A Method to Calculate the Centroid of Areas on a Sphere and its Application to Determining Geographic Centers

This study shows how a method of calculating the coordinates of the centroid of discrete points on a sphere can be generalized to determine the centroid of areas on a sphere by minimizing the surface integral of squared spherical distances. A quadratically convergent algorithm for calculating the coordinates of the centroid is presented. Equations are given to verify that the centroid so determined satisfies the condition that the surface integrals of the components of the associated distance vectors be equal to zero. Evaluating a surface integral over an irregular geographic region is facilitated by using data from a digital elevation model to generate a digital image that, with appropriate editing, can accurately characterize a geographic region for the purpose of determining the location of its geographic center. A case study using the example of North America determines the location of the geographic center of the continent to be at 51.90 N 97.58 W.