A Global G² Spline Space with Improved Geometry Consistency Near Extraordinary Vertices

This paper presents a spline space that fills in extraordinary surface patches of a subdivision scheme with curvature continuous bi-7 degree Bézier surface patches and with the least geometry consistence error among different levels of refinement. The scheme is built upon a bounded curvature subdivision scheme for quadrilateral meshes that produces two-ring extraordinary surface patches near an extraordinary corner position with the least polar artifacts, but the idea can also be extended to any other subdivision schemes for quadrilateral meshes. Explicit G² continuity constraints among the 1st-ring extraordinary surface patches and additional C² continuity constraints among other extraordinary surface patches and between the 2nd-ring extraordinary surface patches and neighboring regular surface patches are derived. Final basis functions of the resulting spline space are produced by minimizing a hybrid objective function of the local surface energy and the desired geometry consistency subjecting to exact satisfaction of the established G²/C² continuity constraints. The resulting basis functions can be stored for efficient future use. Numerical examples show that our method produces quality surfaces with well distributed reflection lines and with the least geometry consistency error compared with an existing G² construction using Catmull–Clark subdivision.