In many temporally ordered data sets, it is observed that the parameters of
the underlying distribution change abruptly at unknown times. The detection of
such changepoints is important for many applications. While this problem has
been studied substantially in the linear data setup, not much work has been
done for angular data. In this article, we utilize the intrinsic geometry of a
torus to introduce the notion of the `square of an angle' and use it to propose
a new measure of variation, called the `curved variance', of an angular random
variable. Using the above ideas, we propose new tests for the existence of
changepoint(s) in the concentration, mean direction, and/or both of these. The
limiting distributions of the test statistics are derived and their powers are
obtained using extensive simulation. It is seen that the tests have better
power than the corresponding existing tests. The proposed methods have been
implemented on three real-life data sets revealing interesting insights. In
particular, our method when used to detect simultaneous changes in mean
direction and concentration for hourly wind direction measurements of the
cyclonic storm `Amphan' identified changepoints that could be associated with
important meteorological events