  We introduce a four-parameter extended family of distributions related to the
wrapped Cauchy distribution on the circle. The proposed family can be derived
by altering the settings of a problem in Brownian motion which generates the
wrapped Cauchy. The densities of this family have a closed form and can be
symmetric 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 the
trigonometric moments. Other topics related to the model, including alternative
derivations and M\"{o}bius transformation, are considered. Discussion of the
symmetric submodels is given. Finally, generalization to a family of
distributions on the sphere is briefly made.
