This post continues the series on Generating Functions. The first installment was discussing the problem about changing dollar into coins. The second post was about degeneracy of energy levels in central field in quantum mechanics. The connection between previous and current problems is not too difficult to see yet the method of generating functions applied to counting “lucky numbers” has its own interesting twist.
Seemingly intimidating task of calculating degeneration of energy levels for quantum particle in central field can be solved easily with generating function.
What’s common between changing dollar into coins, counting “lucky tickets” and degeneracy of the energy levels of quantum harmonic oscillator in 3D?
All of these subjects can be approached with generating function technique.
In recent article in Game Programming Gems 8 on roads modeling , roads are generated as geodesic curves on terrain surface. The idea presented in the article is to modify the standard metric, induced on the terrain surface, with a metric, which penalizes travelling up and down terrain slopes. It was mentioned that a high penalty for slopes may lead to computational instability with roads taking increasingly longer detours. Here we discuss upper bound on the detour length.