In 1569, Gerhardus Mercator, a Flemish cartographer, produced a revolutionary aid to navigation: a map on which paths of constant compass direction (rhumb lines) are straight, and angles are the same as those found on the globe. In other words, he mapped the surface of a sphere onto a rectangle in such a way that angles are preserved. These days, this is called a conformal mapping. The problem he had to solve was how far apart to place the lines of latitude so that the rhumb lines are straight.
But how did Mercator obtain his map? We do not know! But we might expect the task of transforming the map on the curved surface of the globe to a flat map in a way which preserves shapes to be quite hard.
