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

Abstract

Dr. Cai Wen defined in his 1983 paper:- the distance formula between a point x0 and a one-dimensional (1D) interval [a, b];- and the dependence function which gives the degree of dependence of a point with respect to a pair of included 1D - intervals. This 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 non-linear distance between a point and the 2Dand 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 2D-surface 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 one-dimension to 2, 3 and even n-dimensions, therefore more classes of applications will result in consequence.