Corner cutting with trapezoidal augmentation for area-preserving smoothing of polygons and polylines

Recently, a new subdivision method was introduced by the author for smoothing polygons and polylines while preserving the enclosed area [Gordon D. Corner cutting and augmentation: an area-preserving method for smoothing polygons and polylines. Computer Aided Geometric Design 2010; 27(7):551–62]. The new technique, called “corner cutting and augmentation” (CCA), operates by cutting corners with line segments and adding the cut area of each corner to two augmenting structures constructed on the two incident edges; this operation can be iterated as needed. Area is preserved in a local sense, meaning that when a corner is cut, the cut area is added to the other side of the line in immediate proximity to the cut corner. Thus, CCA is also applicable to self-intersecting polygons and polylines, and it enables local control. CCA was originally developed with triangular augmentation, which was called CCA1. This work presents CCA2, in which the augmenting structures are trapezoids. A theoretical result from previous work is used to show that certain implementation restrictions guarantee the existence and the G¹-continuity of the limit curve of CCA2, and also the preservation of convexity. The main difference between CCA1 and CCA2 is that the limit curve of CCA1 does not contain straight line segments, while CCA2 can contain such segments. CCA2 allows the user to determine how closely each iteration follows its previous polygon. Potential applications include computer aided geometric design, an alternative to spline approximation, an aid to artistic design, and a possible alternative to multiresolution curves.     

Harmonic rational Bézier curves, p-Bézier curves and trigonometric polynomials

In a recent article, Ge et al. (1997) identify a special class of rational curves (Harmonic Rational Bézier (HRB) curves) that can be reparameterized in sinusoidal form. Here we show how this family of curves strongly relates to the class of p-Bézier curves, curves easily expressible as single-valued in polar coordinates. Although both subsets do not coincide, the reparameterization needed in both cases is exactly the same, and the weights of a HRB curve are those corresponding to the representation of a circular arc as a p-Bézier curve. We also prove that a HRB curve can be written as a combination of its control points and certain Bernstein-like trigonometric basis functions. These functions form a normalized totally positive B-basis (that is, the basis with optimal shape preserving properties) of the space of trigonometric polynomials {1, sint, cost, …. sinmt, cosmt} defined on an interval of length < π.