NURBS curves are easily generalized into NURBS patches.
One of the first common uses of NURBS was in 3D modelling packages, notably Alias's line starting in the early 1990s. NURBS are useful for a number of reasons, they:
The knot vector is a sequence of parameter values that determine the continuity along the NURBS curve. The number of knots is always equal to the number of control points plus the order of the curve.
The values of the knot vector must be in ascending order: the following vector is valid [0 0 1 2 3] while this one is not [0 0 2 1 3]. Also note that the only significant factor is the ratio of the values to each other: the knot vectors [0 0 1 2 3], [0 0 2 4 6] and [1 1 2 3 4] produce the same curve. There can be no more duplicate values than the degree: knots multiplicity <= degree. For degree 1 NURBS each knot is paired with a control point.
The order of the curve is greater or equal to 2, corresponding to a linear curve (order = 2), a quadratic curve (order = 3) and a cubic curve (order = 4). The curve is represented mathematically by a polynomial of same order, a cubic curve is represented by a degree 3 polynomial which order is 4. Besides, the number of control points must be equal to or greater than the order of the curve.