Deriving And Deducing The Equation Of The Curve Of Quickest Descent

Abstract

In Mathematics, we repeatedly come across the concept that a straight line is the shortest distance between two points. However hardly we find discussions on the curve that takes minimum time. My essay consider this issue not limited to 2-Dimensional but extended to spheres and other complex geometrical shapes and explores the research question тАЬHow do we find the equation to the curve of quickest descent between two places using calculus and justify the same comparatively with different plane curves?тАЭ The investigation aims to derive the equation of the curve of quickest descent. Since time is a crucial element in everyoneтАЩs life, achieving the aim can expunge some of the biggest problems. The extended essay explores the research subject step wise with consideration of certain assumptions, however most of the paramount factors are included in the essay. It investigates different curves such as straight line, parabolic arc, elliptical arc and cycloid on 2D plane taking the example of a ball (negligible mass) allowed to travel along slides of different shapes. Then the result of part 1 is applied to a real life situation where it is proposed that if a road or an underground tunnel were built between the cities of Bhubaneswar, Orissa, India and New Delhi, India, which shape should it take to optimize travel time. Second part of essay examines a larger scope of problem. I employ important identities and formulae such as Beltrami Identity and Great Circle Distance Formula as well as calculus of Variations to derive the equations of the curves especially cycloid and hypocycloid. The essay concludes in the first part that given a small-scale example, the curve of quickest descent is a cycloid. With this result, it further goes on to conclude that Hypocycloid is the Brachistochrone between Bhubaneswar and New Delhi.