Introduction

The problem of determining the expected distance between two randomly and uniformly selected points on the surface of a unit sphere is a fascinating and complex one. Depending on the type of distance considered, either great circle distance or straight line distance, the expected outcomes differ significantly. To explore this topic further, this article will examine both the great circle distance and straight line distance in detail, providing a comprehensive understanding of the expected distances for each case.

Great Circle Distance

The great circle distance on a unit sphere is the shortest path between two points lying on the surface of the sphere, following a great circle (a circle on the surface of the sphere whose center coincides with the sphere's center).

Consider a point on the sphere as a pole. The distance to the equator can be represented as de. The probability of landing on any parallel at a given distance d1 is the same as landing on any circle at a given distance d2 since they have the same circumference. As a result, the average distance from the equator is equal to d1 d2/2 de.

Calculating the Expected Distance

The expected great circle distance is derived based on the symmetry of the unit sphere. For any point P on the sphere, the expected value of the distance to another randomly chosen point Q is the average distance from the point to any other point, which is equal to π/2. This value is obtained by averaging the distances over all possible pairs of points on the sphere, which can be expressed as:

Expected Distance (Great Circle) π/2 ≈ 1.57 for a unit circle, or πr/2 for any circle, where r is the radius of the circle.

Thus, half the time, the great circle distance will be greater than π/2, and the other half of the time, it will be less than π/2.

Direct Line Distance

In contrast to the great circle distance, the direct line distance (Euclidean distance) is the straight-line distance between two points in 3-dimensional space, not constrained to the surface of the sphere. This distance is more complex to calculate due to the spatial geometry involved.

Deriving the Expected Distance

Let's examine the expected direct line distance. For a point on the sphere, the probability of any distance being uniformly selected is different for the case of the direct line distance. For a given distance x from the equator, the distances d1 and d2 are given by:

d1 √(1 - x^2 x), d2 √(1 - x^2 - x)

The overall expected distance can be calculated through integration. The probability distribution for the direct line distance is derived as:

Expected Distance (Direct Line) √(2/3) ≈ 0.8165 for a unit sphere.

This means that, on average, half the time the direct line distance will be greater than √(2/3), and the other half of the time, it will be less.

Conclusion

This article has explored the expected distances between two randomly and uniformly selected points on the surface of the unit sphere, considering both the great circle distance and the direct line distance. The expected great circle distance is π/2, while the expected direct line distance is √(2/3). Understanding these results can provide valuable insights into the geometric properties of the unit sphere and can be useful in various applications, such as in astronomy, logistics, and data science.

Key Takeaways

The great circle distance is the shortest path on the surface of the sphere and has an expected value of π/2. The direct line distance is the straight-line distance through space and has an expected value of √(2/3). Both distances offer unique perspectives on the geometric properties of the sphere.