This paper describes theoretical developments and algorithms involved in the design of a P2 cavity operator to generate anisotropic curved meshes. Both volume and surface are adapted. A high-level approach is chosen, such that the existing P1 cavity operator is used as-is to handle topology. The P2 extension performs the curving process and ensures geometric validity. Volume curvature is based on Riemannian edge length minimization, first requiring a description of the metric field along a Bézier edge: this leads to the proposed high-order extension of the log-Euclidean scheme and differentiation of geometrical quantities in this framework. Surface curvature is based on similar principles, with the added difficulty of CAD or CAD surrogate projection. Numerical results illustrating the P2 cavity operator’s ability to recover curvature, from surface geometry to boundary layers to metric fields are presented. Examples are based on 3D real-world geometries encountered in Computational Fluid Dynamics (CFD). This framework allow us to curve highly anisotropic meshes with around 10 million elements within minutes.