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The length of spirographic curves. (English)
Am. Math. Mon. 121, No. 3, 207-212 (2014).
If a disk $D$ of radius $r$ rolls, without slipping, on the inside of a larger circle $C$ of radius $R$, and if $P$ is a fixed point on $D$ at distance $d \le r$ from its center, then the curve traced by $P$ is called a {\it hypotrochoid}. As is known to anyone familiar with spirographs, varying $R$, $r$, and $d$ results in a rich variety of beautiful curves. Such, and related, curves are discussed by {\it V. Gutenmacher} and {\it N. B. Vasilyev} [Lines and curves. A practical geometry handbook. Basel: Birkhäuser (2004; Zbl 1086.51001)]. Notice when $R=2r$ and $d < r$, one obtains an ellipse. Calling a point on the hypotrochoid with maximal distance from the center of $C$ a {\it top}, and the arc between two consecutive tops a {\it bow}, the author of the paper under review proves that if $d < r$, then the length of a bow is equal to $(R-r)/R$ times the perimeter of the ellipse having semi-axes $r+d$ and $r-d$. Amazingly, he does this $\grave{a}~ la~ Euclid$, without any resort to calculus. It is worth mentioning that when $R=2r$ and $d=r$, then the locus of $P$ turns out, unexpectedly, to be a straight line. This is usually referred to as {\it Copernicus theorem}, as, for example, on page 3 of the afore-mentioned book of Gutenmacher and Vasilyev, and on page 145 of {\it H. Steinhaus}’s book [Mathematical snapshots. 3rd Am. ed., rev. and enl. Reprint. Oxford etc.: Oxford University Press (1983; Zbl 0513.00002)]. However, the theorem was known to the Islamic astronomer Naṣīr al-Dīn al-Ṭūsī (1201‒1274), who stated and proved it in his {\it Commentary on the Almagest} (1247) in the context of his solution for the latitudial motion of the inferior planets. Interested readers are referred to {\it I. N. Veselovsky}’s article [Copernicus and Naṣīr al-Dīn al-Ṭūsī, J. Hist. Astron. 4, No. 2, 128‒130 (1973; \url{doi:10.1177/002182867300400205})]. Copernicus theorem is also referred to as {\it al-Ṭūsī’s couple} and {\it rolling device}.
Reviewer: Mowaffaq Hajja (Irbid)
Classification: G45 A30 M55