Euclidean space conventions

Consider a crisp or fuzzy subset A of the Euclidean space X, a (possibly approximate) symmetry element R, and the associated symmetry operator R. A fixed point of R is chosen as a reference point c e A, and a local Cartesian coordinate system of origin c is specified, with coordinate axes oriented according to the usual conventions with respect to the symmetry operator R, as described for crisp sets in Section XIII. 

Conventional physical space has three dimensions. Any point in this space can represented by three Cartesian coordinates (.X], X2,. x.d. The natural generalization of three dimensional (3-D) physical space is the n dimensional Euclidean space (or Rn ), which can be described as the set of all possible vectors of order n ... 

The Minkowski space-time of special relativity differs from conventional Euclidean space only in the number of dimensions and gives the correct description of all forms of uniform relative motion. However, it fails when applied to accelerated motion, of which circular motion at constant orbital speed is the simplest example. Relativistic contraction only occurs in the direction of motion, but not in the perpendicular radial direction towards the centre of the orbit. The simple Euclidean formula that relates the circumference of the circle to its radius therefore no longer holds. The inevitable conclusion is that relativistic acceleration implies non-Euclidean geometry. 

If / is a reflection plane, then m = 2, and the two half spaces with boundary plane the reflection plane R fulfill the conditions. If / is a C rotation axis or an 5 axis, then m = k, and each segment of the Euclidean space X can be taken as a wedge of edge the C or the Sk axis, and wedge angle In/k. If / is a point of inversion i, then m = 2, and the two half spaces with boundary plane (x, y) fulfill the conditions. The notation P is used for the specification of the actual convention used in the positioning of R with respect to crisp continuum set A and for the partitioning Xo,Xi,Xm-1 of the space X. 

This generalized gradient is based on a Riemannian metric defined on the interior of the concentration space x > 0 X x = 1, fc = 1,. . . , n, which replaces the conventional Euclidean metric. We compare the definitions of the two inner products ... 

The distance between two points in an ultrametric space in the conventional Euclidean sense can be defined as... 

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