Optimal curvature and gradient-constrained directional cost paths in 3-space

Alan Chang (), Marcus Brazil (), J. Rubinstein () and Doreen Thomas ()

Journal of Global Optimization, 2015, vol. 62, issue 3, 507-527

Abstract: In the design of underground tunnel layout, the development cost is often dependent on the direction of the tunnel at each point due to directional ground fracturing. This paper considers the problem of finding a minimum cost curvature-constrained path between two directed points in 3-space, where the cost at every point along the path depends on the instantaneous direction. This anisotropic behaviour of the cost models the development cost of a tunnel in ground with faulting planes that are almost vertical. The main result we prove in this paper is that there exists an optimal path of the form $$\mathcal {C}\mathcal {S}\mathcal {C}\mathcal {S}\mathcal {C}\mathcal {S}\mathcal {C}$$ C S C S C S C (or a degeneracy), where $$\mathcal {C}$$ C represents a segment of a helix with unit radius and $$\mathcal {S}$$ S represents a straight line segment. This generalises a previous result that in the restriction of the problem to the horizontal plane there always exists a path of the form $$\mathcal {C}\mathcal {S}\mathcal {C}\mathcal {S}\mathcal {C}$$ C S C S C or a degeneracy which is optimal. We also prove some key structural results which are necessary for creating an algorithm which can construct an optimal path between a given pair of directed points in 3-space with a prescribed directional cost function. Copyright Springer Science+Business Media New York 2015

Keywords: Computational geometry; Path optimization; Curvature-constrained paths; Anisotropic costs (search for similar items in EconPapers)
Date: 2015
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://hdl.handle.net/10.1007/s10898-014-0243-6 (text/html)
Access to full text is restricted to subscribers.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jglopt:v:62:y:2015:i:3:p:507-527

Ordering information: This journal article can be ordered from
http://www.springer. ... search/journal/10898

DOI: 10.1007/s10898-014-0243-6

Access Statistics for this article

Journal of Global Optimization is currently edited by Sergiy Butenko

More articles in Journal of Global Optimization from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().