Chaikin's Algorithm: Proof of Convergence Chaikin's algorithm is, in some sense, similar to de Casteljau algorithm in that (in the limit) it produces a curve from a set of control points.
There are claims all over the internet that Chaikin's algorithm converges to a quadratic $B$-spline.
Where could I find a proof of convergence? Please note that I am asking for document that is not behind paywall (there should be a freely available draft of a paper for example). Books are fine though.
 A: Let the initial set of points be $\mathbf P_i, i = 0,\dots, n$. Let also $S$ be the B-spline of second degree with knot vector
$$
\mathbf T = \left\{u_{-1}, u_0 = 0, \dots, u_{n+1} = n+1, u_{n+2}\right\}, \quad u_0 \leq 0, \quad u_{n+2} \geq n+1
$$
and $\mathbf P_i$ as its control polygon. $u_{-1}$ and $u_{n+2}$ are boundary knots and can be defined arbitrary.

Chaikin's corner cutting is basically a knot insetion procedure for this specific B-spline. It is a procedure that changes control polygon and knot vector without altering the B-spline shape itself.
There are $n+1$ knot segments $[u_i, u_{i+1}],\; i = 0, 1, \dots, n$ and we're going to insert $n+1$ knots in the middle of each segment.
Insering knot $u = i + \frac{1}{2}$ is done by changing $\mathbf P_i$ into a pair $\mathbf Q_i^- = (1-a_i^-) \mathbf P_{i-1} + a_i^- \mathbf P_i$ and $\mathbf Q_i^+ = a_i^+ \mathbf P_{i} + (1 - a_i^+) \mathbf P_{i+1}$ where 
$$
a_{i}^- = \frac{u-u_{i-1}}{u_{i+1}-u_{i-1}} = \frac{3}{4}\\
a_{i}^+ = \frac{u_{i+2}-u}{u_{i+2}-u_{i}} = \frac{3}{4}\\
$$
The only $\mathbf Q$ that are ill-defined are $\mathbf Q_{0}^-$ and $\mathbf Q_{n}^+$, but those are not used anyway. After inserting all $n+1$ knots and throwing away $u_{-1}, u_{n+2}$ with $\mathbf Q_{0}^-$ and $\mathbf Q_{n}^+$, the knot vector and control polygon become
$$
\mathbf T' = \left\{0, \frac{1}{2}, 1, \dots, n, n + \frac{1}{2}, n+1, \right\}\\
\mathbf Q_0^{+},
\mathbf Q_1^{-},
\mathbf Q_1^{+},
\dots,
\mathbf Q_{n-1}^{-},
\mathbf Q_{n-1}^{+},
\mathbf Q_{n}^{-}.
$$

The knot vector left equally spaced, so one can scale and shift it to become
$$
\mathbf T' = \left\{-1, 0, 1, \dots, 2n-1, 2n, 2n+1 \right\}
$$
The new knot vector and new sett of points describe the same B-spline curve.
We can apply this procedure as many times as we need. Every thime we obtain a new set of points that converge to the actual curve since $\|\mathbf Q_{i+1}-\mathbf Q_i\| \to 0$ and the point $\frac{\mathbf Q_i + \mathbf Q_{i+1}}{2}$ lies on the curve.
