Let cubic Bézier curve $C$ be based on points $p_0, p_1, p_2, p_3$. Suppose $L$ is the polyline through $p_0, p_1, p_2, p_3$. Is there some well known analytical relation between lengths of these two geometrical objects?
This paper has some material that might be relevant:
Specifying the arc length of Bézier curves
John A. Roulier
Computer Aided Geometric Design
Volume 10, Issue 1, February 1993, Pages 25-56
For example he shows that the length of the curve is less than the length of the polyline. The proof is fairly simple.