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A geodesic triangle is formed by the geodesics joining each pair out of three points on a given surface. On the sphere, the geodesics are great circle arcs, forming a spherical triangle. Geodesic triangles in spaces of positive (top), negative (middle) and zero (bottom) curvature.
Geodesics on the sphere are great circles, circles whose center coincides with the center of the sphere. Any two distinct points on a sphere that are not antipodal (diametrically opposite) both lie on a unique great circle, which the points separate into two arcs; the length of the shorter arc is the great-circle distance between the points.
A 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. [7] The number of closed geodesics of length at most L on a smooth topological sphere grows in proportion to L/log ...
Geodesic polyhedra are constructed by subdividing faces of simpler polyhedra, and then projecting the new vertices onto the surface of a sphere. A geodesic polyhedron has straight edges and flat faces that approximate a sphere, but it can also be made as a spherical polyhedron (a tessellation on a sphere ) with true geodesic curved edges on the ...
* For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This property is unique to the sphere. All geodesics of the sphere are closed curves. Geodesics are curves on a surface that give the shortest distance between two points. They are a generalization of the concept of ...
The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. A sphere with a spherical triangle on it.
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an oblate ellipsoid, a slightly flattened sphere. A geodesic is the shortest path between two points on a curved surface, analogous to a straight line on a
Vincenty's formulae are two related iterative methods used in geodesy to calculate the distance between two points on the surface of a spheroid, developed by Thaddeus Vincenty (1975a). They are based on the assumption that the figure of the Earth is an oblate spheroid, and hence are more accurate than methods that assume a spherical Earth, such ...