UBC Mathematics Department

An amusing property of the Catenary (Java animation)

`` A bridge [...] is a very special thing. Haven't you seen how delicate they are in relation to their size? They soar like birds; they extend and embody our finest efforts; and they utilize the curve of heaven. When a catenary of steel a mile long is hung in the clear over a river, believe me, God knows. [...] the catenary, this marvelous graceful thing, this joy of physics, this perfect balance between rebellion and obedience, is God's own signature on earth. I think it pleases Him to see them raised.''

Quoted from Mark Helprin - Winter's Tale. (Copyright © 1983 Mark Helprin). Published by arrangement with Harcourt Brace Jovanovich, Inc.

A catenary (see our on-line Latin dictionary) is a curve described by the hyperbolic cosine function

y = cosh(x).

Such curves also arise as the solution of several problems from the calculus of variations. The name catenary is associated with the curve because it describes the shape formed by a chain or rope freely suspended by its endpoints.

It turns out that a polygon, such as the square above, can "roll" smoothly on a track made of segments of catenaries.

The animation above consists of 26 GIF frames, each approximately 600 bytes in size, derived from a 111 frame Mathematica animation. This was saved as a sequence of PostScript files, each of which was converted into a GIF file using GhostScript, then cropped using xv. The Mathematica program which generated the animation is from "Mathematica in Action" by Stan Wagon (wagon@macalstr.edu), published by W. H. Freeman and Co., New York, 1991; translated into German, Japanese, and Italian. The animation itself is displayed using Sun's demo Animator class, written by Herb Jellinek (jellinek@eng.sun.com)

The animation was produced by Djun M. Kim


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