The image on the Riemann sphere of any circle under a complex rational mapping with numerator
and denominator having degrees no more than 
has length no longer than 
.
Spijker's Lemma
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References
Edelman, A. and Kostlan, E. "How Many Zeros of a Random Polynomial are Real?" Bull. Amer. Math. Soc. 32, 1-37, 1995.Wegert, E. and Trefethen, L. N. "From the Buffon Needle Problem to the Kreiss Matrix Theorem." Amer. Math. Monthly 101, 132-139, 1994.Referenced on Wolfram|Alpha
Spijker's LemmaCite this as:
Weisstein, Eric W. "Spijker's Lemma." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SpijkersLemma.html