Subject: Re: Circle cutting
Date: Mon, 12 May 1997 00:04:09 -0400
From: Alexander Bogomolny
Andy:
The question you asked is not at all simple. I'll give you a couple of
formulas. See if you need more help. Proceed as follows:
First of all assume that no three of your chords pass through the
same point.
Note that the problem admits a different wording. x points on a circle
form a convex polygon. Disregarding x circular segments between
its sides and the circle, the question is into how many regions
a convex polygon is divided by its diagonals. If A(x) is the number
you are after, and B(x) is the answer to the latter problem, then
A(x) = B(x) + x.
The answer to the problem #2 is obtained by computing the total
number of points and the total sum of all angles in thus
obtained regions.
C(x) = x(x-1)(x-2)(x-3)/24 is the number of points of intersection of
all diagonals (this is the hard part!) Let we have r3 triangles, r4
quadrilaterals, etc. Then 3*r3 + 4*r4 + ... = 4*C(x) + x(x-2).
Counting the total of all angles, we have
(r3 + 2*r4 + 3*r5) * Pi = C(x) * 2*Pi + (x-2)*Pi.