  Principal Component Analysis (PCA) is a well-known linear dimension-reduction
technique designed for Euclidean data. In a wide spectrum of applied fields,
however, it is common to observe multivariate circular data (also known as
toroidal data), rendering spurious the use of PCA on it due to the periodicity
of its support. This paper introduces Toroidal Ridge PCA (TR-PCA), a novel
construction of PCA for bivariate circular data that leverages the concept of
density ridges as a flexible first principal component analog. Two reference
bivariate circular distributions, the bivariate sine von Mises and the
bivariate wrapped Cauchy, are employed as the parametric distributional basis
of TR-PCA. Efficient algorithms are presented to compute density ridges for
these two distribution models. A complete PCA methodology adapted to toroidal
data (including scores, variance decomposition, and resolution of edge cases)
is introduced and implemented in the companion R package ridgetorus. The
usefulness of TR-PCA is showcased with a novel case study involving the
analysis of ocean currents on the coast of Santa Barbara.

Methodology