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Minimization of the low-temperature elastic free-energy functional of orientationlly ordered membranes involves independent variation of the membrane-shape, while keeping the orientational order on it (its texture) fixed. We propose an operational, coordinate-independent method for implementing such a variation. Using the Nelson-Peliti formulation of elasticity that emphasizes the interplay between geometry, topology, and thermal fluctuations of orientationally ordered membranes, we minimize the elastic free energy to obtain equations governing their equilibrium shape, together with associated free boundary conditions. Our results are essential for understanding and predicting equilibrium shapes as well as textures of membranes and vesicles; particularly under conditions in which shape deformations are large. [ABSTRACT FROM AUTHOR]