Distance Geometry Article Index for
Distance 		Website Links For
Distance 		 

Information About

Distance Geometry
  CATEGORIES ABOUT DISTANCE GEOMETRY
   
metric geometry
   
SHOPPER'S DELIGHT
 
Shop For:  
Department
Price Minimum:
Price Maximum:
      




INTRODUCTION


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''.

Similarly, suppose one knows

  • the distance from ''A'' to ''B'';

  • the distance from ''A'' to ''C'';

  • the distance from ''A'' to ''D'';

  • the distance from ''B'' to ''C'';

  • the distance from ''B'' to ''D''; and

  • the distance from ''C'' to ''D''.


Knowing only these six numbers, one would like to figure out

  • whether ''A'', ''B'', ''C'', and ''D'' lie on a common straight line;

  • whether ''A'', ''B'', and ''C'' lie on a common line but ''D'' is not on that line (and similarly for any of ''A'', ''B'', and ''C'' in the role of the one exceptional point);

  • whether all four points lie in a common plane;

  • if they lie in a common plane, whether one of them is in the interior of the Triangle formed by the other three, and if so, which one.


Distance geometry includes the solution of such problems.


CAYLEY-MENGER DETERMINANTS


Of particular utility and importance are classifications by means of Cayley-Menger Determinant s, named after Arthur Cayley and Karl Menger :

  • a set Λ (with at least three distinct elements) is called straight if

    • : for any three elements ''A'', ''B'', and ''C'' of Λ holds


:: \det \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} = 0,

  • a set Π (with at least four distinct elements) is called plane if

    • : for any four elements ''A'', ''B'', ''C'' and ''D'' of Π,


:: \det \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} = 0,
: 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,


:: \det \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} = 0,
: but not all quadruples of elements of Φ are plane to each other;
and so on.


SEE ALSO