Calculating the easiest path when walking over a hill

A biomechanics problem that has impacts in archaeology: "which route is easiest for humans walking over hilly terrain". If we know this, we can make an educated guess as to the route that would have been taken between ancient settlements.

In this project I define "easiest" as "the path with the least metabolic cost of walking, i.e. energy expended per unit mass over the entire journey". I then draw on previous research to build up a cost function which relates metabolic cost to speed and gradient.

I show that the optimal path lies somewhere between the path with the least climbing, and the shortest path. I define an optimisation problem which when programmatically solved, finds the easiest route between two points on a simple terrain map.