Shape preservation behavior of spline curves

Abstract

Shape preservation behavior of a spline consists of criterial conditions for preserving convexity, inflection, collinearity, torsion and coplanarity shapes of data polgonal arc. We present our results which acts as an improvement in the definitions of and provide geometrical insight into each of the above shape preservation criteria. We also investigate the effect of various results from the literature on various shape preservation criteria. These results have not been earlier refered in the context of shape preservation behaviour of splines. We point out that each curve segment need to satisfy more than one shape preservation criteria. We investigate the conflict between different shape preservation criteria 1)on each curve segment and 2)of adjacent curve segments. We derive simplified formula for shape preservation criteria for cubic curve segments. We study the shape preservation behavior of cubic Catmull-Rom splines and see that, though being very simple spline curve, it indeed satisfy all the shape preservation criteria.