Localized construction of curved surfaces from polygon meshes: A simple and practical approach on GPU

We present a method for refining n-sided polygons on a given piecewise linear model by using local computation, where the curved polygons generated by our method interpolate the positions and normals of vertices on the input model. Firstly, we construct a Bezier curve for each silhouette edge. Secondly, we employ a new method to obtain C^1 continuous cross-tangent functions that are constructed on these silhouette curves. An important feature of our method is that the cross tangent functions are produced solely by their corresponding facet parameters. Gregory patches can therefore be locally constructed on every polygon while preserving G^1 continuity between neighboring patches. To provide a flexible shape control, several local schemes are provided to modify the cross-tangent functions so that the sharp features can be retained on the resultant models. Because of the localized construction, our method can be easily accelerated by graphics hardware and fully run on the Graphics Processing Unit (GPU).