It is an open problem whether any of these quasigeodesics can be constructed in polynomial time. A version of the theorem of the three geodesics for convex polyhedra states that all polyhedra have at least three simple closed quasigeodesics this can be proved by approximating the polyhedron by a smooth surface and applying the theorem of the three geodesics to this surface. Nevertheless, the theorem of the three geodesics can be extended to convex polyhedra by considering quasigeodesics, curves that are geodesic except at the vertices of the polyhedra and that have angles less than π on both sides at each vertex they cross. In particular, a simple closed geodesic of a convex polyhedron would necessarily bisect the total angular defect of the vertices, and almost all polyhedra do not have such bisectors. Although some polyhedra have simple closed geodesics (for instance, the regular tetrahedron and disphenoids have infinitely many closed geodesics, all simple) others do not. The surface of a convex polyhedron has a metric that is locally Euclidean except at the vertices of the polyhedron, and a curve that avoids the vertices is a geodesic if it follows straight line segments within each face of the polyhedron and stays straight across each polyhedron edge that it crosses. It is also possible to define geodesics on some surfaces that are not smooth everywhere, such as convex polyhedra. (more unsolved problems in computer science) The growth rate of the number of simple closed geodesics, as a function of their length, was investigated by Maryam Mirzakhani. They are encoded analytically by the Selberg zeta function. On compact hyperbolic Riemann surfaces, there are infinitely many simple closed geodesics, but only finitely many with a given length bound. The number of closed geodesics of length at most L on a smooth topological sphere grows in proportion to L/log L, but not all such geodesics can be guaranteed to be simple. Generalizations Ī strengthened version of the theorem states that, on any Riemannian surface that is topologically a sphere, there necessarily exist three simple closed geodesics whose length is at most proportional to the diameter of the surface.
One proof of this conjecture examines the homology of the space of smooth curves on the sphere, and uses the curve-shortening flow to find a simple closed geodesic that represents each of the three nontrivial homology classes of this space. The proof was repaired by Hans Werner Ballmann in 1978.
In 1905, Henri Poincaré conjectured that every smooth surface topologically equivalent to a sphere likewise contains at least three simple closed geodesics, and in 1929 Lazar Lyusternik and Lev Schnirelmann published a proof of the conjecture, which was later found to be flawed.
In particular, a nearly-spherical triaxial ellipsoid has only three simple closed geodesics, its equators. This result stems from the mathematics of ocean navigation, where the surface of the earth can be modeled accurately by an ellipsoid, and from the study of the geodesics on an ellipsoid, the shortest paths for ships to travel. There may be more than three, for instance, the sphere itself has infinitely many. The theorem of the three geodesics says that for surfaces homeomorphic to the sphere, there exist at least three non-self-crossing closed geodesics. A geodesic is said to be a closed geodesic if it returns to its starting point and starting direction in doing so it may cross itself multiple times. The shortest path in the surface between two points is always a geodesic, but other geodesics may exist as well. For instance, on the Euclidean plane the geodesics are lines, and on the surface of a sphere the geodesics are great circles. You can also find the fastest route between multiple point by clicking optimize route.A triaxial ellipsoid and its three geodesicsĪ geodesic, on a Riemannian surface, is a curve that is locally straight at each of its points. If you want to reorder the route you can drag and address on the list into a new position. The tool will get driving directions as you enter addresses. If you want your directions to start and end at the place enter that address twice and place it at the top and the bottom. Since this tool uses google maps to retrieve the driving directions it can find almost any address.Īfter entering all of the locations, place the location which you want to start your trip from at the top, and place the location you want to end the trip at the bottom. This could include the exact address, the place name, or even just the city. To get the driving direction, first enter all of the places you want to visit. Driving Direction to multiple points: Find the fastest route to drive between multiple addresses