Distance geometry
Distance geometry is the branch of mathematics concerned with characterizing and studying sets of points based only on given values of the distances between pairs of points.[1][2][3] More abstractly, it is the study of semimetric spaces and the isometric transformations between them. In this view, it can be considered as a subject within general topology.[4]
Historically, the first result in distance geometry is Heron's formula in 1st century AD. The modern theory began in 19th century with work by Arthur Cayley, followed by more extensive developments in the 20th century by Karl Menger and others.
Distance geometry problems arise whenever one needs to infer the shape of a configuration of points (relative positions) from the distances between them, such as in biology,[4] sensor networks,[5] surveying, navigation, cartography, and physics.
History[edit]
The first result in distance geometry is Heron's formula, from 1st century AD, which gives the area of a triangle from the distances between its 3 vertices. Brahmagupta's formula, from 7th century AD, generalizes it to cyclic quadrilaterals. Tartaglia, from 16th century AD, generalized it to give the volume of tetrahedron from the distances between its 4 vertices.
The modern theory of distance geometry began with Arthur Cayley and Karl Menger.[7] Cayley published the Cayley determinant in 1841,[8] which is a special case of the general Cayley–Menger determinant. Menger proved in 1928 a characterization theorem of all semimetric spaces that are isometrically embeddable in the n-dimensional Euclidean space .[9][10] In 1931, Menger used distance relations to give an axiomatic treatment of Euclidean geometry.[11]
Leonard Blumenthal's book[12] gives a general overview for distance geometry at the graduate level, a large part of which is treated in English for the first time when it was published.
There are many applications of distance geometry.[3]
In telecommunication networks such as GPS, the positions of some sensors are known (which are called anchors) and some of the distances between sensors are also known: the problem is to identify the positions for all sensors.[5] Hyperbolic navigation is one pre-GPS technology that uses distance geometry for locating ships based on the time it takes for signals to reach anchors.
There are many applications in chemistry.[4][12] Techniques such as NMR can measure distances between pairs of atoms of a given molecule, and the problem is to infer the 3-dimensional shape of the molecule from those distances.
Some software packages for applications are: