  We present canonical quantiles and depths for directional data following a
distribution which is elliptically symmetric about a direction $\mu$ on the
sphere $\mathcal{S}^{d-1}$. Our approach extends the concept of Ley et al. [1],
which provides valuable geometric properties of the depth contours (such as
convexity and rotational equivariance) and a Bahadur-type representation of the
quantiles. Their concept is canonical for rotationally symmetric depth
contours. However, it also produces rotationally symmetric depth contours when
the underlying distribution is not rotationally symmetric. We solve this lack
of flexibility for distributions with elliptical depth contours. The basic idea
is to deform the elliptic contours by a diffeomorphic mapping to rotationally
symmetric contours, thus reverting to the canonical case in Ley et al. [1]. A
Monte Carlo simulation study confirms our results. We use our method to
evaluate the ellipticity of depth contours and for trimming of directional
data. The analysis of fibre directions in fibre-reinforced concrete underlines
the practical relevance.
