Deformation principle as foundation of physical geometry and its   application to space-time geometry
  Physical geometry studies mutual disposition of geometrical objects and points in space, or space-time, which is described by the distance function d, or by the world function \sigma =d^{2}/2. One suggests a new general method of the physical geometry construction. The proper Euclidean geometry is described in terms of its world function \sigma_E. Any physical geometry G is obtained from the Euclidean geometry as a result of replacement of the Euclidean world function \sigma_E by the world function \sigma of G. This method is very simple and effective. It introduces a new geometric property: nondegeneracy of geometry. Using this method, one can construct deterministic space-time geometries with primordially stochastic motion of free particles and geometrized particle mass. Such a space-time geometry defined properly (with quantum constant as an attribute of geometry) allows one to explain quantum effects as a result of the statistical description of the stochastic particle motion (without a use of quantum principles). 
Yuri A. Rylov