COMPUTER AIDED GEOMETRIC DESIGN zyxwvutsrqponmlkjihg ELSEVIEK Computer Aided Geometric Design 11 (1994) 425-450 Curvature-sign-type boundary conditions in parametric cubic-spline interpolation P.D. Kaklis*, N.S. Sapidis ** zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK Ship zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Design Laboratory, Department of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Herron Polytechneiou Street, Zografou, Athens I5 773, Greece Received September 1992; revised August 1993 Abstract In this paper we prove a necessary and sufficient condition ensuring that the following problem possesses a solution: construct a twice-continuous cubic parametric spline, which interpolates a given set of planar points with a given parametrization and satisfies curvature- sign- ty pe boundary conditions, i.e., the sign of the curvature is prescribed at the boundary points of the data. Based on this solvability condition, we also derive and numerically test a simple interactive algorithm that solves the above problem. 1. Introduction Let 27 = {I, = (xm,ym)=,m = 1 , . . . , N} be a set of points in the plane with I, # I,+l,m = l,.. .,N- 1, let U = {u1,u2 ,..., UN} be a set of ordered real numbers with ~1 < 2.42 < . . . < UN and, finally, let C3 (D;U) be the family of C2- continuous cubic parametric splines Q(u) = (Ql(u),Q2(u))‘,u E [u~,uN], which interpolate the data set D with parametrization U, i.e., Q (u, ) = I,, m = 1 , . . . , N. If additional suitable conditions are to be satisfied at the boundary points 1, and IN of V, then the family C3 (D;U) contains only one element. Typical boundary conditions are the so-called type-1 boundary conditions: Q(ul) = to, Q( UN) = tN, the so-called type-11 boundary conditions: o( ~1) = SO, 0 (UN) = s,V and, in the case of closed data (I1 = IN), the periodic * Corresponding author. Email: kaklis@esprit.naval.ntua.gr ** Email: sapidis@esprit.naval.ntua.gr 0167-8396/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved SSDI0167-8396(93)E0038-F