Documentation

# cscvn

“Natural” or periodic interpolating cubic spline curve

## Syntax

```curve = cscvn(points) ```

## Description

`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  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.

## Examples

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 ```

## Algorithms

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).

## References

 E. T. Y. Lee. “Choosing nodes in parametric curve interpolation.” Computer-Aided Design 21 (1989), 363–370.