NURBS: Important Properties

Given a set of n+1 control points P₀, P₁, ..., P_n, each of which is associated with a non-negative weight w_i (i.e., P_i has weight w_i >= 0), and a knot vector U = { u₀, u₁, ..., u_m } of m+1 knots, the NURBS curve of degree p is defined as follows:

where R_i,p(u) is defined as follows:

The reason a different notation is used is because we want to rewrite the definition of a NURBS curve in a form as close to that of a B-spline curve as possible. In the above notation, R_i,p(u)'s are NURBS basis functions.

Important Properties of NURBS Basis Functions

1. R_i,p(u) is a degree p rational function in u
2. Nonnegativity -- For all i and p, R_i,p(u) is nonnegative
   
3. Local Support -- R_i,p(u) is a non-zero on [u_i,u_i+p+1)
   Since N_i,p(u) is non-zero on [u_i,u_i+p+1), so is R_i,p(u). Note that we assume w_i's are non-negative.
4. On any knot span [u_i, u_i+1), at most p+1 degree p basis functions are non-zero, namely: R_i-p,p(u), R_i-p+1,p(u), R_i-p+2,p(u), ..., and R_i,p(u)
   
5. Partition of Unity -- The sum of all non-zero degree p basis functions on span [u_i, u_i+1) is 1:
   
6. If the number of knots is m+1, the degree of the basis functions is p, and the number of degree p basis functions is n+1, then m = n + p + 1 :
   
7. Basis function R_i,p(u) is a composite curve of degree p rational functions with joining points at knots in [u_i, u_i+p+1 )
   
8. At a knot of multiplicity k, basis function R_i,p(u) is C^p-k continuous.
   Therefore, increasing multiplicity decreases the level of continuity, and increasing degree increases continuity.
9. If w_i = c for all i, where c is a non-zero constant, R_i,p(u) = N_i,p(u)
   Therefore, B-spline basis functions are special cases of NURBS basis functions when all weights become a non-zero constant. We have mentioned the special of c = 1.

Important Properties of NURBS Curves

1. NURBS curve C(u) is a piecewise curve with each component a degree p rational curve
   Actually, each component is a rational Bézier curve.
2. Equality m = n + p + 1 must be satisfied
   
3. A clamped NURBS curve C(u) passes through the two end control points P₀ and P_n
4. Strong Convex Hull Property: the NURBS curve is contained in the convex hull of its control points. Moreover, if u is in knot span [u_i,u_i+1), then C(u) is in the convex hull of control points P_i-p, P_i-p+1, ..., P_i
   We have made it very clear that all weights must be non-negative. If some of them are negative, the strong convex hull property or even the convex hull property will not hold. In the following, the left figure is a NURBS curve of degree 2 with n = 2, m = 5 and the first three and last three knots clamped. The weights of the two control points at both ends are 1's and the weight of the middle control point is 0.5. This is actually an elliptic arc. The curve segment lies in the convex hull.
   

   The middle figure has the weight of the middle control point set to zero. Since this control point has no effect, the result is the line segment determined by the endpoints. It still lies in the convex hull.

   If the weight is changed to -0.5, the curve segment is not contained in the convex hull and hence the convex hull property fails.

5. Local Modification Scheme: changing the position of control point P_i only affects the curve C(u) on interval [u_i, u_i+p+1)
   This follows from the local modification scheme property of B-spline basis functions. Recall that R_i,p(u) is non-zero on interval [u_i, u_i+p+1). If u is not in this interval, since R_i,p(u) is zero and R_i,p(u)P_i has no effect in computing p(u). On the other hand, if u is in the indicated interval, R_i,p(u) is non-zero, and if R_i,p(u)P_i is changed, so does C(u).

   This local modification scheme is very important to curve design, because we can modify a curve locally without changing the shape in a global way. Moreover, if fine-tuning curve shape is required, one can insert more knots (and therefore more control points) so that the affected region could be restricted to a very narrow one. We shall talk about knot insertion later.

6. C(u) is C^p-k continuous at a knot of multiplicity k
   If u is not a knot, C(u) is in the middle of a curve segment of degree p and is therefore infinitely differentiable. If u is a knot in the non-zero domain of R_i,p(u), since R_i,p(u) is only C^p-k continuous, so does C(u).
7. Variation Diminishing Property:
   The variation diminishing property also holds for NURBS curves. If the curve is contained in a plane (resp., space), this means no straight line (resp., plane) intersects a NURBS curve more times than it intersects the curve's control polyline.
8. B-spline Curves and Bézier Curves Are Special Cases of NURBS Curves
   If all weights are equal, a NURBS curve becomes a B-spline curve. If furthermore n = p (i.e., the degree of a B-spline curve is equal to n, the number of control points minus 1) and there are 2(p + 1) = 2(n + 1) knots with p + 1 of them clamped at each end, this NURBS curve reduces to a Bézier curve.
9. Projective Invariance
   If a projective transformation is applied to a NURBS curve, the result can be constructed from the projective images of its control points. This is a nice property. When we want to apply a geometric or even projective transformation to a NURBS curve, this property guarantees that we can apply the transformation to control points and the transformed NURBS curve is defined by the transformed control points. Therefore, we do not have to transform the curve.

   Note that Bézier curves and B-spline curves only satisfy the affine invariance property rather than this projective invariance property. This is because only NURBS curves involve projective transformations.