A spatial version of the theorem of the angle of circumference

Abstract

We try a generalization of the theorem of the angle of circumference to a version in three-dimensional Euclidean space and ask for pairs (ɛ, φ) of planes passing through two (different skew) straight lines e ∋ ɛ and f ∋ φ such that the angle α enclosed by ɛ and φ is constant. It turns out that the set of all such intersection lines is a quartic ruled surface Φ with e ∪ f being its double curve. We shall study the surface Φ and its properties together with certain special appearances showing up for special values of some shape parameters such as the slope of e and f (with respect to a fixed plane) or the angle α.