Holditch Theorem and Steiner Formula for the Planar Hyperbolic Motions

The Steiner formula and the Holditch Theorem for one-parameter closed planar Euclidean motions 1, 7] were expressed by H.R. Müller 9] under the one-parameter closed planar motions in the complex sense. In this paper, in analogy with complex motions as given by Müller 9], the Steiner formula and the mixture area formula are obtained under one parameter hyperbolic motions. Also Holditch theorems were expressed in the hyperbolic sense. The classical Holditch Theorem: If the endpoints A, B of a segment of fixed length are rotated once on an oval, then a given point X of this segment, with , describes a closed, not necessarily convex, curve. The area of the ring-shaped domain bounded by the two curves is πab, 1, 7].