  We define a regularized size-shape distortion (quality) measure for curved
high-order elements on a Riemannian space. To this end, we measure the
deviation of a given element, straight-sided or curved, from the stretching,
alignment, and sizing determined by a target metric. The defined distortion
(quality) is suitable to check the validity and the quality of straight-sided
and curved elements on Riemannian spaces determined by constant and point-wise
varying metrics. The examples illustrate that the distortion can be minimized
to curve (deform) the elements of a given high-order (linear) mesh and try to
match with curved (linear) elements the point-wise stretching, alignment, and
sizing of a discrete target metric tensor. In addition, the resulting meshes
simultaneously match the curved features of the target metric and boundary.
Finally, to verify if the minimization of the metric-aware size-shape
distortion leads to meshes approximating the target metric, we compute the
Riemannian measures for the element edges, faces, and cells. The results show
that, when compared to anisotropic straight-sided meshes, the Riemannian
measures of the curved high-order mesh entities are closer to unit.
Furthermore, the optimized meshes illustrate the potential of curved
$r$-adaptation to improve the accuracy of a function representation.
