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Given a map with genus g>0, Heawood showed in 1890 that the maximum number N_u of colors necessary to color a map (the chromatic number) on an unbounded surface is N_u = ...

A projection which maps a sphere (or spheroid) onto a plane. Map projections are generally classified into groups according to common properties (cylindrical vs. conical, ...

The mean tetrahedron volume V^_ is the average volume of a tetrahedron in tetrahedron picking within some given shape. As summarized in the following table, it is possible to ...

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 ...

In the case of a general surface, the distance between two points measured along the surface is known as a geodesic. For example, the shortest distance between two points on ...

A projective plane, sometimes called a twisted sphere (Henle 1994, p. 110), is a surface without boundary derived from a usual plane by addition of a line at infinity. Just ...

The set R union {infty}, obtained by adjoining one improper element to the set R of real numbers, is the set of projectively extended real numbers. Although notation is not ...

An algebraic surface of surface order 4. Unlike cubic surfaces, quartic surfaces have not been fully classified. Examples of quartic surfaces include the apple surface, ...

The Schwarz triangles are spherical triangles which, by repeated reflection in their indices, lead to a set of congruent spherical triangles covering the sphere a finite ...

The spherical curve obtained when moving along the surface of a sphere with constant speed, while maintaining a constant angular velocity with respect to a fixed diameter ...