Properties of Waves

From NURBS Curves to Surfaces

By now you know the basics of applying your knowledge of curves to surfaces. You first extend the curve equations to two dimensions and then you compute partial derivatives that will be used to generate the tangent vectors. These tangent vectors are used to compute the surface normal vectors. As you can see in Equation 9.1, the equation for a NURBS surface is essentially the same as Equation 8.1 with the added weighting factor.

(9.1) Equation for NURBS Surfaces. 

Like B-spline surfaces, the knot vectors in the u and v directions are independent. There can be different knot values, different orders, and so on. However, the weight values are properties of the control points. Therefore, the weight for a particular point is the same in both the u and v directions. This is why you see two independent knot vectors but not two independent weight vectors. This is shown graphically in Figure 9.1. Each control point has a weight and there are independent sets of basis functions in the u and v directions.

Figure 9.1: The building blocks of a NURBS surface.

The partial derivatives of the equations in Equation 9.1 look quite intimidating, but they are derived using the same basic approaches outlined in Appendix A, "Derivative Calculus," and first seen in Chapter 7, "Basic Surface Concepts and Bezier Surfaces." Equation 9.2 shows the partial derivatives of Equation 9.1 with respect to u and v.

The equations are not nearly as frightening as they may seem. If you are uncomfortable with the equations, refer back to Appendix A and Chapter 5, "NURBS," and then apply the quotient rule to Equation 9.1. Writing out the equations should make them clearer. Also, the programmatic implementation should shed more light on how the numbers actually come together.

(9.2) Partial derivatives of Equation 9.1.