UBC Mathematics Department
`` 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 functiony = 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 (firstname.lastname@example.org), 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 (email@example.com)
The animation was produced by Djun M. Kim