Bezier basis functions can be determined using recursion:
$B_{i,p} = (1-t)B_{i,p-1}+tB_{i-1,p}$
So for a quadratic bezier basis, we get:
$1-2t+t^2$
$2t-2t^2$
$t^2$
So for a quadratic bezier curve we simply take linear combinations of these basis functions and we get a curve in space.. But the problem is, if we introduce another control point, like in the image below, then we elevate the order so..
Enter B-splines! These are just piece-wise continuous bezier curves. But how can we derive the B-spline basis functions without using deBoor-cox recursion formula?
Let's consider this example:
How do I go about imposing $C^1$ continuity and partition of unity to obtain the basis functions for this example?
We can imagine that, for this example, we want to manually derive the B-spline basis functions corresponding to the knot vector: [0 0 0 .5 1 1 1]