Managing the shape of planar splines by their control polygons

Abstract

A practical formulation of the shape-preservation theorem is presented. It states how the control points should be positioned to obtain a prescribed curve shape and avoid shape anomalies. The concept of hyperconvex polygons is introduced. These polygons can be intersected by straight lines at three points, and are therefore not covered by previous versions of the theorem. The extended theorem addresses all the possible control-point configurations, and is proved for important classes of planar B-splines and beta2-splines.