Journal of Formalized Mathematics
Volume 15, 2003
University of Bialystok
Copyright (c) 2003 Association of Mizar Users

## On the Segmentation of a Simple Closed Curve

Andrzej Trybulec
University of Bialystok

### Summary.

The main goal of the work was to introduce the concept of the segmentation of a simple closed curve into (arbitrary small) arcs. The existence of it has been proved by Yatsuka Nakamura [21]. The concept of the gap of a segmentation is also introduced. It is the smallest distance between disjoint segments in the segmentation. For this purpose, the relationship between segments of an arc [24] and segments on a simple closed curve [21] has been shown.

This work has been partially supported by the CALCULEMUS grant HPRN-CT-2000-00102 and TYPES grant IST-1999-29001.

#### MML Identifier: JORDAN_A

The terminology and notation used in this paper have been introduced in the following articles [29] [35] [10] [3] [2] [32] [1] [13] [8] [9] [7] [4] [34] [25] [33] [22] [20] [28] [15] [26] [27] [18] [6] [12] [30] [19] [14] [16] [17] [23] [5] [24] [21] [11] [31]

#### Contents (PDF format)

1. Preliminaries
2. The Euclidean Distance
3. On the Distance between Subsets of a Euclidean Space
4. On the Segments
5. The Concept of a Segmentation
6. The Segments of a Segmentation
7. The Diameter of a Segmentation
8. The Concept of the Gap of a Segmentation

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Received August 18, 2003

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