Control points in nurbs surfaces

Question

So I've read a lot about nurbs recently and completely understand nurbs curves ( even wrote a small library for it ). But I'm having some problem with surfaces. I can see that I need two sets of control points. My problem is that what the difference between points in these two sets is? Can anybody briefly explain it or give me some link that does?

Answer 1

I think my favorite way of understanding NURBS surfaces (if you already understand NURBS curves) is beads on a wire.

So, let's look at the much simpler example of a Bezier surface (I assume if you understand NURBS curves you understand Bezier curves).

A cubic Bezier curve has 4 control points. Imagine a Bezier curve which is just a smooth horizontal curve. You can compute any point on that curve given a parameter value (usually this is called t).. just plug t into the parametric equation of the curve, and a point is produced.

Now imagine you have 4 horizontal Bezier curves, each one is above the other. If you plug the same parameter value into all 4 curves, you get 4 points, one for each curve. Those are the beads on the wires. Let's call the parameter value for the horizontal curves 's'.

Take those 4 "bead" points and treat them as the control points of a vertical curve. Evaluate that curve at another parameter value (this one we'll call 't', like usual) and it will give you a point. That point is on the surface. Specifically, that's the point P(s,t).

So, given a 4x4 grid of control points, you can use beads on a wire to compute points on the surface. As s changes, the beads sweep out different curves along the wire.. the set of all those curves is the surface.

You can do the exact same thing with Nurbs curves.. you just need a knot vector for s, another knot vector for t, and a grid of control points.

Answer 2

For a NURBS surface, you dont need two sets of control points, you need a 2 dimensional grid or mesh of control points. This mesh will have n rows and m columns, and each point in the mesh will have an x, y and z co-ordinate as well as a w value, the NURBS weight for that point.