What is the term for the approximation of the Earth's shape based on a mathematical equation?

The term that best describes the approximation of the Earth's shape based on a mathematical equation is "ellipsoid." An ellipsoid is a mathematically defined surface that approximates the shape of the Earth by assuming it is slightly flattened at the poles and bulging at the equator. This model is essential for many calculations in GIS, cartography, and geodesy, as it provides a more accurate representation of geographical coordinates and distances over the Earth's surface compared to a simple sphere.

While the geoid is also a representation of the Earth's shape, it is more complex and refers to the mean sea level extended under land, taking into account variations in gravity across the Earth. A sphere presents a simplified three-dimensional shape that lacks the necessary detail to represent the Earth accurately. Although "spheroid" can sometimes be used interchangeably with "ellipsoid," in the context of precise mathematical definitions and applications, particularly in GIS, "ellipsoid" is the more commonly accepted term for the specific mathematical model that represents the Earth's shape.