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From: Han de Bruijn on 10 May 2010 09:47
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.

The equations of the two circles are, without imposing restrictions on
generality:

(x + d/2)^2 + y^2 = R1^2 ; (x - d/2)^2 + y^2 = R2^2 ; 0 < R1 < R2

It is assumed that (R2 - R1) < d < (R2 + R1) : non-trivial case only.

The areas of the circles as well as the area of their intersection is
given: A1, A2, AO . We know that R1 = sqrt(A1/Pi) , R2 = sqrt(A2/Pi)
and AO < A1 < A2 . The problem is to find d .

The Original Posting as well as my solution is given at the bottom of
the following web page (together with other goodies):

http://hdebruijn.soo.dto.tudelft.nl/jaar2010/

- Circle-Circle Intersection

There are three Delphi Projects involved with this:

- Project1 for displaying a bunch of different (non) intersections:
http://hdebruijn.soo.dto.tudelft.nl/jaar2010/eieren/eieren0.htm

- Project2 to display the shape of the area AO as a function of (d):
http://hdebruijn.soo.dto.tudelft.nl/jaar2010/eieren/eierenD.htm

- Project3 to solve the problem as formulated in the OP. Regula Falsi
has been employed as a (fast converging) iteration method :
http://hdebruijn.soo.dto.tudelft.nl/jaar2010/eieren/eierenB.htm
http://hdebruijn.soo.dto.tudelft.nl/jaar2010/eieren/eierenC.htm
http://en.wikipedia.org/wiki/False_position_method

Disclaimer: especially the latter (Project3) may be not fail-safe for
all values of user input. Suggestions for improving this are welcome.

Mathematics. What else?

Han de Bruijn
From: Ludovicus on 12 May 2010 11:33
On May 5, 10:25 pm, 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.

If the circle of radius R1 have its centre in the origin , then
it is necesary to solve the following equations by trials on D:

1.- x^2 + y^2 = R1^2 ; (x - D)^2 + y^2 = R1^2 ---> x = (R1^2 -
R2^2 + D^2)/2D

2.- cos(a) = x / R1 ; cos(b) = (D - x)/ R2 ; a = arcos(x/R1) ; b =
arcos[(D-x)/R2]

3. A1 = R1^2(2a - sin^2(2a))/2 ; A2 = R2^2(2b - sin^2(2b))/2

4.- A = A1 + A2

Solution for R1 = 12 ; R2 = 10 ; A = 100 ---> D = 13.63
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