The geodesic on an oblate spheroid can be computed analytically, although the resulting expression is much more unwieldy than for a simple sphere. A spheroid with equatorial radius and polar radius can be specified parametrically by(1)(2)(3)where . Using the second partial derivatives(4)(5)(6)(7)(8)(9)gives the geodesics functions as(10)(11)(12)(13)(14)(15)where(16)is the ellipticity.Since and and are explicit functions of only, we can use the special form of the geodesic equation(17)(18)(19)where is a constant depending on the starting and ending points. Integrating gives(20)where(21)(22) is an elliptic integral of the first kind with parameter , and is an elliptic integral of the third kind.Geodesics other than meridians of an oblate spheroid undulate between two parallels with latitudes equidistant from the equator. Using the Weierstrass sigma function and Weierstrass zeta function, the geodesic on the oblate spheroid..
A great circle is a section of a sphere that contains a diameter of the sphere (Kern and Bland 1948, p. 87). Sections of the sphere that do not contain a diameter are called small circles. A great circle becomes a straight line in a gnomonic projection (Steinhaus 1999, pp. 220-221).The shortest path between two points on a sphere, also known as an orthodrome, is a segment of a great circle. To find the great circle (geodesic) distance between two points located at latitude and longitude of and on a sphere of radius , convert spherical coordinates to Cartesian coordinates using(1)(Note that the latitude is related to the colatitude of spherical coordinates by , so the conversion to Cartesian coordinates replaces and by and , respectively.) Now find the angle between and using the dot product,(2)(3)(4)The great circle distance is then(5)For the Earth, the equatorial radius is km, or 3963 (statute) miles. Unfortunately, the flattening..
A geodesic mapping between two Riemannian manifolds is a diffeomorphism sending geodesics of into geodesics of , whose inverse also sends geodesics to geodesics (Ambartzumian 1982, p. 26).
The spherical distance between two points and on a sphere is the distance of the shortest path along the surface of the sphere (paths that cut through the interior of the sphere are not allowed) from to , which always lies along a great circle.For points and on the unit sphere, the spherical distance is given bywhere denotes a dot product.
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..
In a rectangular room (a cuboid) with dimensions , a spider is located in the middle of one wall one foot away from the ceiling. A fly is in the middle of the opposite wall one foot away from the floor. If the fly remains stationary, what is the shortest total distance (i.e., the geodesic) the spider must crawl along the walls, ceiling, and floor in order to capture the fly? The answer, , can be obtained by "flattening" the walls as illustrated above. Note that his distance is shorter than the the spider would have to travel if first crawling along the wall to the floor, then across the floor, then up one foot to get to the fly. The puzzle was originally posed in an English newspaper by Dudeney in 1903 (Gardner 1958).A twist to the problem can be obtained by a spider that suspends himself from strand of cobweb and thus takes a shortcut by not being forced to remain glued to a surface of the room. If the spider attaches a strand of cobweb to the wall at his starting..
A geodesic on a paraboloid(1)(2)(3)has differential parameters defined by(4)(5)(6)(7)(8)(9)The geodesic is then given by solving the Euler-Lagrangedifferential equation(10)As given by Weinstock (1974), the solution simplifies to(11)
The great sphere on the surface of a hypersphere is the three-dimensional analog of the great circle on the surface of a sphere. Let be the number of reflecting spheres, and let great spheres divide a hypersphere into four-dimensional tetrahedra. Then for the polytope with Schläfli symbol ,
An ellipsoid can be specified parametrically by(1)(2)(3)The geodesic parameters are then(4)(5)(6)When the coordinates of a point are on the quadric(7)and expressed in terms of the parameters and of the confocal quadrics passing through that point (in other words, having , , , and , , for the squares of their semimajor axes), then the equation of a geodesic can be expressed in the form(8)with an arbitrary constant, and the arc length element is given by(9)where upper and lower signs are taken together.