Cayley–Menger determinant

The distance geometry problem is that of characterization and study of sets of points based only on given values of the distances between member pairs. Therefore distance geometry has immediate relevance where distance values are determined or considered, such as biology, sensor network,surveying, cartography, and physics.

The distance geometry problem (DGP) is that of finding the coordinates of a set of points by using the distances between some pairs of such points. There exists nowadays a large community that is actively working on this problem, because there are several real-life applications that can lead to the formulation of a DGP. As an example, an interesting application is that of locating sensors in telecommunication networks. In such a case, the positions of some sensors are known (which are called anchors) and some of the distances between sensors (which can be anchors or not) are also known: the problem is to identify the positions in space for all sensors.

An interesting application arises in biology. Experimental techniques are able to estimate distances between pairs of atoms of a given molecule, and the problem becomes the one of identifying the three-dimensional conformation of the molecule, i.e. the positions of all its atoms. In this field, the main interest is on proteins, because discovering their three-dimensional conformation allows us to get clues about the function they are able to perform. The implications in related fields, such as biomedicine and drug design, are evident. When dealing with biological molecules, the DGP is generally referred to as molecular DGP (MDGP).

In the following, even if the article considers in general the DGP, the MDGP will be used as an example.

A straight line is the shortest path between two points. Therefore the distance from A to B is no bigger than the length of the straight-line path from A to C plus the length of the straight-line path from C to B. This fact is called the triangle inequality. If that sum happens to be equal to the distance from A to B, then the three points A, B, and C lie on a straight line, with C between A and B.

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