Cubic Spline : Start/End Segment interpolation

Question

I'm doing Spline interpolation in C++. I've used code from here: http://tehc0dez.blogspot.ch/2010/04/nice-curves-catmullrom-spline-in-c.html (the code is also linked on that page, it's up on github). The app is working just fine for closed contours, since it's copying the first three points to the end.

But in my case I need to be able to make an "open" shape - or rather line-, where the first and last points are not connected.

It is my understanding that since the Catmull-Rom spline is cubic, I won't be able to calculate the interpolated points for the first and last segment without adding any additional points.

I read that a common method to interpolate the points in those two segments is to use Quadratic Interpolation.

Unfortunately I can't wrap my head around how to do this. I've found out how to do quadratic Bezier approximation, but this is not what I want to do since I don't want to introduce any additional support points.

I found this site: http://dafeda.wordpress.com/2010/09/01/newtons-divided-difference-polynomial-quadratic-interpolation/ Which explains quite nicely how to do Quadratic interpolation. But I don't know how to adapt this for my case, where I want to calculate a new Point rather than just y.

Any help would be appreciated. Thanks !

Answer 1

Managed to implement a decent solution thanks to the formula found here: http://www.doc.ic.ac.uk/~dfg/AndysSplineTutorial/Parametrics.html

They also provide a nice Java applet to check out the different parameters.

For my problem I set the t1 value to 0.5 and check if t is above/below this threshold, since I only want to draw one segment of the curve! Works out nicely.

Answer 2

The usual way to do this is to add a second copy of your two end points... So if you have a spline passing through A-B-C-D then you will calculate the spline A-A-B-C-D-D.