What is the difference between Tautochrone curve and Brachistochrone curve as both are cycloid?
If possible, show some reference please?
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Mathematically, they both are the same curve but they arise from slightly different but related problems.
While the Brachistochrone is the path between two points that takes shortest to traverse given only constant gravitational force, the Tautochrone is the curve where, no matter at what height you start, any mass will reach the lowest point in equal time, again given constant gravity.
These origins can be seen in the names:
Greek:
ταὐτό (tauto) the same
βράχιστος (brachistos) the shortest
χρόνος (chronos) time
Both problems are solved via Variational Calculus.
Here an illustration of the Tautochrone from Wikipedia (by Claudio Rocchini):
By comparison, this is the problem you are trying to solve with a Brachistochrone (Maxim Razin on Wikipedia):
