Both envelopes of the central surface are shown. The principal planes of the hyperboloid are represented by scribe lines. The central surface of a surface Phi is the location of the principal centers of curvature of Phi and, at the same time, the envelope of the surface normal to Phi (cf. focal surface).

The model was developed under the direction of Brill and Klein.

The central surface of the single-sheet hyperboloid is a 12th-order surface. It has three plane return edges, namely two hyperbolas and an ellipse, which lie in three mutually perpendicular planes. Furthermore, the central surface has a double curve of the 24th order.

See Cayley, On the Centro-Surface of an Ellipsoid. Cambridge, Philos. Transactions, vol. XII, p. 319ff.; Salmon / Fiedler, Analytical Geometry of Space, vol. 1, article 207 and vol. 2, article 244, 2nd ed.