Evaluating curve points

http://en.wikipedia.org/wiki/Bézier_curve

Catmull-Rom splines pass through all the control points. I think that's what you need.

If you don't mind a bit of math, you can always use Newton polynomials to build a n-degree polynomial that interpolates n+1 points. I prefer them over Lagrange polynomials or using Neville's scheme simply because they take O(n^2), whereas Newton takes O(n) + some
preprocessing time.

I would post an example on how to do this, but I can't get the equation tags to show the formulas and the explanation text. It does involve building a divided difference table to get a polynomial. You can either build a set of parametric equations to evaluate at any parameter, or you can put that parameter back in the divided difference table and compute it from there (faster for sure).

My preferred method for this problem is natural cubic splines.

My preferred method for this problem is natural cubic splines.

Thanks for posting that! I actually recently did a tweener based on cubic splines, but didn't know that article existed (so I had to do all the boring algebra).

@OP: Just to add another spline type into the mix in this thread, there's also Kochanek–Bartels splines (kb splines).

Yeah, the natural cubic spline approach is better than mine if you want the curves. This way, you get cubic Bezier curves that are C² with each other. With Newton polynomials, you'd end up having to convert from polynomial basis to Bernstein basis and then choose a simple knot vector to get it to a NURBS.

I believe Lagrange Interpolation is also a valid method.

I believe Lagrange Interpolation is also a valid method.

This is a bad idea. Look at this example from the same Wikipedia page you linked.

You'll be wanting a curve fron the hermite family, eg cardinal spline, kochanek-bartels, or catmull-rom.