Minkowski sum of semi-convex domains in R 2:
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| Format: | Book |
| Language: | English |
| Published: |
Warszawa
Polska Akad. Nauk, Inst. Matematyczny
2002
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| Series: | Dissertationes mathematicae
411 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 55 S. graph. Darst. |
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Titel: Minkowski sum of semi-convex domains in R 2
Autor: Choi, Sung Woo
Jahr: 2002
CONTENTS
1. Introduction.
2. Curves.
3. The class W.
4. Sectors and domains.
5. Virtual boundary.
6. Minkowski sum of domains.
7. Minkowski sum of semi-convex domains
8. Maximality of semi-convexity.
9. Closedness of semi-convexity.
10. Conclusion.
References.
14
17
21
25
33
45
48
53
54
8
5
Abstract
The Minkowski sum of two sets A, B in Rn is defined to be the set of all points of the form a + b
for a € A and b 6 B. Due to its fundamental nature, the Minkowski sum is an important object
in many practical application areas such as image processing, geometric design, robotics, etc.
However, compared to the simplicity of the definition, a Minkowski sum of plane domains can
have quite complicated topological and geometric features in general. This is the case even when
the summands are relatively simple. For example, even if the summands are homeomorphic to
the unit disk, their Minkowski sum need not be.
We first introduce natural curve classes called Minkowski classes., and show that the set of
all planar domains, called .M-domains, whose boundaries consist of a finite number of curves
in a Minkowski class M, is closed under Minkowski sum. Then we introduce the notion of
semi-convexity for plane domains, which extends convexity, and show that the Minkowski sum
of semi-convex .M-domains is homeomorphic to the unit disk for any Minkowski class A4. We
also show that, in some sense, the semi-convexity is the weakest condition ensuring that the
Minkowski sum is homeomorphic to the unit disk. It is also shown that the set of all semi-convex
.M-domains is closed under Minkowski sum for any Minkowski class M. These results reveal a
new topological behavior of Minkowski sum.
2000 Mathematics Subject Classification: 52A30, 51M05, 68U07, 26E05.
Key words and phrases: Minkowski sum, convolution, semi-convex, real-analytic, Lojasiewicz's
structure theorem, Brunn-Minkowski theory.
Received 19.2.2001; revised 18.3.2002. |
| any_adam_object | 1 |
| author | Choi, Sung Woo |
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| discipline | Mathematik |
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| publishDate | 2002 |
| publishDateSearch | 2002 |
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| publisher | Polska Akad. Nauk, Inst. Matematyczny |
| record_format | marc |
| series | Dissertationes mathematicae |
| series2 | Dissertationes mathematicae |
| spelling | Choi, Sung Woo Verfasser aut Minkowski sum of semi-convex domains in R 2 Sung Woo Choi Warszawa Polska Akad. Nauk, Inst. Matematyczny 2002 55 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Dissertationes mathematicae 411 Convex domains Minkowski geometry Dissertationes mathematicae 411 (DE-604)BV000003039 411 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010187016&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Choi, Sung Woo Minkowski sum of semi-convex domains in R 2 Dissertationes mathematicae Convex domains Minkowski geometry |
| title | Minkowski sum of semi-convex domains in R 2 |
| title_auth | Minkowski sum of semi-convex domains in R 2 |
| title_exact_search | Minkowski sum of semi-convex domains in R 2 |
| title_full | Minkowski sum of semi-convex domains in R 2 Sung Woo Choi |
| title_fullStr | Minkowski sum of semi-convex domains in R 2 Sung Woo Choi |
| title_full_unstemmed | Minkowski sum of semi-convex domains in R 2 Sung Woo Choi |
| title_short | Minkowski sum of semi-convex domains in R 2 |
| title_sort | minkowski sum of semi convex domains in r 2 |
| topic | Convex domains Minkowski geometry |
| topic_facet | Convex domains Minkowski geometry |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010187016&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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