In the framework of a generalized fully anisotropic Bardeen-Cooper-Schrieffer-type quantum theory, the low-temperature, low-field coherence-length tensor is derived from first principles. Anisotropy of the Fermi surface, the gap parameter, and the energy band structure in the effective-mass approximation is included. For lattice parameters and effective-mass ratios suitable for the high-${T}_{c}$ superconductors the coherence lengths along the principal axes show a marked anisotropy exceeding 20:1. The coherence length along the $c$ axis is found to be of the order of the lattice spacing $c$, suggesting marginal two dimensionality.