Distance Geometry - Discussion

Discussion

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.