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I have learnt about cycloids and have a related question:

What is parametric equation of a locus of a fixed point of a circle rolling along an ellipse in $\mathbb{R}^2$?

Ng Chung Tak
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Daniel K
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  • Have you looked around the MSE for any similar questions? There isn't just one... – Mr Pie May 19 '18 at 07:59
  • The length $L$ of an ellipse is $L\approx 2\pi\sqrt{\dfrac{a^2+b^2}{2}}$ and your curve probably it will cover the entire region between the two ellipses of axes $ a, b $ and $ a + 2r, b + 2r $ ($r=$ radius of the circle) because the quotient $\dfrac {L} {2\pi r}$ is in general irrational. It would be interesting a case where this quotient is an integer in which case there would be a nice curve. – Ataulfo May 19 '18 at 10:50
  • Rotation of rolling circle of radius $a$ is rolled ellipse contact length ( expressed in elliptic integrals from ends of,say major axis) divided by $a$. Calculated rotated components have to be added on to ellipse coordinates. – Narasimham May 19 '18 at 13:46

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Using clockwise convention,

\begin{align} z &= a\sin \theta+bi\cos \theta \tag{ellipse contact} \\ n &= \frac{iz'}{|z'|} \tag{unit normal vector} \\ &= \frac{b\sin \theta+ai\cos \theta}{\sqrt{a^2\cos^2 \theta+b^2\sin^2 \theta}} \\ e &= \frac{\sqrt{a^2-b^2}}{a} \\ s &= a\int_{0}^{\theta} \sqrt{1-e^2\sin^2 \theta} \, d\theta \tag{arc length} \\ &= aE(\theta,e) \\ c &= z+rn \tag{centre of circle} \\ \frac{z-c}{w-c} &= e^{is/r} \tag{angle rolled by circle} \\ w &= c+(z-c) e^{-is/r} \tag{required locus} \\ &= z+r(1-e^{-is/r})n \end{align}

Ng Chung Tak
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