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Route choice in hilly terrain

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journal contribution
posted on 12.12.2014, 13:29 by Anthony Kay
The orienteering route choice problem involves finding the fastest route between two given points, with running speed determined by various properties of the terrain. In this study, I consider only the effect of climbing or descending on running speed. If a runner's pace p (the reciprocal of speed) varies linearly with gradient m, the straight-line route always is fastest. However, a nonlinear formulation for p(m), with d2p/dm2 > 0, will more accurately model runners’ capabilities. As a result, critical gradients may exist for ascent and/or descent, such that optimal routes will never ascend or descend more steeply than the critical gradient. I review and propose several formulations for the pace function p(m) and calculate their critical gradients. In principle, the Euler–Lagrange equation can be used to find optimal routes between arbitrary points on any topography where the height can be expressed as a smooth function of horizontal coordinates. I obtain first integrals of this equation for idealized landforms: hillsides with straight contours and axisymmetric hills. Next, optimal routes are computed for various combinations of start- and endpoints on these landforms based on various pace functions. These routes are classified as either subcritical or maximal steepness: The former ascends or descends less steeply than the critical gradient; the latter takes the line of steepest ascent where it is not steeper than the critical gradient, but follows a curve at the critical gradient where the slope is steeper. In some cases, the optimal route zigzags up or down a hill along sections of a critical-gradient curve.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

GEOGRAPHICAL ANALYSIS

Volume

44

Issue

2

Pages

87 - 108 (22)

Citation

KAY, A., 2012. Route choice in hilly terrain. Geographical Analysis, 44 (2), pp. 87 - 108.

Publisher

Wiley / © The Ohio State University

Version

AM (Accepted Manuscript)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2012

Notes

This is the accepted version of the following article: KAY, A., 2012. Route choice in hilly terrain. Geographical Analysis, 44 (2), pp. 87 - 108, which has been published in final form at: http://dx.doi.org/10.1111/j.1538-4632.2012.00838.x

ISSN

0016-7363

Language

en

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