A model of a hyperboloid of one sheet and its asymptotic cone

Abstract

In this article is described the construction of a thread model of a hyperboloid of one sheet ( H ) and its asymptotic cone ( C ). It ia simple to make, requiring only cardboard and thread, and can be made collapsible and of pocket size if desired. The model consists of two hinged pieces of cardboard (intersecting planes π and ) on which are drawn circles S H , respectively in which the planes meet H , and the concentric circles S C , respectively in which the planes meet C . A number of generators of the same system on H are now represented by threads joining S H and , and the corresponding parallel generators of C are represented by threads joining S C and . In order to ensure that these generators are well spaced, those of C are taken at equal eccentric angles apart in a principal elliptic section. The main theorem used in the design is that if l a generator of C , then the tangent plane to C at points of l meets H in two generators both of which are parallel to l