I need a closed form equation for a curve.  This curve can be
visualized by by placing the centers of a series of circles along a
sine wave, and then "draping" a continuous curve along these circles. 
I believe that the curve is tangent to each circle where it touches
it.  Notice that the resulting curve looks something like a cycloid.

A simple normal offset doesn't seem to always work, since, depending
on the radius of the circles, certain discontinuities arise.

Here is a specific case for the sine wave and the circle which I'd like you to use:

y=5*sin(9*theta) (in degrees)
x²+y²=25

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Request for Question Clarification by mathtalk-ga on 13 Apr 2005 19:19 PDT

Hi, birdbrain-ga:

Help me visualize this more clearly, please, exactly what is needed here.

If I were to place the center of a circle of constant radius 5, per
your example, along every point of a sine wave of amplitude 5 but very
rapid frequency (short period), then the "upper boundary" of the
region covered by these infinitely many circles would be a continuous
and periodic curve.

It would have a "kink" (discontinuous first derivative) at the
midpoints between the periodic maxima y = 10 that coincide with the
arguments where the sine wave peaks.  These kinks would be the minima
for the boundary curve and would coincide with the sine wave
"valleys".

So the description reduces to a "piecewise" formula that holds between those kinks.

There probably is no more of a "closed form" for this curve than there
is for a sine wave itself, so I'm not sure to what extent I'd be able
to satisfy your requirement here.  It's possible that a power series
expansion whose coefficients are determined by a recurrance of some
kind is as close as I'd be able to come.

regards, mathtalk-ga

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Clarification of Question by birdbrain-ga on 13 Apr 2005 19:52 PDT

Hi Mathtalk-

Yes, it seems like you understand the problem well, and have
visualized it properly.  Could you provide a closed form equation of
the curve between the discontinuities (y=-10 and y=30, right?) which I
could just repeat?  Perhaps it would make sense to phase shift it by
10, such that y ranges from 0 to 40.

It would be most helpful if you could abstract the solution to
substitue n for 9.  I need the solution to work for values of n from 6
to 9. (I think n=9 is the "most discontinuous" case, though, since
it's the highest frequency sine wave).

Thanks!