Cardinal Splines Part 1

When anyone hears the term ‘cardinal spline’ for the first time, the most common question is why the name ‘cardinal?’  Yes, that was my first question way back in the day 🙂  As it happens, there is a subtle relation between the spline and the cardinal series [1].  I’ll leave it to interested readers to pursue the history and mathematics of the series as an aside.

I believe it was Schoenberg who said that the cardinal spline bridges the gap between linear splines and the cardinal series [2].  Again, I’ll leave the details to those who want to either read Schoenberg’s book or pursue the matter to whatever degree it is documented online.  In this series, we will be interested in how to construct cardinal splines and get them into Degrafa.

First, let’s back up and look at cubic Hermite interpolation.  Quadratic Hermite interpolation was discussed in the Quadratic Hermite Curve series (see the Degrafa page for links to the entire series).  This case involved interpolating two points with a quadratic polynomial.  The three degrees of freedom were resolved by forcing the curve to interpolate the two points and assigning a start tangent.  This selection inferred an end tangent as shown below.

Start and End Tangents for a polynomial curve
Start and End Tangents for a polynomial curve

Instead of selecting the start tangent, T0 and forcing the end tangent, T1, what if we allowed both tangents to be variable?  This introduces an extra degree of freedom, allowing a cubic curve to be constructed.  The cubic curve has more flexibility, but requires both T0 and T1 to be initially specified.

The natural question is why not develop a cubic Hermite spline?  We could, although it requires two initial parameter selections.  Most designers prefer to have a single element of control or have everything automatically done for them.  An alternative approach is to examine specialized cases of Hermite interpolation where the tangents are automatically chosen.

In part 2, we will look at one such specification and how this method produces a computationally efficient interpolant with an adjustable tension parameter for shape control.


[1]  Higgins, J.R., “Five Short Stories About the Cardianal Series”, Bulletin of the American Mathematical Society, Vol. 12, No. 1, 1985.

[2] Schoenberg, I.J., “Cardinal Spline Interpolation”, The Mathematics Research Center, Univ. of Wisconsin-Madison, Regional Conference Series in Applied Mathematics, Capital City Press, 1993.

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