Brachistochrone

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A brachistochrone is the curve of fastest descent from $A$ to $B$. It is described by parametric equations which are simple to derive:

A cycloid is the path traced by a point on a rolling circle: [asy] draw((0,0)--(9,0)); draw(Circle((0,1),1)); dot((0,0)); draw((0,1)--(3,1),Arrow); [/asy]

If the radius of the circle is $a$ and the center of the circle is moving at a speed of $a$ units per second, then it moves $2\pi a$, or one revolution, every $2\pi$ seconds (in other words, it revolves 1 radian per 1 second). Then the x-coordinate of the center relative to the ground is: $at$ the x-coordinate of the point relative to the center is: $a\cos(-\frac{\pi}{2}-t)=-a\sin t$ so the x-coordinate of the point is: $a(t-\sin t)$

Similarly, the y-coordinate of the center relative to the ground is: $a$ the y-coordinate of the point relative to the center is: $a\sin(-\frac{\pi}{2}-t)=-a\cos t$ so the y-coordinate of the point is: $a(1-\cos t)$

Since a brachistochrone is an upside-down cycloid, we reverse the sign of y: $x=a(t-\sin t)$ $y=a(\cos t-1)$ This is a brachistochrone starting at $(0,0)$.

If you want it to start at $(x_0,y_0)$ you just shift it: $x-x_0=a(t-\sin t)$ $y-y_0=a(\cos t-1)$

If you want it to go through $(c,d)$ you need to solve for $a$: $c-x_0=a(t-\sin t)$ $d-y_0=a(\cos t-1)$

I recommend first solving for $t$ in terms of $a$ using $d-y_0=a(\cos t-1)$, then substituting into $c-x_0=a(t-\sin t)$.

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