Point-normal convexity preserving curve interpolation

Abstract

This paper studies the notion of convexity preservation in the context of point-normal interpolation in 2D. It is proved that $G^1$-continuous piecewise generalized cubic curves in tension are always capable of preserving the convexity implied by the data with finite tension parameters. Focus is placed on the variable degree polynomials and it is proved that there always exist a $G^1$-continuous cubic spline that interpolates the data and is also convexity preserving.