The joint modeling of angular and linear observations is crucial as data of this nature
are prevalent in multiple disciplines, for example the joint modeling of wind direction and another
climatological variable such as wind speed or air temperature, the direction an animal moves and the
distance moved, or wave direction and wave height. Hence, there is a need for developing flexible
distributions on the hyper-disc, which has support of the interior of the hyper-sphere, as it allows
for modeling the combination of angular and linear observations. This paper addresses this need by
developing flexible distributions for the disc that have the ability to capture any inherent bimodality
present in the data. A new class of bivariate distributions is proposed which has support on the unit
disc in two dimensions that includes, as a special case, the existing Möbius distribution on the disc.
This class is obtained by expressing the density function in a general form using a measurable function
termed as generator. Special cases of this generator are considered to demonstrate the flexibility.
By applying a conformal mapping to the generator function a new Möbius distribution class emanates.
This class of bivariate distributions on the disc is the first to account for bimodality and skewness
present in the data. The flexible behavior of the proposed models in terms of bimodality and skewness
is graphically demonstrated. Preliminary evidential analysis of the wind data observed at Marion
Island reveals the absence of unimodality in the data. The fit of the proposed models, which account
for bimodality, to the Marion Island wind data were evaluated analytically and visually.