I have been working on polynomial splines for several years. Right now it is mainly spherical splines.

There are several methods for solving the problem of scattered data interpolation. Domain could be a line, a plane or the unit sphere.

There are local and global methods, and in one of the global methods we minimize an energy functional subject to interpolation and

smoothness conditions. If you are interested, look at some examples.

Univariate splines

Pictures below illustrate approximation of cos(x²) on the interval [0, 2π] using two methods. Blue curves are smoothing splines, red curve is an interpolating spline and green curves are graphs of the original function. In the second plot weights for the smoothing spline are 10^-3 and in the third one they are 10^-4.

Bivariate planar splines

For my technical report I studied tensor-product planar cubic and quintic smoothing splines.

In this case the spline solution does not interpolate given data, it approximates it within given parameters. The functional to be minimized is a weighted sum of least square and thin plate energy terms.

For the following pictures data values were measured a total of 60 grid-points. The measurements were erroneous and therefore smoothing of the data was required.

Circular splines

Circular splines are a lot of fun. In a picture below you can see an interpolation of arccosine function by a univariate spline. Data points are originally located on the unit circle. We change cartesian coordinates to polar and construct a univariate spline on an interval [0, 2π]. Note that, by choosing 0 and 2π to be our first and last nodes correspondingly we avoided poising continuity and smoothness conditions on the circular spline at the location corresponding to these angles. This is done on purpose, since the function to be approximated is not differentiable at 0. There are total of four data locations. On the next picture there are 20 locations at which data is collected. However none correspond to the 0 angle, and therefore the spline function passes smoothly through this angle.

Spherical splines

Spherical splines interpolate or approximate data on the unit sphere. In the left picture below the green sphere is our domain, the red stars correspond to the values to be approximated and blue lines indicate the directions of the unit vectors corresponding to the locations on the unit sphere. In the right picture you can see the minimal energy interpolating cubic spline.