Global Parameterization with Periodic Coordinates

We present a new globally smooth parameterization method for surfaces of arbitrary topology. Our method does not require any prior partition into charts nor any cutting. The chart layout (i.e., the topology of the base complex) and the parameterization emerge simultaneously from a global numerical optimization process. Given two orthogonal piecewise linear vector fields, our method computes two piecewise linear periodic functions, aligned with the input vector fields, by minimizing an objective function. The bivariate function they define is a smooth parameterization almost everywhere, except in the vicinity of the singular points of the vector field, where both the vector field and the derivatives of the parameterization vanish. Our method can construct quasi-isometric parameterizations at the expense of introducing additional singular points in non-developable regions where the curl of the input vector field is non-zero. We also propose a curvature-adapted parameterization method, that minimizes the curl and removes those additional singular points by adaptively scaling the parameterization. In addition, the same formalism is used to allow smoothing of the control vector fields. We demonstrate the versatility of our method by using it for quaddominant remeshing and T-spline surface fitting. For both applications, the input vector fields are derived by estimating the principal directions of curvatures. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Boundary representations.