Shortest line

A curve in a metric space, joining two points of this space, whose length does not exceed that of any other curve joining these points. On the plane, the shortest lines are segments of straight lines, while on the sphere they are arcs of great semi-circles. In Riemannian spaces, geodesic lines (cf. Geodesic line) are shortest lines in small sections; the length of the corresponding small section of a geodesic line can be estimated in relation to the curvature and the topology of the space. Shortest lines play an important role in the geometry of surfaces in the large and for metrics not subject to any regularity conditions. Thus, these concepts are fundamental and simple in the axiomatic construction of general spaces with an internal metric: Many intrinsic and extrinsic geometrical properties of general convex surfaces can be proved using the properties of shortest lines (cf. Convex surface).