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Geometrical Class and Degree for Surfaces in Three-Space

by Thomas F. Banchoff and Nicolaas H. Kuiper

In this paper, we examine embeddings and immersions *f*:M -> E^3,
where M is a smooth and polyhedral closed surface. We define the
concept of a *transversal line*, *geometric degree* of a surface,
and *algebraic degree* of a surface. We then relate the geometric
degree, algebraic degree, and Euler characteristic. The concept
of *tightness* for a surface is defined, we this is also
related to the degree of a surface. Finally, several generalizations
into higher dimensional space are given.