Corner cutting and augmentation: An area-preserving method for smoothing polygons and polylines

Abstract

A new subdivision method is presented for smoothing polygons and polylines while preserving the enclosed area. 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. The concept of a “bounding hull” of a convex polygon is defined and used to show that certain implementation restrictions guarantee the existence and the G 1 -continuity of the limit curve, and also the preservation of convexity when the original polygon is convex. CCA allows the definition of various profiles which determine how closely each iteration follows its previous polygon. Potential applications include computer aided geometric design, an alternative to spline approximation, as an aid to artistic design, and for some applications, as a potential alternative to multiresolution curves.