Whale Math

Introduction

A reader asked a question that I answered in a comment response, but others may be interested so I will include my response as a post. One of my most read blog posts is about the amount of vertical deviation that exists between a level line and the Earth’s surface. A reader of that post asked a related question. She measures the distance to sea life from an observation point on a cliff and was wondering how to compute the difference in distance between the arc length and the horizontal distance to an object on the water. This is a nice exercise in geometry.

Background

Figure 1 illustrates the measurement scenario. H represents the height of her observation site above the ocean. θ represents the angle measured by her theodolite.

Figure 1: Observation Scenario for Sea Life.

Figure 2 illustrates the basic geometrical aspects of the problem. R represents the radius of the Earth. D represents the distance to the object.

Figure 2: Basic Geometry of the Observation Problem.

Analysis

Figure 3 illustrates my analysis. The question posed wanted me to assume a cliff height H = ~100 meters and an observation angle θ = 45°.

Figure 3: Analysis of the Observation Problem.

The horizontal distance to the object is 100 meters and the arc length to the object is 100.00076 meters. This is a very small difference.

Conclusion

The amount of distance error introduced by the curvature of the Earth for short distances is very small and would be difficult to measure. However, the error could be significant for long distance measurements.