Route Choice in Hilly Terrain

Abstract

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. El problema de trazo y selección de ruta en terrenos accidentados (orienteering route choice problem) consiste en encontrar la ruta más rápida entre dos puntos dados, cuando la velocidad de desplazamiento está en función a varias propiedades del terreno. En este estudio, el autor sólo considera el efecto de subir o bajar sobre la velocidad de desplazamiento. Si el paso p de un individuo (la inversa de la velocidad) varía linealmente con la pendiente m, la ruta en línea recta siempre es la más rápida. Sin embargo, una formulación no lineal de p (m), con D2P / dm 2> 0, resulta en un modelo más preciso de las capacidades de los individuos que se desplazan por el terreno. Como resultado, es posible que existan gradientes críticos tanto para el movimiento de ascenso como el de descenso. En dicho caso, las rutas óptimas nunca ascenderán o descenderán por gradientes más pronunciados que las identificadas comode gradiente crítico. El autor revisa y propone varias fórmulas para una función de paso p (m) y calcula los gradientes críticos respectivos. En principio, la ecuación de Euler‐Lagrange puede ser utilizada para trazar las rutas óptimas entre puntos arbitrarios en cualquier topografía en donde la altura pueda ser expresada como una función suavizada de las coordenadas horizontales. El autor calcula las integrales de esta ecuación para dos formas terrestres idealizadas: laderas con contornos rectos, y colinas c...

Importance of descending skill for performance in fell races: a statistical analysis of race results

“…These pace functions may be valid for the elite runners who achieve record times; for any other runner, we might expect the pace function to be a constant multiple of that for elite runners, but only if running performance at all gradients is purely a function of metabolism. In this case, route choice decisions of the sort examined by Kay (2012a) would be identical for all runners. However, by comparing results of physiological experiments on mountain runners on an inclined treadmill with data from mountain races, Minetti et al (2002) found that, whereas uphill running pace does appear to depend only on metabolism, downhill pace in races is considerably slower than might be expected purely on the basis of metabolism.…”

Investigating the Influence of Different DEMs on GIS-Based Cost Distance Modeling for Site Catchment Analysis of Prehistoric Sites in Andalusia

“…Originally unrelated to these archaeological questions, William Naismith proposed a rule in 1892 that related human walking speed to the slope of the terrain [32], which was refined later by Aitken [33] and Langmuir [34]. This formula is implemented in a cost distance tool in GRASS GIS [35], but this work applies a different formula that was established later.…”

Pace and Critical Gradient for Hill Runners: An Analysis of Race Records

“…Pace is defined here as time per unit horizontal distance (as shown on a map), and is a more convenient variable for route choice studies than speed. In an earlier study of the route choice problem (Kay, 2012) we proposed several pace functions, derived in a rather ad hoc manner from various data sources including race results. The present work is a more careful statistical study of results from hill races over a wide range of uphill and downhill gradients, with the objective of deriving a pace function p(m) that should at least be valid for a "typical" elite male hill runner.…”

“…Second, the formulation of the route choice problem by Kay (2012), based on a continuous representation of space, requires a twice-differentiable pace function (although this would not be necessary for a route choice algorithm such as that of Collischonn and Pilar (2000) which uses a discretisation of space). Third, Kay (2012) reasoned that d 2 p/dm 2 should be non-negative: a route at constant gradient should never be slower than a route with the same distance and total ascent but involving some steeper and some less steep terrain.…”