Learn more about Spline Toolbox

Example: A Spline Curve

As another simple example,

points = .95*[0 -1 0 1;1 0 -1 0]; 
sp = spmak(-4:8,[points points]);

provides a planar, quartic, spline curve whose middle part is a pretty good approximation to a circle, as the plot on the next page shows. It is generated by a subsequent

plot(points(1,:),points(2,:),'x'), hold on 
fnplt(sp,[0,4]), axis equal square, hold off

Insertion of additional control points would make a visually perfect circle.

Here are more details. The spline curve generated has the form , with -4:8 the uniform knot sequence, and with its control points the sequence with . Only the curve part between the parameter values 0 and 4 is actually plotted.

To get a feeling for how close to circular this part of the curve actually is, we compute its unsigned curvature. The curvature at the curve point of a space curve can be computed from the formula

in which is the Euclidean length of the 3-vector a, and is the cross product of the two 3-vectors a and b, and and are the first and second derivative of the curve with respect to the parameter used. We treat our planar curve as a space curve in the -plane, hence obtain the maximum and minimum of its curvature at 21 points as follows:

t = linspace(0,4,21);zt = zeros(size(t));
dsp = fnder(sp); dspt = fnval(dsp,t); ddspt = fnval(fnder(dsp),t);
kappa = abs(dspt(1,:).*ddspt(2,:)-dspt(2,:).*ddspt(1,:))./...
   (sum(dspt.^2)).^(3/2);
[min(kappa),max(kappa)] 

ans = 
     1.6747    1.8611

So, while the curvature is not quite constant, it is close to 1/radius of the circle, as we see from the next calculation:

1/norm(fnval(sp,0)) 

ans = 
     1.7864 

Spline Approximation to a Circle; Control Points Are Marked x

Construction

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