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 [1] centripetal scheme, i.e., as accumulated square root of chord length:

$${\displaystyle \sum _{i<j}\sqrt{\Vert \text{points}(:,i+1)-\text{points }(:,i)\Vert {}_2}}$$

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.

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

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