Distinct Distances on Curves Via Rigidity

作者：Charalambides Marcos^*

来源：Discrete & Computational Geometry, 2014, 51(3): 666-701.

DOI：10.1007/s00454-014-9586-5

• Distinct distances
• Algebraic curves
• Structural rigidity
• Erdos distance problem

It is shown that points on a real algebraic curve of degree in always determine distinct distances, unless the curve is a straight line or the closed geodesic of a flat torus. In the latter case, there are arrangements of points which determine distinct distances. The method may be applied to other quantities of interest to obtain analogous exponent gaps. An important step in the proof involves understanding the structural rigidity of certain frameworks on curves.

• 出版日期2014-4