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In Magnus Wenninger's Spherical models, polyhedra are given geodesic notation in the form {3,q+} b,c, where {3,q} is the Schläfli symbol for the regular polyhedron with triangular faces, and q-valence vertices.
Geodetic latitude and geocentric latitude have different definitions. Geodetic latitude is defined as the angle between the equatorial plane and the surface normal at a point on the ellipsoid, whereas geocentric latitude is defined as the angle between the equatorial plane and a radial line connecting the centre of the ellipsoid to a point on the surface (see figure).
In some cases they have geometric realizations. An example is the Szilassi polyhedron, a toroidal polyhedron that realizes the Heawood map. In this case, the polyhedron is much less symmetric than the underlying map, but in some cases it is possible for self-crossing polyhedra to realize some or all of the symmetries of a regular map.
On an n-dimensional Riemannian manifold, the geodesic equation written in a coordinate chart with coordinates is: + = where the coordinates x a (s) are regarded as the coordinates of a curve γ(s) in and are the Christoffel symbols.
Several extensions of the theorem are known, in which the polyhedron that realizes a given graph has additional constraints; for instance, every polyhedral graph is the graph of a convex polyhedron with integer coordinates, or the graph of a convex polyhedron all of whose edges are tangent to a common midsphere.
The sum of the distances from any point in the interior of a regular polyhedron to the sides is independent of the location of the point (this is an extension of Viviani's theorem.) However, the converse does not hold, not even for tetrahedra. [2]
3D model of a regular icosahedron. The insphere of a convex polyhedron is a sphere inside the polyhedron, touching every face. The circumsphere of a convex polyhedron is a sphere that contains the polyhedron and touches every vertex.
Latitude and longitude coordinates on maps are usually expressed in degrees. In the given forms of the formulae below, one or more values must be expressed in the specified units to obtain the correct result. Where geographic coordinates are used as the argument of a trigonometric function, the values may be expressed in any angular units ...