Holditch Theorem and Steiner Formula for the Planar Hyperbolic Motions

Abstract

The Steiner formula and the Holditch Theorem 1 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.In analogy with complex numbers C = R[i], z = x + iy ∈ C, x, y ∈ R, i 2 = −1, hyperbolic numbers (perplex numbers or split-complex numbers or double numbers)are composed of a real "x" and hyperbolic imaginary (or unipotent or hallucinatory) "y" part, [2-6, 8, 10, 12, 13]. Thus the hyperbolic numbers H = R[j] := {z = x + jy | x, y ∈ R, j 2 = 1} are just the real numbers extended to include the unipotent j in the same way that the complex numbers C := R[i] are the real numbers extended to include the imaginary i. The collection of all hyperbolic number is called the Hyperbolic plane H (or split-complex plane). The hyperbolic point z = x + jy can be represented 1 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 AX = a, XB = b, describes a closed, not necessarily convex, curve. The area of the ring-shaped domain bounded by the two curves is πab, [1,7].