Fit a curve (spline?) through a solid in 3D and map a value onto it

Question

I have a series of xyz data in a unit cube. It forms a sort of tapered & curved cylinder. It will begin and end at 0,0,0 and 1,1,1. I want some sort of best fit curve down the 'axis' of this shape. And I need to take a new xyz point and map/project it onto this best fit curve. If I was expecting a straight line, I'd use the first principal component, but in this case I'm almost certain that I want some kind of 3d spline. I'm not after a fit to the surface. To be clear, there is no response variable; it's just xyz data points, so a model is not what I need.

Can someone give me some suggestions for the name of what I want to do? I need some keywords to get me going. I'm working in R so if you can suggest a package even better. Alternatively, perhaps something in Matlab would work. But mostly I need to know what this process is called. Thanks.

Answer 1

Sounds like a job for a principal curve. In R try the princurve package based on Trevor Hastie's original code. The cems package would be worth a look too as it fits principal curves via a different approach, but it has been archived on CRAN (or was last time I looked) because it was failing checks there. cems is still there but you need to build it yourself.

You can look at a description of principal curves in Hastie et al (2010) Elements of Statistical Learning on page 541. The PDF is freely available.