  A generic family of distributions, defined on the surface of a curved torus
is introduced using the area element of it. The area uniformity and the maximum
entropy distribution are identified using the trigonometric moments of the
proposed family. A marginal distribution is obtained as a three-parameter
modification of the von Mises distribution that encompasses the von Mises,
Cardioid, and Uniform distributions as special cases. The proposed family of
the marginal distribution exhibits both symmetric and asymmetric, unimodal or
bimodal shapes, contingent upon parameters. Furthermore, we scrutinize a
two-parameter symmetric submodel, examining its moments, measure of variation,
Kullback-Leibler divergence, and maximum likelihood estimation, among other
properties. In addition, we introduce a modified acceptance-rejection sampling
with a thin envelope obtained from the upper-Riemann-sum of a circular density,
achieving a high rate of acceptance. This proposed sampling scheme will
accelerate the empirical studies for a large-scale simulation reducing the
processing time. Furthermore, we extend the Uniform, Wrapped Cauchy, and
Kato-Jones distributions to the surface of the curved torus and implemented the
proposed bivariate toroidal distribution for different groups of protein data,
namely, $\alpha$-helix, $\beta$-sheet, and their mixture. A marginal of this
proposed distribution is fitted to the wind direction data.
