## Generalizations of the Distance and Dependent Function in Extenics to 2D, 3D, and n-D

**Authors:** Florentin Smarandache

Dr. Cai Wen defined in his 1983 paper:
- the distance formula between a point x0 and a one-dimensional (1D) interval ;
- and the dependence function which gives the degree of dependence of a point with respect to
a pair of included 1D-intervals.
His paper inspired us to generalize the Extension Set to two-dimensions, i.e. in plane of real
numbers R2 where one has a rectangle (instead of a segment of line), determined by two
arbitrary points A(a1, a2) and B(b1, b2). And similarly in R3, where one has a prism determined by
two arbitrary points A(a1, a2, a3) and B(b1, b2, b3). We geometrically define the linear and nonlinear
distance between a point and the 2D- and 3D-extension set and the dependent function
for a nest of two included 2D- and 3D-extension sets. Linearly and non-linearly attraction point
principles towards the optimal point are presented as well.
The same procedure can be then used considering, instead of a rectangle, any bounded 2Dsurface
and similarly any bounded 3D-solid, and any bounded n-D-body in Rn.
These generalizations are very important since the Extension Set is generalized from onedimension
to 2, 3 and even n-dimensions, therefore more classes of applications will result in
consequence.
Introduction.

**Comments:** 17 Pages.

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### Submission history

[v1] 2012-06-04 23:37:54

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