This paper focuses on hallucinating a facial shape from a low-resolution 3D facial shape. Firstly, we give a constrained conformal embedding of 3D shape in R2, which establishes an isomorphic mapping between curved facial surface and 2D planar domain. With such conformal embedding, two planar representations of 3D shapes are proposed: Gaussian curvature image (GCI) for a facial surface, and surface displacement image (SDI) for a pair of facial surfaces. The conformal planar representation reduces the data complexity from 3D irregular curved surface to 2D regular grid while preserving the necessary information for hallucination. Then, hallucinating a low resolution facial shape is formalized as inference of SDI from GCIs by modeling the relationship between GCI and SDI by RBF regression. The experiments on USF HumanID 3D face database demonstrate the effectiveness of the approach. Our method can be easily extended to hallucinate those category-specific 3D surfaces sharing with similar geometric structures.