I'm trying to manually (i.e. without fancy equations like the Cox-DeBoor recurrence relation) deduce the quadratic uniform B-spline basis functions.
In the book "Curves and Surfaces for Computer Graphics" it is explained, but there is one step that is not clear to me. Summary of the process:
A quadratic curve depends on 3 control points (say $\mathbf{P_0}$, $\mathbf{P_1}$ and $\mathbf{P_2}$ -- or $\mathbf{P_1}$, $\mathbf{P_2}$ and $\mathbf{P_3}$).
For each control point there is a basis/blending/weight function -- $a(t)$, $b(t)$ and $c(t)$, such that
$\begin{align}Segment_1 = a(t)\mathbf{P_0} + b(t)\mathbf{P_1} + c(t)\mathbf{P_2}\\Segment_2 = a(t)\mathbf{P_1} + b(t)\mathbf{P_2} + c(t)\mathbf{P_3}\end{align}$
The basis functions are of course quadratic polynomials (defined on the unit interval $[0, 1]$)
$\begin{align}a(t) = a_0 + a_1t + a_2t^2\\b(t) = b_0 + b_1t + b_2t^2\\c(t) = c_0 + c_1t + c_2t^2\end{align}$
Now, the requirements for a quadratic B-spline is that two consecutive segments join with $C^1$ continuity, i.e.
$\begin{align}a(1)\mathbf{P_0}+b(1)\mathbf{P_1}+c(1)\mathbf{P_2} &= a(0)\mathbf{P_1} + b(0)\mathbf{P_2}+c(0)\mathbf{P_3}\\a'(1)\mathbf{P_0}+b'(1)\mathbf{P_1}+c'(1)\mathbf{P_2} &= a'(0)\mathbf{P_1} + b'(0)\mathbf{P_2}+c'(0)\mathbf{P_3}\end{align}$
Now the author says that, because the four points $\mathbf{P_i}$ are arbitrary,
$\begin{align}a(1)&=0\\b(1)&=a(0)\\c(1)&=b(0)\\c(0)&=0\end{align}$
And something similar for the derivatives of the basis functions,
$\begin{align}a'(1)&=0\\b'(1)&=a'(0)\\c'(1)&=b'(0)\\c'(0)&=0\end{align}$
Now, I see that this is *a* solution (that is valid for all combinations of four points), but isn't it the case that for each individual case of four points there are other solutions (e.g. such that $a(1) \neq 0$)?
Besides these 8 equations, there is the partition-of-unity property of the basis functions such that there are now 9 equations to solve the 9 unknown coefficients of the basis functions ($a_0$, $a_1$, $a_2$, $b_0$, $b_1$, $b_2$, $c_0$, $c_1$, $c_2$). This results in the familiar quadratic basis functions
$\begin{align}B_{02} &= \frac{1}{2}(t^2-2t+1) \\ B_{12} &= \frac{1}{2}(-2t^2+2t+1)\\ B_{22} &= \frac{1}{2}t^2\end{align}$
So in conclusion: I thought that using these requirements, a unique curve could be defined, but it seems that there exist other solutions?
Could somebody shine some light on this method for deducing the basis functions?