Probabilistic basis for creating random points and the cosine model

Abstract

This article addresses different techniques for generating random points on the globe. The longitude (X) and latitude (Y) are understood as continuous random variables, contained within the intervals between -180° and 180° and between -90° and 90°, respectively. Two possibilities are contemplated, both considering these two variables as independent. The first one uses uniform distributions for both variables (uniform model), and the second one uses a uniform distribution for longitude and a cosine distribution for latitude (cosine model). The probability distribution functions are presented for the marginal distributions of X and Y, as well as the joint distribution - random vector (X, Y). The properties of expectation and variance for the marginal distributions are also provided, and the independence of the joint distributions is demonstrated. Considering the entire globe, it was observed that the uniform model tends to generate a lower point density per area in the equatorial region and higher at the poles, while the cosine model tends to generate a variable density around a constant average. Thus, the cosine model corrects a bias caused by the non-Euclidean geometry of the Earth's shape. A demonstration carried out for two areas in the Brazilian territory showed that, using cosine distribution, the southern portion of Rio Grande do Sul would have 14.73% less chance in point generation compared to the equatorial region. This difference occurs due to the variation in the extent of the parallels along the latitudes.