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Path: utzoo!watmath!watnot!watdaisy!ghgonnet
From: ghgonnet@watdaisy.UUCP (Gaston H Gonnet)
Newsgroups: net.math
Subject: Re: Polar clarification. (bi-square-cles)
Message-ID: <7496@watdaisy.UUCP>
Date: Fri, 22-Nov-85 09:45:26 EST
Article-I.D.: watdaisy.7496
Posted: Fri Nov 22 09:45:26 1985
Date-Received: Sat, 23-Nov-85 00:14:57 EST
References: <268@epicen.UUCP> <325@ihnet.UUCP> <1713@uwmacc.UUCP>
Distribution: net
Organization: U of Waterloo, Ontario
Lines: 34

There was a land where they designed the roads in a very bumpy
way, so bumpy as to make life miserable to everybody.  This
was the consequence of a new minister of transportation, who
(as opposed to most in the trade) knew something.  He once
heard about the sqrt(2) not being a rational and that the
functions sinh(x) and cosh(x) were not the result of someone
lisping sin(x) and cos(x).  

To apply his knowledge he designed roads which had a cross
section given by

	y(x) = sinh(x) + sqrt(2)*(1-cosh(x)),

but since this would give problems for large values of x,
he just restricted it to 0 <= x <= ln(sqrt(2)+1).  To make
long distances he just mirrored the pattern and repeated it.

Needles to say that motorists were not very happy, neither
cyclists, until they discovered that the town blockhead (who
used to ride his "bi-square-cle" (bicycle with square wheels))
was riding at perfect level.  (His wheels had internal radius
of 1 unit (or side=2)).

Of course the idea spread rapidly, and once the smoothness
problem was solved, they found that to maintain constant speed
their pedalling was uneven, and if they were pedalling evenly
then their speed was uneven.  Surprisingly this was solved
accidentally by Mr Meathead (bi-square-cle repairman) when
he "fixed" the gears of Mr Blockhead.

	What shape of gears did Mr. Meathead designed that
will maintain constant speed at level ride (angular speed
of the wheels themselves is of no importance) for a constant
pedalling speed?