On the Upper and Lower Approximations of the Curve1

The articles [28], [32], [2], [15], [1], [5], [6], [4], [31], [16], [29], [17], [27], [13], [3], [25], [26], [10], [11], [8], [30], [14], [20], [18], [12], [23], [22], [24], [7], [9], [19], and [21] provide the notation and terminology for this paper. In this paper n denotes a natural number. Let C be a simple closed curve. The functor UpperAppr(C) yields a sequence of subsets of the carrier of E2 T and is defined as follows: (Def. 1) For every natural number i holds (UpperAppr(C))(i) = UpperArc(L̃(Cage(C, i))). The functor LowerAppr(C) yielding a sequence of subsets of the carrier of E2 T is defined as follows: (Def. 2) For every natural number i holds (LowerAppr(C))(i) = LowerArc(L̃(Cage(C, i))). Let C be a simple closed curve. The functor NorthArc(C) yielding a subset of E2 T is defined by: (Def. 3) NorthArc(C) = Li UpperAppr(C). The functor SouthArc(C) yields a subset of E2 T and is defined by: (Def. 4) SouthArc(C) = Li LowerAppr(C).