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We introduce a four-parameter extended family of distributions related to thewrapped Cauchy distribution on the circle. The proposed family can be derivedby altering the settings of a problem in Brownian motion which generates thewrapped Cauchy. The densities of this family have a closed form and can besymmetric or asymmetric depending on the choice of the parameters.Trigonometric moments are available, and they are shown to have a simple form.Further tractable properties of the model are obtained, many by utilizing thetrigonometric moments. Other topics related to the model, including alternativederivations and M\"{o}bius transformation, are considered. Discussion of thesymmetric submodels is given. Finally, generalization to a family ofdistributions on the sphere is briefly made.

An extended family of circular distributions related to wrapped Cauchy distributions via Brownian motion