Abstract
This chapter focuses on geometric concepts, such as projective differential geometric methods, sphere geometries, line geometry, and non-Euclidean geometries. It illustrates the applications of the respective geometries in geometric modeling. Projective differential geometry is based on properties of curves or surfaces, which are invariant under re-parameterization, re-normalization, and projective mappings. Line geometry investigates the set of lines in three-space. Special emphasis is put on a general important principle—namely, the simplification of a geometric problem by application of an appropriate geometric transformation. The chapter describes how to apply curve algorithms for computing with special surfaces, such as developable surfaces, canal surfaces, and ruled surfaces. An appropriate geometric transformation can map an arbitrary rational surface onto a rational surface.
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