Shortest paths for line segments

Christian Icking, Günter Rote, Emo Welzl, Chee Yap

Research output: Contribution to journalArticle

Abstract

We study the problem of shortest paths for a line segment in the plane. As a measure of the distance traversed by a path, we take the average curve length of the orbits of prescribed points on the line segment. This problem is nontrivial even in free space (i.e., in the absence of obstacles). We characterize all shortest paths of the line segment moving in free space under the measure d 2, the average orbit length of the two endpoints. The problem of d 2 optimal motion has been solved by Gurevich and also by Dubovitskij, who calls it Ulam's problem. Unlike previous solutions, our basic tool is Cauchy's surface-area formula. This new approach is relatively elementary, and yields new insights.

Original languageEnglish (US)
Pages (from-to)182-200
Number of pages19
JournalAlgorithmica (New York)
Volume10
Issue number2-4
DOIs
StatePublished - Oct 1993

Fingerprint

Line segment
Shortest path
Orbits
Free Space
Surface area formula
Orbit
Ulam's Problem
Cauchy
Path
Curve
Motion

Keywords

  • Cauchy's surface-area formula
  • Motion planning
  • Optimal motion
  • Ulam's problem

ASJC Scopus subject areas

  • Applied Mathematics
  • Safety, Risk, Reliability and Quality
  • Software
  • Computer Graphics and Computer-Aided Design
  • Computer Science Applications
  • Computer Science(all)

Cite this

Shortest paths for line segments. / Icking, Christian; Rote, Günter; Welzl, Emo; Yap, Chee.

In: Algorithmica (New York), Vol. 10, No. 2-4, 10.1993, p. 182-200.

Research output: Contribution to journalArticle

Icking, C, Rote, G, Welzl, E & Yap, C 1993, 'Shortest paths for line segments', Algorithmica (New York), vol. 10, no. 2-4, pp. 182-200. https://doi.org/10.1007/BF01891839
Icking, Christian ; Rote, Günter ; Welzl, Emo ; Yap, Chee. / Shortest paths for line segments. In: Algorithmica (New York). 1993 ; Vol. 10, No. 2-4. pp. 182-200.
@article{7ac5204aabd64767a52244377becfdab,
title = "Shortest paths for line segments",
abstract = "We study the problem of shortest paths for a line segment in the plane. As a measure of the distance traversed by a path, we take the average curve length of the orbits of prescribed points on the line segment. This problem is nontrivial even in free space (i.e., in the absence of obstacles). We characterize all shortest paths of the line segment moving in free space under the measure d 2, the average orbit length of the two endpoints. The problem of d 2 optimal motion has been solved by Gurevich and also by Dubovitskij, who calls it Ulam's problem. Unlike previous solutions, our basic tool is Cauchy's surface-area formula. This new approach is relatively elementary, and yields new insights.",
keywords = "Cauchy's surface-area formula, Motion planning, Optimal motion, Ulam's problem",
author = "Christian Icking and G{\"u}nter Rote and Emo Welzl and Chee Yap",
year = "1993",
month = "10",
doi = "10.1007/BF01891839",
language = "English (US)",
volume = "10",
pages = "182--200",
journal = "Algorithmica",
issn = "0178-4617",
publisher = "Springer New York",
number = "2-4",

}

TY - JOUR

T1 - Shortest paths for line segments

AU - Icking, Christian

AU - Rote, Günter

AU - Welzl, Emo

AU - Yap, Chee

PY - 1993/10

Y1 - 1993/10

N2 - We study the problem of shortest paths for a line segment in the plane. As a measure of the distance traversed by a path, we take the average curve length of the orbits of prescribed points on the line segment. This problem is nontrivial even in free space (i.e., in the absence of obstacles). We characterize all shortest paths of the line segment moving in free space under the measure d 2, the average orbit length of the two endpoints. The problem of d 2 optimal motion has been solved by Gurevich and also by Dubovitskij, who calls it Ulam's problem. Unlike previous solutions, our basic tool is Cauchy's surface-area formula. This new approach is relatively elementary, and yields new insights.

AB - We study the problem of shortest paths for a line segment in the plane. As a measure of the distance traversed by a path, we take the average curve length of the orbits of prescribed points on the line segment. This problem is nontrivial even in free space (i.e., in the absence of obstacles). We characterize all shortest paths of the line segment moving in free space under the measure d 2, the average orbit length of the two endpoints. The problem of d 2 optimal motion has been solved by Gurevich and also by Dubovitskij, who calls it Ulam's problem. Unlike previous solutions, our basic tool is Cauchy's surface-area formula. This new approach is relatively elementary, and yields new insights.

KW - Cauchy's surface-area formula

KW - Motion planning

KW - Optimal motion

KW - Ulam's problem

UR - http://www.scopus.com/inward/record.url?scp=0010223411&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0010223411&partnerID=8YFLogxK

U2 - 10.1007/BF01891839

DO - 10.1007/BF01891839

M3 - Article

AN - SCOPUS:0010223411

VL - 10

SP - 182

EP - 200

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 2-4

ER -