Imagine a ball of radius \(a/4\) rolling around the inside of a circle of radius \(a\text{.}\) The curve traced by a point \(P\) painted on the inner circle (that's the blue curve in the figures below) is called an astroid 1 . We shall find its equation.
Define the angles \(\theta\) and \(\phi\) as in the figure in the left below.
That is
the vector from the centre, \(O\text{,}\) of the circle of radius \(a\) to the centre, \(Q\text{,}\) of the ball of radius \(a/4\) is \(\frac{3}{4}a\big(\cos\theta,\sin\theta\big)\) and
the vector from the centre, \(Q\text{,}\) of the ball of radius \(a/4\) to the point \(P\) is \(\frac{1}{4}a\big(\cos\phi,-\sin\phi\big)\)
As \(\theta\) runs from 0 to \(\frac{\pi}{2}\text{,}\) the point of contact between the two circles travels through one quarter of the circumference of the circle of radius \(a\text{,}\) which is a distance \(\frac{1}{4}(2\pi a)\text{,}\) which, in turn, is exactly the circumference of the inner circle. Hence if \(\phi=0\) for \(\theta=0\) (i.e. if \(P\) starts on the \(x\)-axis), then for \(\theta=\frac{\pi}{2}\text{,}\)\(P\) is back in contact with the big circle at the north pole of both the inner and outer circles. That is, \(\phi=\frac{3\pi}{2}\) when \(\theta=\frac{\pi}{2}\text{.}\) (See the figure on the right above.) So \(\phi=3\theta\) and \(P\) has coordinates
which is surprisingly simple, considering what we went through to get here.
There remains the danger that there could exist points \((x,y)\) obeying the equation \(x^{2/3}+y^{2/3}=a^{2/3}\) that are not of the form \(x= a\cos^3\theta,\ y=a\sin^3\theta\) for any \(\theta\text{.}\) That is, there is a danger that the parametrized curve \(x= a\cos^3\theta,\ y=a\sin^3\theta\) covers only a portion of \(x^{2/3}+y^{2/3}=a^{2/3}\text{.}\) We now show that the parametrized curve \(x= a\cos^3\theta,\ y=a\sin^3\theta\) in fact covers all of \(x^{2/3}+y^{2/3}=a^{2/3}\) as \(\theta\) runs from \(0\) to \(2\pi\text{.}\)
First, observe that \(x^{2/3}=\big(\root 3\of x\big)^2\ge 0\) and \(y^{2/3}=\big(\root 3\of y\big)^2\ge 0\text{.}\) Hence, if \((x,y)\) obeys \(x^{2/3}+y^{2/3}=a^{2/3}\text{,}\) then necessarily \(0\le x^{2/3}\le a^{2/3} \) and so \(-a\le x\le a\text{.}\) As \(\theta\) runs from \(0\) to \(2\pi\text{,}\)\(a\cos^3\theta\) takes all values between \(-a\) and \(a\) and hence takes all possible values of \(x\text{.}\) For each \(x\in[-a,a]\text{,}\)\(y\) takes two values, namely \(\pm{[a^{2/3}-x^{2/3}]}^{3/2}\text{.}\) If \(x=a\cos^3\theta_0=a\cos^3(2\pi-\theta_0)\text{,}\) the two corresponding values of \(y\) are precisely \(a\sin^3\theta_0\) and \(-a\sin^3\theta_0=a\sin^3(2\pi-\theta_0)\text{.}\)
The name “astroid” comes from the Greek word “aster”, meaning star, with the suffix “oid” meaning “having the shape of”. The curve was first discussed by Johann Bernoulli in 1691–92.
