Abstract.
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.
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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 \(\overline{AX} = a, \overline{XB} = b\), describes a closed, not necessarily convex, curve. The area of the ring-shaped domain bounded by the two curves is πab, [1, 7].
An erratum to this article is available at http://dx.doi.org/10.1007/s00006-011-0278-4.
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Yüce, S., Kuruoğlu, N. Holditch Theorem and Steiner Formula for the Planar Hyperbolic Motions. AACA 19, 155–160 (2009). https://doi.org/10.1007/s00006-008-0131-6
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DOI: https://doi.org/10.1007/s00006-008-0131-6