Hi Mike,

I have been looking at your equations and am confused on equation number

5.      x = (r1 + (R - x)/(R * r1)) * sin(atan(x/p(x))

How can you define x with x in the right side of the equation?
Thanks,
Arie

"Mike Williams" <mik### [at] nospamplease> wrote in message
news:5og### [at] econymdemoncouk...
> Wasn't it Arie L. Stavchansky who wrote:
> >Okay, this note may get a little complicated so forgive me if I confuse
you.
> >
> >I have written a macro that makes stacks of rings of spheres as seen in
the
> >attached image.  The macro adjusts the radius of the spheres in a ring
> >dependant on the radius of the ring because I wanted to make sure that
all
> >spheres are tangential (touching each other at a tangent).
> >
> >Now, what I would like to do is at the top of the "cylinder of spheres"
is
> >start to make the rings, but so that every stack of it fits the profile i
> >have drawn in.  Figuring out what the radius of the ring should be and
the y
> >value of the ring is very difficult for me--especially since the sphere's
> >radius changes as the ring's radius gets smaller.  I have attempted to
> >figure this out and in the second attached image you can see my initial
> >approach.
> >
> >Initially I have thought to define a "path" for the profile by a
polynomial
> >equation.  I would then input an x-coordinate value into the function
with
> >the equation and I would get a y-coordinate as an output.  The problem is
> >that when you actually go and make this a macro, how can I determine the
> >change in the x value so it will give me the correct y value and STILL
have
> >the sphere be tangential?   In other words how do I determine the
> >x-coordinate to input into the function that gives me the appropriate y
> >value?  Well I imagined that the change in x could be (as shown in the
> >diagram):
> >
> >radius1*(sin theta1) + radius2(sin theta2)
> >
> >I have figured out how to determine the radius for each sphere of any
ring
> >size, but I simply can not figure out how to determine the "thetas" of
which
> >to use for my equation.  Am I way off?  Is there an easier way to do
this?
> >
> >F. Audet has put up his Apple code which does something similar, and I
have
> >looked at it, but I am trying to understand how I can figure the math so
> >that I can become more adept at making what I want.  Thanks for your time
> >and help I appreciate it a great deal since it has helped me realize the
> >images I see in my head :)
>
> I think it gets rather messy, but here are a few pointers:-
>
> Firstly, note that the green lines in your diagram are parallel, so
> theta1 and theta2 are equal. So
>
> 1.      x = (r1 + r2)*sin theta
>
> Now consider the ratio of r2 to r1. Because there are the same number of
> spheres in each ring, the size of the spheres decreases by the same
> ratio as the radius of the ring. Call the radius of the first ring "R",
> then
>
> 2.      r2/r1 = (R-x)/R
>
> The polynomial path that you are trying to follow provides the
> relationship between y and x
>
> 3.      y = p(x)
>
> And we can see that
>
> 4.      tan theta = x/y
>
> The values of r1 and R are known, so we now have a system of four
> simultaneous equations involving four unknowns (x, y, r2, theta).
> Reducing these gives (if my algebra is correct)
>
> 5.      x = (r1 + (R - x)/(R * r1)) * sin(atan(x/p(x))
>
> That's a horribly messy equation, but the only unknown in there is x, so
> it should be soluble. The method of solution may well depend on the
> nature of p(x). Good luck!
>
> Once you've calculated x, obtain y from #3, r2 from #2, and theta (not
> that you need it) from #4.
>
> --
> Mike Williams
> Gentleman of Leisure

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