“Natural” or periodic interpolating cubic spline curve
curve = cscvn(points)
curve = cscvn(points)
returns
a parametric variational, or natural, cubic spline curve (in ppform) passing
through the given sequence points(:j), j =
1:end. The parameter value t(j)
for the jth point is chosen by Eugene Lee's [1] centripetal scheme,
i.e., as accumulated square root of chord length:
If the first and last point coincide (and there are no other repeated points), then a periodic cubic spline curve is constructed. However, double points result in corners.
The following provides the plot of a questionable curve through some points (marked as circles):
points=[0 1 1 0 -1 -1 0 0; 0 0 1 2 1 0 -1 -2]; fnplt(cscvn(points)); hold on, plot(points(1,:),points(2,:),'o'), hold off
Here is a closed curve, good for 14 February, with one double point:
c=fnplt(cscvn([0 .82 .92 0 0 -.92 -.82 0; .66 .9 0 ... -.83 -.83 0 .9 .66])); fill(c(1,:),c(2,:),'r'), axis equal
The break sequence t
is determined as
t = cumsum([0;((diff(points.').^2)*ones(d,1)).^(1/4)]).';
and csape
(with either periodic or variational
end conditions) is used to construct the smooth pieces between double
points (if any).
[1] E. T. Y. Lee. “Choosing nodes in parametric curve interpolation.” Computer-Aided Design 21 (1989), 363–370.