If someone is familiar with this kind of thing, I'd like to know a little more about the Jaynes solution described. It seems to me that if the circle were translated off to some distant portion of the plane, it would be very unlikely that any method of choosing chords would have the same distributional properties on the original circle and on the translated circle, because the chords are all originally restricted to intersect the first circle and, given two circles in a plane, there are many lines that intersect one but not both circles. Will Jaynes' method really shade the translated circle the same way it will shade the original? My intuition says the translated one should be shaded more heavily in a sort of "band" parallel to the line connecting the origins of the two circles, or possibly in a crescent pattern with more shading closer to the original.
It's easy to see that the second method can be applied so it works independently of the position and size of the circle. Let's start by reformulating it slightly: instead of a "random radius" we will use a "random diameter".
> Choose a diameter of the circle, choose a point on the diameter and construct the chord through this point and perpendicular to the diameter.
In the general case, choose a line going through the origin.
=> In the circle, there is a diameter parallel to this line.
Then choose a point on the line, and construct the perpendicular line at that point. (To be able to choose a point uniformly, you have to consider a finite segment but this is not a problem as long as the location/size of the circle is bounded).
=> This perpendicular line might not intersect the circle, but when it does it determines a chord that goes through a point chosen uniformly on the diameter.
This generalized method can be used to cover uniformly multiple circles at the same time, but obviously not every line will intersect every circle.
("choose a random chord intersecting a circle")
If someone is familiar with this kind of thing, I'd like to know a little more about the Jaynes solution described. It seems to me that if the circle were translated off to some distant portion of the plane, it would be very unlikely that any method of choosing chords would have the same distributional properties on the original circle and on the translated circle, because the chords are all originally restricted to intersect the first circle and, given two circles in a plane, there are many lines that intersect one but not both circles. Will Jaynes' method really shade the translated circle the same way it will shade the original? My intuition says the translated one should be shaded more heavily in a sort of "band" parallel to the line connecting the origins of the two circles, or possibly in a crescent pattern with more shading closer to the original.