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.
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]
-
Preliminaries
-
The Euclidean Distance
-
On the Distance between Subsets of a Euclidean Space
-
On the Segments
-
The Concept of a Segmentation
-
The Segments of a Segmentation
-
The Diameter of a Segmentation
-
The Concept of the Gap of a Segmentation
Bibliography
- [1]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
Journal of Formalized Mathematics,
1, 1989.
- [2]
Grzegorz Bancerek.
The ordinal numbers.
Journal of Formalized Mathematics,
1, 1989.
- [3]
Grzegorz Bancerek.
Sequences of ordinal numbers.
Journal of Formalized Mathematics,
1, 1989.
- [4]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
Journal of Formalized Mathematics,
1, 1989.
- [5]
Jozef Bialas and Yatsuka Nakamura.
The theorem of Weierstrass.
Journal of Formalized Mathematics,
7, 1995.
- [6]
Leszek Borys.
Paracompact and metrizable spaces.
Journal of Formalized Mathematics,
3, 1991.
- [7]
Czeslaw Bylinski.
Binary operations.
Journal of Formalized Mathematics,
1, 1989.
- [8]
Czeslaw Bylinski.
Functions and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
- [9]
Czeslaw Bylinski.
Functions from a set to a set.
Journal of Formalized Mathematics,
1, 1989.
- [10]
Czeslaw Bylinski.
Some basic properties of sets.
Journal of Formalized Mathematics,
1, 1989.
- [11]
Czeslaw Bylinski and Piotr Rudnicki.
Bounding boxes for compact sets in $\calE^2$.
Journal of Formalized Mathematics,
9, 1997.
- [12]
Agata Darmochwal.
Compact spaces.
Journal of Formalized Mathematics,
1, 1989.
- [13]
Agata Darmochwal.
Finite sets.
Journal of Formalized Mathematics,
1, 1989.
- [14]
Agata Darmochwal.
The Euclidean space.
Journal of Formalized Mathematics,
3, 1991.
- [15]
Agata Darmochwal and Yatsuka Nakamura.
Metric spaces as topological spaces --- fundamental concepts.
Journal of Formalized Mathematics,
3, 1991.
- [16]
Agata Darmochwal and Yatsuka Nakamura.
The topological space $\calE^2_\rmT$. Arcs, line segments and special polygonal arcs.
Journal of Formalized Mathematics,
3, 1991.
- [17]
Agata Darmochwal and Yatsuka Nakamura.
The topological space $\calE^2_\rmT$. Simple closed curves.
Journal of Formalized Mathematics,
3, 1991.
- [18]
Alicia de la Cruz.
Totally bounded metric spaces.
Journal of Formalized Mathematics,
3, 1991.
- [19]
Stanislawa Kanas, Adam Lecko, and Mariusz Startek.
Metric spaces.
Journal of Formalized Mathematics,
2, 1990.
- [20]
Jaroslaw Kotowicz.
Convergent real sequences. Upper and lower bound of sets of real numbers.
Journal of Formalized Mathematics,
1, 1989.
- [21]
Yatsuka Nakamura.
On the dividing function of the simple closed curve into segments.
Journal of Formalized Mathematics,
10, 1998.
- [22]
Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto.
Preliminaries to circuits, I.
Journal of Formalized Mathematics,
6, 1994.
- [23]
Yatsuka Nakamura and Andrzej Trybulec.
Adjacency concept for pairs of natural numbers.
Journal of Formalized Mathematics,
8, 1996.
- [24]
Yatsuka Nakamura and Andrzej Trybulec.
A decomposition of simple closed curves and the order of their points.
Journal of Formalized Mathematics,
9, 1997.
- [25]
Takaya Nishiyama and Yasuho Mizuhara.
Binary arithmetics.
Journal of Formalized Mathematics,
5, 1993.
- [26]
Beata Padlewska.
Locally connected spaces.
Journal of Formalized Mathematics,
2, 1990.
- [27]
Beata Padlewska and Agata Darmochwal.
Topological spaces and continuous functions.
Journal of Formalized Mathematics,
1, 1989.
- [28]
Konrad Raczkowski and Pawel Sadowski.
Topological properties of subsets in real numbers.
Journal of Formalized Mathematics,
2, 1990.
- [29]
Andrzej Trybulec.
Tarski Grothendieck set theory.
Journal of Formalized Mathematics,
Axiomatics, 1989.
- [30]
Andrzej Trybulec.
A Borsuk theorem on homotopy types.
Journal of Formalized Mathematics,
3, 1991.
- [31]
Andrzej Trybulec.
On the minimal distance between set in Euclidean space.
Journal of Formalized Mathematics,
14, 2002.
- [32]
Andrzej Trybulec.
Subsets of real numbers.
Journal of Formalized Mathematics,
Addenda, 2003.
- [33]
Andrzej Trybulec and Czeslaw Bylinski.
Some properties of real numbers operations: min, max, square, and square root.
Journal of Formalized Mathematics,
1, 1989.
- [34]
Wojciech A. Trybulec.
Pigeon hole principle.
Journal of Formalized Mathematics,
2, 1990.
- [35]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
Received August 18, 2003
[
Download a postscript version,
MML identifier index,
Mizar home page]