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In this short note, we present a refined approximation for the log-ratio ofthe density of the von Mises$(\mu,\kappa)$ distribution (also called thecircular normal distribution) to the standard (linear) normal distribution whenthe concentration parameter \k{appa} is large. Our work complements the one ofHill (1976), who obtained a very similar approximation along with quantilecouplings, using earlier approximations by Hill & Davis (1968) ofCornish-Fisher type. One motivation for this note is to highlight theconnection between the circular and linear normal distributions through theircircular variance and (linear) variance.

A bridge between the circular and linear normal distributions