C ^∞ smooth freeform surfaces over hyperbolic domains

Constructing smooth freeform surfaces of arbitrary topology with higher order continuity is one of the most fundamental problems in shape and solid modeling. This paper articulates a novel method to construct C ^∞ smooth surfaces with negative Euler numbers based on hyperbolic geometry and discrete curvature flow. According to Riemann uniformization theorem, every surface with negative Euler number has a unique conformal Riemannian metric, which induces Gaussian curvature of - 1 everywhere. Hence, the surface admits hyperbolic geometry. Such uniformization metric can be computed using the discrete curvature flow method: hyperbolic Ricci flow. Consequently, the basis function for each control point can be naturally defined over a hyperbolic disk, and through the use of partition-of-unity, we build a freeform surface directly over hyperbolic domains while having C ^∞ property. The use of radial, exponential basis functions gives rise to a true meshless method for modeling freeform surfaces with greatest flexibilities, without worrying about control point connectivity. Our algorithm is general for arbitrary surfaces with negative Euler characteristic. Furthermore, it is C ^∞ continuous everywhere across the entire hyperbolic domain without singularities. Our experimental results demonstrate the efficiency and efficacy of the proposed new approach for shape and solid modeling.