Brachistochrone: why the cycloid wins

What you are seeing: three frictionless beads released simultaneously from $A = (0, 0)$ all slide down to $B = (4, -2)$ under uniform gravity. The cyan bead follows the cycloid (Bernoulli's brachistochrone solution); the orange bead follows the straight line; the yellow bead follows a circular arc tangent to the horizontal at $A$. The cycloid arrives first; the line, despite being shortest, is slowest.

The cycloid is parametrized by $x(\theta) = R (\theta - \sin\theta)$, $y(\theta) = -R (1 - \cos\theta)$, with $R$ chosen so the endpoint lies on the curve. Bernoulli's trick (1696) applies Snell's law to a stack of infinitesimal layers whose refractive index decreases with depth in just the right way to make $v(y) \propto y^{1/2}$ match the energy conservation law. The optimal path then traces out a cycloid.