Density of circles centered on like circumference

[Loren Booda]

Imagine a circle of given radius. Construct all circles of equivalent radii whose centers constitute that initial circumference. Can you derive a probability density that describes the overall distribution of points from those resultant circles?

 

[Tron3k]

Unless I'm misinterpreting you, I would think that every point covered by these many circles except for the center point would have been overlapped exactly twice.

 

[tacman]

The density of circles centered on a given circumference can be thought of as a measure of how densely packed the circles are on that circumference. In other words, it represents the likelihood of finding a circle centered at any point on the circumference.

To derive a probability density for this distribution, we first need to consider the total number of circles that can be constructed on a given circumference. This can be calculated by dividing the circumference by the diameter of the circle, which gives us the number of times the circle can fit into the circumference. This number can then be used to determine the probability of finding a circle centered at any point on the circumference.

For example, if we have a circle with a radius of 1 unit, the circumference would be 2π units. If we divide this by the diameter of the circle (which is also 2 units), we get a total of π circles that can be constructed on the circumference. This means that the probability of finding a circle centered at any point on the circumference is 1/π.

We can generalize this for any given radius by using the formula N = C/d, where N is the total number of circles, C is the circumference, and d is the diameter. This gives us a probability density function of 1/d, which is inversely proportional to the diameter of the circle.

In summary, the probability density for the distribution of circles centered on a given circumference is determined by the inverse of the diameter of the circle. This means that as the diameter increases, the probability of finding a circle centered at any point on the circumference decreases.