  Tubular and membranous shapes display a wide range of morphologies that are
difficult to analyze within a common framework. By generalizing the classical
Helfrich energy of biomembranes, we model them as solutions to a curvature
optimization problem in which the principal curvatures may play asymmetric
roles. We then give a novel phase-field formulation to approximate this
geometric problem, and study its Gamma-limsup convergence. This results in an
efficient GPU algorithm that we validate on well-known minimizers of the
Willmore energy; the software for the implementation of our algorithm is freely
available online. Exploring the space of parameters reveals that this
comprehensive framework leads to a wide continuum of shape textures. This first
step towards a unifying theory will have several implications, in biology for
quantifying tubular shapes or designing bio-mimetic scaffolds, but also in
computer graphics or architecture.
