<pre>
The pasture is the circular area around M with the radius of 1.
The goat is nailed on to B and has a length of r.
These circles intersect in C resp C' (mirrowed to the x-axis).

<pre>
                   || . .    C
                   ||     . /|| .
                   ||   .    ||.   .
                   ||.    /  ||       .
                  .||        ||         .
                 . ||    /   ||          .
                .  ||        ||            .
                .  ||   /    ||              .
              .    ||    1   ||         r     .
              .    ||  /     ||h               .
              .    ||        ||                 .
            .      || /      ||                  .
            .      ||        ||                  .
            .      ||/delta  ||             beta 
          --A------M--------H-----1-------------B------
                   ||  1-q             q
                   ||
</pre>

Let's determine some distances:

<pre>
   h^2 + q^2 <code> r^2               q </code> (r^2)/2
                         ->
   h^2 + (1-q)^2 <code> 1             h </code> r*sqrt(1-(r^2)/4)
</pre>


The overlapping area of the two circles is composed of:

<pre>
   the circle segment BCC'
   plus the circle segment MCC'
   minus the 2 triangles MBC and MBC'(double counted by BCC' and MCC')

       2'''beta                2'''delta
   F = ------ ''' pi ''' r^2  +  ------- * pi  -  sqrt(1-(r^2)/4)
        360                    360
</pre>

Transformed to arcus    with

<pre>
   cos(beta)  <code> (r^2/2)r    </code> r/2
   cos(delta) <code> (1-r^2/2)/1 </code> 1-r^2/2

   F = r^2 ''' acos(r/2)  +  acos(1-r^2/2)  -  r ''' sqrt(1-r^2/4)
   <code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code><code></code>=
</pre>

Now you can determine the required radius by a numerical approximation. For
F = pi/2 the goat's length computes to 1.15872847302 (meters, miles or feet).

-------------------------------------------------------------------
Wolfgang Schumacher
schumach@cs.tu-berlin.de               2B or not 2B = FF
-------------------------------------------------------------------
</pre>
