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A New Look at the So-Called Trammel of Archimedes

Apostol, Tom M. and Mnatsakanian, Mamikon A. (2009) A New Look at the So-Called Trammel of Archimedes. American Mathematical Monthly, 116 (2). pp. 115-133. ISSN 0002-9890. https://resolver.caltech.edu/CaltechAUTHORS:20090901-094318840

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Abstract

The paper begins with an elementary treatment of a standard trammel (trammel of Archimedes), a line segment of fixed length whose ends slide along two perpendicular axes. During the motion, points on the trammel trace ellipses, and the trammel produces an astroid as an envelope that is also the envelope of the family of traced ellipses. Two generalizations are introduced: a zigzag trammel, obtained by dividing a standard trammel into several hinged pieces, and a flexible trammel whose length may vary during the motion. All properties regarding traces and envelopes of a standard trammel are extended to these more general trammels. Applications of zigzag trammels are given to problems involving folding doors. Flexible trammels provide not only a deeper understanding of the standard trammel but also a new solution of a classical problem of determining the envelope of a family of straight lines. They also reveal unexpected connections between various classical curves; for example, the cycloid and the quadratrix of Hippias, curves known from antiquity.


Item Type:Article
Additional Information:© 2009 The Mathematical Association of America.
Issue or Number:2
Record Number:CaltechAUTHORS:20090901-094318840
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20090901-094318840
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:15512
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:14 Sep 2009 21:46
Last Modified:03 Oct 2019 00:58

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