B-spline Curves:

Supposed that you have to represent a very complex, curved shape approximated by, say, 20 points. If we design this shape by using a Bezier, we shall require a degree 19 curve. Such curves are hard to maintain/modify and compute. We would desire to work with lower degree functions for computational reasons. Additionally, depicting a curve by a single high degree function is not good while we modify shapes - as each control point has some effect on the shape of the curve. Therefore, changing the location of P18 will change the shape (a little) near the control point P₂. But frequently, in design, we would like more "localized" control on shape modification. Now we learn a method to solve both of the above problems.

The solution to the problem is the following: divide the overall shape into smaller segments, and represent each segment with a low order polynomial. The math's of this simple idea is not too simple, but we will look at the basics at least. The figure below illustrated this idea, where the curve C (u) is built up of 3 segments, defined as C_i (u), i = 1, 2, 3, the domain of C_i is over u_i - 1 î X_ î _ X_i, and u₀ = 0 < u₁ < u₂ < u₃. Now we may use any of the conventional (e.g. power basis, or Ferguson, Bezier etc.) mechanisms to explain each curve segment C_i, but to be useful, we should ensure that at the intermediate vertices, called as the breakpoints, the curves maintain some level of continuity

Figure: B-Spline Curve

There are several ways to implement the above idea.