As such, to fully model the surface of the earth, we would require an infinite number of points.
Contour lines are the simplest and most familiar to most people.
In a computer, they are typically a line with an attached elevation attribute. In this form, they are of little use except as a static reference. These are raster representations of the earth's surface. The assumption is that there is a regular grid of spot heights which represent the average elevation for each raster cell. The amount depends on the spatial resolution.