Archive for February 27th, 2008

How to climb hills

February 27, 2008

800px-lombardstreet.jpgClimbing a gently sloping hill is straightforward:  to get to the top, you just climb straight up the hill.  Climbing a steep hill is a different matter.  If you were to go straight up the side of the hill, following the shortest possible path to your destination, you would be stymied trying to climb a slope too steep for your legs to ascend.  The solution is to tack: climb the hill on an oblique path, one whose slope you can manage, and use switchbacks when necessary to continue toward your intended destination.  It follows that there is a critical slope where one’s strategy shifts from the direct attack to the construction of switchbacks.

An article by Marcos Llobera and Tim J. Sluckin in the Journal of Theoretical Biology (volume 249, 21 Nov 2007, pp 206-217) presents a simple model for human hill-climbing behavior, based on the notion of minimizing energy expended per unit of distance travelled in the desired direction.  They show that their optimization model predicts the expected switchback behavior of hill-climbers, and yields a good fit to observed human behavior.  In particular, their model predicts that the critical slope for hill climbing is significantly greater than the corresponding slope for descending hills, an asymmetry which has consequences for the evolution of systems hiking trails in hilly terrain. 

While L. C. Young is relegated to a single sentence and a citation, much of their analysis has a similar flavor to his generalized curves as solutions to variational problems.  Young used the example of a sailboat attempting to sail into the wind as a motivating example for his theory;  Young was interested in finding the optimal path subject to a constraint (a maximal slope in the hillclimbing context). Llobera and Sluskin arrive at a comparable conclusion by seeking to locally minimize effort expended.

[I first heard about this study at Yahoo! News, who in turn were channelling; a quick web search reveals a flurry of postings on this, possibly the earliest a press release from the Univ of Washington.]