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From: Phil on 5 May 2010 22:25
I am trying to determine the equation to give the distance (d) between
the centers of two circles of known area (AreaA, AreaB) such that
their overlap (AreaO) comprises a desired area. For instance with
Circle A with an area of 314 units^2 (radius=10) and Circle B with an
area of 452 units^2 (radius = 12), how far apart must their centers be
to have an overlap of 100 units. I reviewed the info on
http://mathworld.wolfram.com/Circle-CircleIntersection.html but was
unable to twist those formulas into a usable form. (If I knew the
address of my high school algebra teacher, I would send him a note of
apology).

The cases where there is no overlap, or 100% overlap is covered, I am
looking for an equation to cover the intermediate state.

Trying to automate the production of accurate Venn Diagrams for 2, and
then 3 circles.
From: Gerry Myerson on 5 May 2010 22:36
In article
<3fb25eeb-8e3b-46f4-8a60-8d3bc6baa3d5(a)l6g2000vbo.googlegroups.com>,
Phil <lornapix(a)gmail.com> wrote:

> I am trying to determine the equation to give the distance (d) between
> the centers of two circles of known area (AreaA, AreaB) such that
> their overlap (AreaO) comprises a desired area. For instance with
> Circle A with an area of 314 units^2 (radius=10) and Circle B with an
> area of 452 units^2 (radius = 12), how far apart must their centers be
> to have an overlap of 100 units. I reviewed the info on
> http://mathworld.wolfram.com/Circle-CircleIntersection.html but was
> unable to twist those formulas into a usable form. (If I knew the
> address of my high school algebra teacher, I would send him a note of
> apology).

Notice that even in the simple special case treated at that webpage,
where the two circles have the same area and the overlap is supposed
to be half that area, they get an equation
pi / 2 = 2 arccos (d / 2) - (d / 2) sqrt (4 - d^2)
which can't be solved algebraically, only numerically.
There's going to be a limit to the usability of the formulas,
no matter how skillfully you twist them.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Greg Neill on 6 May 2010 00:05
Phil wrote:
> I am trying to determine the equation to give the distance (d) between
> the centers of two circles of known area (AreaA, AreaB) such that
> their overlap (AreaO) comprises a desired area. For instance with
> Circle A with an area of 314 units^2 (radius=10) and Circle B with an
> area of 452 units^2 (radius = 12), how far apart must their centers be
> to have an overlap of 100 units. I reviewed the info on
> http://mathworld.wolfram.com/Circle-CircleIntersection.html but was
> unable to twist those formulas into a usable form. (If I knew the
> address of my high school algebra teacher, I would send him a note of
> apology).
>
> The cases where there is no overlap, or 100% overlap is covered, I am
> looking for an equation to cover the intermediate state.
>
> Trying to automate the production of accurate Venn Diagrams for 2, and
> then 3 circles.

It doesn't appear to me that you're going to find an
exact inverse for the area function so that you can
calculate the circle center separation directly.

You might consider doing a few iterations of a binary
search on the area function to zero-in on the value,
or if you have the patience, find the derivative of the
function and use a Newton's Method iteration.


From: Han de Bruijn on 6 May 2010 05:01
On May 6, 4:25 am, Phil <lorna...(a)gmail.com> wrote:
> I am trying to determine the equation to give the distance (d) between
> the centers of two circles of known area (AreaA, AreaB) such that
> their overlap (AreaO) comprises a desired area.  For instance with
> Circle A with an area of 314 units^2 (radius=10) and Circle B with an
> area of 452 units^2 (radius = 12), how far apart must their centers be
> to have an overlap of 100 units.  I reviewed the info onhttp://mathworld.wolfram.com/Circle-CircleIntersection.htmlbut was
> unable to twist those formulas into a usable form. (If I knew the
> address of my high school algebra teacher, I would send him a note of
> apology).
>
> The cases where there is no overlap, or 100% overlap is covered, I am
> looking for an equation to cover the intermediate state.
>
> Trying to automate the production of accurate Venn Diagrams for 2, and
> then 3 circles.

If the _latter_ is your objective, why not try instead Karnaugh maps?

http://en.wikipedia.org/wiki/Karnaugh_map

Because Karnaugh maps are equivalent to Venn diagrams. Moreover, up to
four (4) variables presents no problem. Unless you like the challenge,
of course and BTW, it's impossible to have an area proportional to the
number of elements in the corresponding set with Karnaugh diagrams.

Han de Bruijn
From: Han de Bruijn on 6 May 2010 05:08
On May 6, 11:01 am, Han de Bruijn <umum...(a)gmail.com> wrote:
> On May 6, 4:25 am, Phil <lorna...(a)gmail.com> wrote:
>
> > I am trying to determine the equation to give the distance (d) between
> > the centers of two circles of known area (AreaA, AreaB) such that
> > their overlap (AreaO) comprises a desired area.  For instance with
> > Circle A with an area of 314 units^2 (radius=10) and Circle B with an
> > area of 452 units^2 (radius = 12), how far apart must their centers be
> > to have an overlap of 100 units.  I reviewed the info onhttp://mathworld.wolfram.com/Circle-CircleIntersection.htmlbutwas
> > unable to twist those formulas into a usable form. (If I knew the
> > address of my high school algebra teacher, I would send him a note of
> > apology).
>
> > The cases where there is no overlap, or 100% overlap is covered, I am
> > looking for an equation to cover the intermediate state.
>
> > Trying to automate the production of accurate Venn Diagrams for 2, and
> > then 3 circles.
>
> If the _latter_ is your objective, why not try instead Karnaugh maps?
>
> http://en.wikipedia.org/wiki/Karnaugh_map
>
> Because Karnaugh maps are equivalent to Venn diagrams. Moreover, up to
> four (4) variables presents no problem. Unless you like the challenge,
> of course and BTW, it's impossible to have an area proportional to the
> number of elements in the corresponding set with Karnaugh diagrams.

Hmm, I'm not so sure anymore about the latter remark .. The gist of an
alternative, anyway, is to employ rectangles instead of circles.

Han de Bruijn
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