 Given a metric, a differential formula for the distance ds between neighboring points (which is everywhere the same), the curvature can be found. If, for example, the formula for distance is that for a map of the sphere on the plane, the curvature will be found to be the reciprocal of the square of the radius of the sphere \(1/R^2\). A plane has the same Gaussian curvature as a cylinder, or also as something close to a cone (that one may form from a cylinder, without breaking, stretching, or folding); they are locally isometric; although curved, the cylinder has zero curvature in Gauss’s technical sense, just as does the plane, which is why printing from a rotating drum is possible. Gauss investigated when one surface can be mapped onto another in such a way that distances are not altered, showing that a necessary condition for this to happen is that the curvatures at corresponding points are the same. And this also implies that the same figures can be constructed on all of these surfaces; the minimal lines or curves of shortest length between two points (geodesics) on one surface correspond to those on the others, so that angles, areas, etc. will be the same. Gauss had found a deeper way to look into the geometry of a surface, and this gave birth to modern differential geometry.