Expected Distance Between Randomly Selected Points on a Circle and a Sphere

Understanding the expected distance between two randomly selected points on a circle or a sphere is essential in various fields including geometry, probability, and statistics. This article delves into the mathematical derivations and provides insights into how these distances behave differently depending on whether the points are on a circle or a sphere.

Circle

The expected distance between two randomly selected points on a circle with radius r is given by 2r/3. This result can be derived by considering the random variable X representing the distance between the two points. The probability density function of X is f_X x / πr^2 for 0 ≤ x ≤ 2r. The expected value is calculated as:

[E[X] ∫_{0}^{2r} x · (x/πr^2) dx (2r^2/πr^2) 2r/3]

Sphere

The expected distance between two randomly selected points on the surface of a sphere with radius r is 4r/3π. The key difference here is that the probability density function of the distance X is f_X x^2 / 2r^3 for 0 ≤ x ≤ 2r. This accounts for the different surface areas of a sphere compared to a circle. The expected value is integrated as:

[E[X] ∫_{0}^{2r} x · (x^2 / 2r^3) dx 4r/3π]

Understanding the Proofs

Circle: When considering the circle, the distance between points A and B is represented by the length of the segment joining them, which is 2r sin(α/2), where α is the angle between OA and OB. The average value of this segment is calculated by integrating over all possible angles, resulting in an expected distance of 4r/π.

Sphere: On a sphere, the length of the path between points A and B depends on the arc joining them. The shortest path is the geodesic arc between A and B. Given that one point is the North pole, and using spherical polar coordinates where θ is the longitude and ? is the latitude, the density for ? given θ is 1/2 cos(?). For this problem, it is better to measure ? from the North pole, so the density is 1/2 sin(?). The expected distance from the North pole is:

[int_0^{π} {2 sin(?/2) / 2 sin(?) d?} int_0^{π} {2 sin^2(?/2) cos(?/2) d?} left[ frac{4}{3} sin^3(?/2) right]_0^{π}

Thus, the expected distance between two points is 1 1/3 times the radius.

Real-World Applications

The results from the circle and the sphere have significant real-world applications. In navigation, the shortest distance between two points on the Earth's surface (assuming the Earth is a sphere) can be calculated using the great-circle distance, which is a fundamental concept in geodesy. Similarly, in the field of statistics, understanding the probability density functions of distributions can help in making accurate predictions and models.

Conclusion

The expected distance between two randomly selected points on a circle is 2r/3, whereas on a sphere it is 4r/3π. These distances are derived from different probability density functions, highlighting the importance of the geometric properties of the surface in determining these expected values. By grasping these concepts, one can gain a deeper understanding of the inherent differences between planar and spherical geometries.