Continuity conditions for spline curves

Abstract

The conditions for continuity of direction and curvature at a knot in a vector-valued spline are derived. A method of normalising tangent vector magnitudes at knots is suggested. Many examples are displayed of closed spline curves constructed to pass through a series of knots, continuous in slop and curvature, with the segments normalised. Some of the figures represent shoe components; all the figures generated are acceptable, whether or not the defining knots are evenly spaced.