A geodesic is a locally length-minimizing curve. Equivalently, it is a path that a particle which is not accelerating would follow. In the plane, the geodesics are straight lines. On the sphere, the geodesics are great circles (like the equator). The geodesics in a space depend on the Riemannian metric, which affects the notions of distance and acceleration.Geodesics preserve a direction on a surface (Tietze 1965, pp. 26-27) and have many other interesting properties. The normal vector to any point of a geodesic arc lies along the normal to a surface at that point (Weinstock 1974, p. 65).Furthermore, no matter how badly a sphere is distorted, there exist an infinite number of closed geodesics on it. This general result, demonstrated in the early 1990s, extended earlier work by Birkhoff, who proved in 1917 that there exists at least one closed geodesic on a distorted sphere, and Lyusternik and Schnirelmann, who proved in 1923 that..
The distance between two points is the length of the path connecting them. In the plane, the distance between points and is given by the Pythagorean theorem,(1)In Euclidean three-space, the distance between points and is(2)In general, the distance between points and in a Euclidean space is given by(3)For curved or more complicated surfaces, the so-called metric can be used to compute the distance between two points by integration. When unqualified, "the" distance generally means the shortest distance between two points. For example, there are an infinite number of paths between two points on a sphere but, in general, only a single shortest path. The shortest distance between two points is the length of a so-called geodesic between the points. In the case of the sphere, the geodesic is a segment of a great circle containing the two points.Let be a smooth curve in a manifold from to with and . Then , where is the tangent space of at . The curve..
Given a unit line segment , pick two points at random on it. Call the first point and the second point . Find the distribution of distances between points. The probability density function for the points being a (positive) distance apart (i.e., without regard to ordering) is given by(1)(2)where is the delta function. The distribution function is then given by(3)Both are plotted above.The raw moments are then(4)(5)(6)(7)(Uspensky 1937, p. 257), giving raw moments(8)(9)(10)(11)(OEIS A000217), which are simply one over thetriangular numbers.The raw moments can also be computed directly withoutexplicit knowledge of the distribution(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)The th central moment is given by(28)The values for , 3, ... are then given by 1/18, 1/135, 1/135, 4/1701, 31/20412, ... (OEIS A103307 and A103308).The mean, variance, skewness,and kurtosis excess are therefore(29)(30)(31)(32)The..