Abstract
Compared to G k continuity, C k continuity simplifies the construction of functions on surfaces and their refinement, e.g. to solve differential equations on the surface. The new class of almost everywhere parametrically C 2 free-form surfaces provide such a parameterization. For example, a new bi-6 construction combines a fast-contracting C 2 guided subdivision surface with a tiny multi-sided G 1 cap. The cap is chosen to be smaller than any refinement anticipated for geometric modeling or computing on surfaces. Fast contraction means that one subdivision step shrinks the remaining hole more than three steps of Catmull-Clark subdivision. This yields smooth surfaces consisting of a finite number of pieces that are suitable for engineering practice. Both the subdivision construction and the cap are guided by a reference surface. This guide conveys the basic shape, but has a different structure and lower smoothness.
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