brando@linus.UUCP (Thom Brando) (08/30/83)
Here's another vote for net.puzzle. Now to the matter at hand!
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* What portion of a circle does a chord "chosen at random" span? *
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The interesting thing about this puxxle is not calculating the
"portion" of the circle, which is a rather standard mathematical
exercise, but in seeing how many different ways you can chose a
chord "at random", assuming that we agree that we are talking about
chosing chords which are UNIFORMLY distributed in SOME space. As
a former Monte Carlo programmer extraordinaire, I had the following
thoughts as I lay sleeplessly in bed at 3 this morning:
1) Fix one endpoint of the chord and generate the other
endpoint randomly around the circumference of the circle.
The average chord in this case will bisect the circle,
but the average length of a chord will, of course, be
something less than the diameter of the circle.
2) Generate the length of the chord randomly between 0 and
2*r.
3) Fix one endpoint of the chord and generate the angle of
the chord with respect to the tangent to the circle (at
the point you have fixed) between 0 and pi (or pi/2).
4) Draw a diameter, which we assume to bisect the chord we
will generate, and randomly generate between 0 and r the
distance from either end of the diameter you have drawn
to the chord you are generating.
5) Randomly choose a point interior to or on the perimeter
of the circle, then randomly choose a direction.
6a,b,...) Randomly choose a point exterior to the circle, then
randomly choose a direction. Throw out a direction if
it doesn't intersect the circle from the point you have
already chosen. (Note: you have a variety of options
here -- you can fix the distance of the exterior point
from the perimeter of the circle, or fix the maximum
distance of the point from the circle, or fix a minimum
and maximum distance, or leave everything unbounded.
I would hazard a guess that the relative portions, how-
ever you define them, would approach 1/2 as the ratio
of (distance from exterior point to circle) / (radius of
circle) grew (without bound, of course!).)
7) Randomly choose the relative portions of the circle
into which your chord will divide the circle!
I'm mostly thinking in terms of choosing the length of the
chord, since that will determine the "portion", which you can
interpret as being a portion of the circumference or a portion of
the area.
I think all 7 of the methods above should yield 7 different
relative "portions" of the circle. Those skeptics among you
could easily verify that with pencil and paper or simulation, if
you are so inclined.
Thom Brando, m/s E020, MITRE Corp, Bedford MA 01730, 617-271-3156
{allegra,decvax,ihnp4,utzoo,uw-beaver}!linus!brando (UUCP)