"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 *j*th point is chosen by Eugene Lee's [1] centripetal scheme,
i.e., as accumulated square root of chord length:

$$\sum _{i<j}\sqrt{\Vert \text{points}(:,i+1)-\text{points}(:,i)\Vert {}_{2}}$$

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

[1] E. T. Y. Lee. "Choosing nodes in
parametric curve interpolation." *Computer-Aided
Design* 21 (1989), 363–370.

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