Piecewise polynomial spaces and geometric continuity of curves

Summary We say that a curve has geometric continuity if its curvatures and Frenet frame are continuous. In this paper we introduce spaces of piecewise polynomials which can be used to model space curves which have geometric continuity. We show that the basic theoretical properties of ordinary spline functions also hold for these spaces. These results extend and unify recent work on Beta-splines and Nu-splines which are used as a design tool in computer-aided geometric design of free form curves and surfaces.This work was initiated when the first author was on Sabbatical at Thomas J. Watson IBM Research Center, and was partially supported by the U.S.-Israel Binational Foundation, grant no. 86-00243/1.