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I have read through the "The Dirty Little Secrets of NURBS" article by Pilot3d (http://www.pilot3d.com/NurbSecrets.htm) and was intrigued by the surface located control points.

It does explain that each control has a respective surface point but it doesn't go as far as to explain how they are found and how moving a surface control point translates to the original control points. If I had to guess, you would find the surface control points by looking for the point on the surface when the contribution from a control point is at its maximum. Not sure about converting changes back to the original control points

user2272296
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I've somewhat figured this out just by thinking about it.

If you consider the general NURBS equation:

nurbs equation

Lets say C(u_pi) is the point on the surface associated with a control point (how you decide this is technically arbitrary but it seems that the surface point closest to the control point will produce the best results) and that you would like to move it by a vector M.

So now you need to find the new P_i that takes into account this translation. If we take the general equation and subtract the contributions from all the control points except P_i (the control point we are interested in) then we get the following equation (assuming all weights are 1):

N_i,n * P_i + M = N_i,n * (P_i+P_idelta)

Then we can quite easily see that:

M = N_i,n*P_idelta

And hence you can control the shape of a NURBS surface by moving points on the surface rather than control points. The disadvantage of this method is that nearby surface points will also move but not at the rate. You can quite easily control the spread of the effect by the spreading the delta across several control points.

user2272296
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