Hilbert's fourth problem
In mathematics, Hilbert's fourth problem in the 1900 Hilbert problems was a foundational question in geometry. In one statement derived from the original, it was to find geometries whose axioms are closest to those of Euclidean geometry if the ordering and incidence axioms are retained, the congruence axioms weakened, and the equivalent of the parallel postulate omitted. A solution was given by Georg Hamel.
The original statement of Hilbert, however, has also been judged too vague to admit a definitive answer.
Hilbert discusses the existence of non-Euclidean geometry and non-Archimedean geometry, as well as the idea that a 'straight line' is defined as the shortes path between two points. He mentions how congruence of triangles is necessary for Euclid's proof that a straight line in the plane is the shortest distance between two points. He summarizes as follows:
The theorem of the straight line as the shortest distance between two points and the essentially equivalent theorem of Euclid about the sides of a triangle, play an important part not only in number theory but also in the theory of surfaces and in the calculus of variations. For this reason, and because I believe that the thorough investigation of the conditions for the validity of this theorem will throw a new light upon the idea of distance, as well as upon other elementary ideas, e. g., upon the idea of the plane, and the possibility of its definition by means of the idea of the straight line, the construction and systematic treatment of the geometries here possible seem to me desirable.
A gnomonic map projection of the sphere displays all great circles as straight lines, resulting in any line segment on a gnomonic map showing the shortest route between the segment's two endpoints. This is achieved by casting surface points of the sphere onto a tangent plane, each landing where a ray from the center of the earth passes through the point on the surface and then on to the plane.
This projection allows one to give a spherical metric to the portion of the plane it maps onto.
Klein disk model
In geometry, Klein disk model is a model of 2-dimensional hyperbolic geometry in which points are represented by the points in the interior of the unit disk and lines are represented by the chords, straight line segments with endpoints on the boundary circle.
- Hilbert, David, "Mathematische Probleme" Göttinger Nachrichten, (1900), pp. 253-297, and in Archiv der Mathematik und Physik, (3) 1 (1901), 44-63 and 213-237. Published in English translation by Dr. Maby Winton Newson, Bulletin of the American Mathematical Society 8 (1902), 437-479   doi:10.1090/S0002-9904-1902-00923-3 . [A fuller title of the journal Göttinger Nachrichten is Nachrichten von der Königl. Gesellschaft der Wiss. zu Göttingen.]
- Paiva, JC Álvarez. "Hilbert’s fourth problem in two dimensions." MASS selecta (2003): 165-183.
- Busemann, Herbert (1976). "Problem IV. Desarguesian spaces". In Browder, Felix E.. Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics. XXVIII. American Mathematical Society. pp. 131–141. ISBN 0-8218-1428-1. Zbl 0352.50010.