approximating a b-spline with a single bezier curve of any degree

Question

suppose we have b-spline composed of several bezier curves with known control points and nodes and is not necessarily smooth.

Is there a way to create a single bezier curve of any degree that will fairly accurately approximate the b-spline?

I checked Is it possible to convert a B-Spline into a Bezier curve? and it is suggested that there are "suitable methods" to approximate the spline. Does anyone know what the possible methods are and how they work?

thank you in advance

Answer 1

Suppose your original curve is $F:[0,1] \to \mathbb{R}^3$. To construct an approximating Bezier curve of degree $m$, proceed as follows:

1. Choose $m+1$ parameter values $t_0, \ldots, t_m$ in the interval $[0,1]$.
2. Calculate $m+1$ points $Q_i = F(t_i)$ on your original spline curve.
3. Calculate the (unique) Bezier curve $B:[0,1] \to \mathbb{R}^3$ that interpolates the points $Q_0, \ldots, Q_m$ at parameter values $t_0, \ldots, t_m$. In other words, find $B$ such that $B(t_i) = Q_i$ for $i=0,1,\ldots,m$.

More about step #3:

If $\phi_0, \ldots, \phi_m$ are the Bernstein polynomials of degree $m$, then we can express $B$ in the form $$ B(t) = \sum_{i=0}^m \phi_i(t)P_i $$ where $P_0, \ldots, P_m$ are the control points of the curve. So, we have to find $P_0, \ldots, P_m$ such that $$ \sum_{i=0}^m \phi_i(t_j)P_i = Q_j \quad\quad (j=0, 1, \ldots, m) $$ This is a nice simple system of linear equations that we can easily solve to find $P_0, \ldots, P_m$.

More about step #1:

The parameter values $t_0, \ldots, t_m$ should be distinct. It is tempting to make them equally spaced in the interval $[0,1]$, but this is a really bad idea, because it will often lead to a Bezier curve with wild oscillations. You need the $t_i$ values to be more closely spaced near $0$ and near $1$. A common approach is to space them like the zeros (or extrema) of Chebyshev polynomials, as outlined here. Since we're working on the interval $[0,1]$, not the interval $[-1,1]$ traditionally used in approximation theory, our parameter values should be: $$ t_i = \tfrac12 \cos\left( \tfrac{i\pi}{m} \right) + \tfrac12 \quad\quad (i = 0,1,\ldots,m) $$