Highest and lowest knot values:

The highest and lowest knot values are known as the extreme values of the knots. All other values of knots are known as interior values. If all of the interior knots are spaced consistently among the extreme values, the resulting B-spline is known as a uniform B- spline. Or else, it is known as a Non-uniform B-spline (NUBS).

A degree-p B-spline curve is described following :

here,    {P_i} are the n control points,

{N_i, p (u)} are degree -p B-spline basis functions over the knot vector along m intervals U = {a, ........ , a, u_{p + 1}, . . . , u_{m - p - 1}, b, . . . , b}

(a) Extreme values, a and b, are repeated (p + 1) times each. There are (m + 1) knots.

(b) In general, the knot vector is, non-uniform and non-periodic.

As before, we described some 'nice' properties of B-splines, before going through some instance.

1. If n = p, and U = {0, . . . , 0, 1, . . . , 1}, C (u) is a Bezier curve.

2. For a curve of degree p, along (n + 1) control points and (m + 1) knots, m = n + p + 1.

3. For a < u < b, C (a) = P₀, and C (b) = P_n.

4. Affine Invariance : Affine transformations of the coordinate system do not alter the shape of the B-spline curve.

5. Strong Convex Hull Property :

          (a) The curve has in the convex hull of its control points.

         (b) If u_i î_X  X_i + 1, and p î_L P - p - 1, then C (u) is enclosed in    the convex hull of the control points P_i - p, . . . , P_i.

This result might be derived from the non-negativity and partition of unity properties of the basis functions.

6. Local Modification: Moving P_i changes C (u) just in the interval [u_i, u_{i + p + 1}), as outside this interval, Ni, p (u) = 0.

7. A B-spline of degree-1 is identical to the control polygon. Since the degree of the B-spline is lowered, it comes closer and closer to the control polygon.