Abstract
This chapter is devoted to two- and three-dimensional manifolds of lines — so-called line congruences and line complexes. We have already encountered them in their simplest forms: Chap. 3 deals with linear complexes and linear congruences of lines, and in Sec. 6.3 we considered the three-dimensional manifold of lines of constant slope. Here we study line congruences and complexes from the viewpoint of projective and Euclidean differential geometry, together with applications in curve and surface theory in three-space. We show how the set of normals of a surface in Euclidean space — which is a line congruence — arises naturally in collision problems in five-axis milling. Further, we study algebraic and rational congruences and complexes, and their relations to geometrical optics.
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