Distance geometry

Distance geometry is the 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 in surveying, cartography and physics.

Of particular utility and importance are classifications by means of Cayley-Menger determinants:

• a set Λ (with at least three distinct elements) is called straight if
for any three elements A, B, and C of Λ holds
[itex] \det \left( \begin{bmatrix}
0 & d(AB)^2 & d(AC)^2 & 1 \\
d(AB)^2 &    0    & d(BC)^2 & 1 \\
d(AC)^2 & d(BC)^2 &       0 & 1 \\
1 &       1 &       1 & 0

\end{bmatrix} \right) = 0, [itex]

• a set Π (with at least four distinct elements) is called plane if
for any four elements A, B, C and D of Π,
[itex] \det \left( \begin{bmatrix}
0 & d(AB)^2 & d(AC)^2 & d(AD)^2 & 1 \\
d(AB)^2 &    0    & d(BC)^2 & d(BD)^2 & 1 \\
d(AC)^2 & d(BC)^2 &       0 & d(CD)^2 & 1 \\
d(AD)^2 & d(BD)^2 & d(CD)^2 &       0 & 1 \\
1 &       1 &       1 & 1       & 0

\end{bmatrix} \right) = 0, [itex]

but not all triples of elements of Π are straight to each other;
• a set Φ (with at least five distinct elements) is called flat if
for any five elements A, B, C, D and E of Φ,
[itex] \det \left( \begin{bmatrix}
0 & d(AB)^2 & d(AC)^2 & d(AD)^2 & d(AE)^2 & 1 \\
d(AB)^2 &    0    & d(BC)^2 & d(BD)^2 & d(BE)^2 & 1 \\
d(AC)^2 & d(BC)^2 &       0 & d(CD)^2 & d(CE)^2 & 1 \\
d(AD)^2 & d(BD)^2 & d(CD)^2 &       0 & d(DE)^2 & 1 \\
d(AE)^2 & d(BE)^2 & d(CE)^2 & d(DE)^2 &       0 & 1 \\
1 &       1 &       1 & 1       &       1 & 0

\end{bmatrix} \right) = 0, [itex]

but not all quadruples of elements of Φ are plane to each other;

and so on.

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