ternary subdivision

“…In binary subdivision literature, the construction of subdivision schemes reproducing conic sections has been addressed extensively [3][4][5][6][7][8][9][10]. Conversely, in the ternary context, there is still no available scheme for exactly generating these families of curves.…”