Geometric Continuity

This chapter focuses on geometric continuity with emphasis on a constructive definition for piecewise parametrized surfaces. The chapter presents examples showing the need for a notion of continuity different from matching of Taylor expansions as in the case of functions. It also defines geometric continuity for parametric curves, and for surfaces, first along edges, then around points, and finally for a whole complex of patches which is called a G k free-form surface spline. G k characterizes a relation between specific maps while C k continuity is a property of the resulting surface. The composition constraint on reparametrizations and the vertex-enclosure constraints are highlighted. The chapter also covers alternative definitions and approaches to generating free-form surface splines, and briefly discusses geometric continuity in the context of implicit representations and of generalized subdivision. The generic construction of G k free-form surface splines is also explained pointing to some low degree constructions.

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