Report Number: CS-TR-85-1047
Institution: Stanford University, Department of Computer Science
Title: Smooth, easy to compute interpolating splines
Author: Hobby, John D.
Date: January 1985
Abstract: We present a system of interpolating splines with first and
approximate second order geometric continuity. The curves are
easily computed in linear time by solving a system of linear
equations without the need to resort to any kind of
successive approximation scheme. 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 scaling, rotation, and reflection,
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.
http://i.stanford.edu/pub/cstr/reports/cs/tr/85/1047/CS-TR-85-1047.pdf