Smooth, easy to compute interpolating splines

Abstract

We present a system of interpolating splines with first-order and approximate second-order geometric continuity. The curves are easily computed in linear time by solving a diagonally dominant, tridiagonal system of linear equations. Emphasis is placed on the need to find aesthetically pleasing curves in a wide range of circumstances; favorable results are obtained even when the knots are very unequally spaced or widely separated. The curves are invariant under translation, rotation, and scaling, and the effects of a local change fall off exponentially as one moves away from the disturbed knot. Approximate second-order continuity is achieved by using a linear "mock curvature" function in place of the actual endpoint curvature for each spline segment and choosing tangent directions at knots so as to equalize these. This avoids extraneous solutions and other forms of undesirable behavior without seriously compromising the quality of the results. The actual spline segments can come from any family of curves whose endpoint curvatures can be suitably approximated, but we propose a specific family of parametric cubics. There is freedom to allow tangent directions and "tension" parameters to be specified at knots, and special "curl" parameters may be given for additional control near the endpoints of open curves. © 1986 Springer-Verlag New York Inc.