Z najmanj truda na Šmarno goro! - Jaklič, Gašper (Obzornik za matematiko in fiziko, 2012)
Finding a curve on the surface with constraints is a difficult problem frequently encountered in civil engineering at road and railway construction. In this article, an optimal mountain ascent (in the sense of energy consumption) is considered. Discrete terrain data are given. A smooth terrain description is constructed using macroelements. An energy functional which depends on terrain inclination and on path length, is defined. By computing energy on the boundaries of polynomial patches, the problem transforms into a discrete problem of finding the cheapest path on a mesh of polynomial curves. Numerical results indicate that the resulting paths are good approximations of the natural rutes. Some examples on real data are presented.
Keywords: Dijkstra's algorithm; Dijkstrov algoritem; energijski funkcional; energy functional; interpolacijski problem; interpolation problem; macroelements; makroelementi; matematika; mathematics; numerical analysis; numerična analiza; polinomske krivulje; polynomial curves