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Optimal Road Problem

Chernowitz, D (2014) Optimal Road Problem. Bachelor's Thesis, Mathematics.

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We consider the problem of paving a path through a mountainous terrain. An optimal path is one that balances the requirement of a route to be both as short as possible and as flat as possible. A new form of Riemannian metric, $tilde{S}$, is proposed to weigh the length of a path, such as to penalize vertical transport more than horizontal, reflecting the cost and difficulty of traffic to change altitude. A geodesic $boldsymbol{gamma}$ of this metric is then a candidate for an optimal path. A number of numerical solution techniques to calculate geodesics are explored, most notably a functional iteration using Christoffel's symbols of the geodesic equation, and geodesic evolution to minimize geodesic curvature. Finally, statistical analyses are done on the resulting geodesics of this weighted metric through a simulated hilly landscape. The metric performs well in balancing horizontal versus vertical travel economically, however it does not prevent a high maximum steepness over the course of the path.

Item Type: Thesis (Bachelor's Thesis)
Degree programme: Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 15 Feb 2018 07:56
Last Modified: 15 Feb 2018 07:56

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