## 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.

**Download:** **PDF**

### Submission history

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

**Unique-IP document downloads:** 142 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary.
In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution.
Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*