Route Choice in Hilly Terrain

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 we consider only the effect of climbing or descent on running speed. If a runner’s pace p (the reciprocal of running speed) varies linearly with gradient m, the straight-line route is always fastest. This may be a reasonable approximation at gentle gradients, but various studies have suggested that a nonlinear formulation for p(m), with d2p/dm2 > 0, is more appropriate. As a result, there is are critical gradients for both ascent and descent, such that optimal routes will never climb or descend more steeply than the critical gradient. Thus it may be necessary to zigzag when climbing or descending a hill where the slope is steeper than the runner’s critical gradient. In general, 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. Results are presented for idealised landforms: a hillside with straight contours and various axisymmetric hills. If the slope varies, the general tendency is for optimal routes to curve so that the variation of the gradient along the route is less than on a straight-line route.