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I am researching about the brachistochrone curve, which is the inverse of the cycloid. The equation for the cycloid is : \begin{cases} x = b(t - \sin\;t) \\ y = b(1 - \cos\;t) \end{cases}

Based on this, I graphed the brachistochrone with the following equation: \begin{cases} x = b(t - \sin\;t) \\ y = b(-1 - \cos\;t) \end{cases}

However, when I tried to plot a point and algebraically solve for the 2 variables, with $t$ being measured in radians, I wasn't able to solve it and I need help.

For example, I was wondering how to find the $b$ and $t$ value algebraically that would satisfy an x value of $20$ and a y value of $-5$. Here is a diagram of the situation.

tl;dr How do you solve this algebraically: \begin{cases} b(t - \sin\;t) = 20 \\ b(\cos\;t - 1) = -5 \end{cases}

Any help would be greatly appreciated!

Sichun Xu
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1 Answers1

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\begin{cases} b(t-sin\;t)=20 \\ b(1-cos\;t)=-5 \end{cases} $$4(1-\cos(t))+(t-\sin(t)=0$$ Appart the trivial root $t=0$ leading to $b=\infty$ which isn't acceptable, there is no closed form for the solution of this equation. The root cannot be expressed on the form of a combination of a finite number of elementary functions.

You have to use numerical calculus to compute an approximate value : $t\simeq -4.376072413012...$ and $b\simeq -3.75936382988...$

Before all, don't forget to prove that the system of equations has only one real solution with $t\neq 0$.

JJacquelin
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