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You can solve the brachistochrone problem using Fermat's principle, as shown in the 3blue1brown video GP is referring to. Since you're looking for the shortest
by taejo 8y ago
You can solve the brachistochrone problem using Fermat's principle, as shown in the 3blue1brown video GP is referring to. Since you're looking for the shortest time, if you can construct a lens where the speed of light is proportional to the speed of the bead on a wire, then the shortest-time wire is the path that light would take by Fermat's principle, and then you can use (an infinitesimal version of) Snell's law to find the direction of the wire at each height.
- speleo 8y agoYes, but they were saying: >For example, the fastest distance between two points is not a straight line, it's the cycloid, specifically the Brachistochrone curve [2]. This is the path light follows. Which is not true for free space, or any space with a constant index of refraction.
- taejo 8y agoOh yes, they do seem to be confused! espeed, if you happen to read this, the brachistochrone is the fastest path for something accelerated by a constant force (e.g. gravity near the surface of the earth).