Generating an Ellipse from Rotating Vectors

Roland Deaux highlights this beauty:

If two vectors issued from a fixed point \(O\) have given but different magnitudes, and if they rotate about \(O\) with constant and opposite angular velocities \(\omega\) and \(-\omega\), the fourth vertex of the parallelogram having the two vectors for a pair of adjacent sides describes an ellipse (\(E\)) of center \(O\).

The theorem is in his book Introduction to the Geometry of Complex Numbers on p. 75 of Howard Eves' English translation available in a cheap Dover reprint, an unbelievable bargain at $9.99.^[1] The book was originally published in French in 1947.

The point \(O\) is the center of the ellipse, the intersection of the major and minor axes. See an animation below illustrating the theorem.

This animation was produced by Gemini.^[2] To get the most out of it, first play around with the sliders: the blue one \((R_1)\) adjusts the length of the blue vector, the green one \((R_2)\) adjusts the length of the green vector, and the purple one \((\beta)\) adjusts the angle between the two vectors. After making some adjustments, click the blue "START ANIMATION" button at the top left to animate the ellipse. Note the sliders can be re-adjusted in mid-animation to generate a new ellipse. The animation can be paused at any time by clicking the red "STOP ANIMATION" button at the top left.

Deaux goes on to point out that if \(a\) and \(b\) represent the original vectors as complex numbers, then this implies that the equation for the ellipse in the complex plane is:
\begin{align*}
z=ae^{i\omega t} + be^{-i\omega t},
\end{align*}
where \(t\) runs over all real values.

Let \(a\) and \(b\) to be positive real numbers, illustrated in the animation by setting \(\beta=0\) in the third slider. Running the animation with \(\beta=0\) results in an ellipse with axes parallel to the coordinate axes. To prove this partial result, let \(a = R_1, \; b = R_2\), where \(R_1\) and \(R_2\) are real and \(R_1 > R_2\). Then:
\begin{align*}
z(t) &= R_1 e^{i\omega t} + R_2 e^{-i\omega t}\\[0.7em]
&= R_1(\cos{(\omega t)}+i\sin{(\omega t)}) + R_2(\cos{(-\omega t)}+i\sin{(-\omega t)})\\[0.7em]
&= R_1(\cos{(\omega t)}+i\sin{(\omega t)}) + R_2(\cos{(\omega t)}-i\sin{(-\omega t)})\\[0.7em]
&= \underbrace{(R_1+R_2)\cos{(\omega t)}}_{\Large x(t)}+i\underbrace{(R_1-R_2)\sin{(\omega t)}}_{\Large y(t)}.
\end{align*}
Squaring \(x(t)\) and \(y(t)\) results in:
\begin{align*}
\left( \frac{x(t)}{R_1+R_2} \right)^2 = \cos^2{\omega t}, \; \left(\frac{y(t)}{R_1-R_2} \right)^2 = \sin^2{\omega t}\\[1.5em]
\therefore \left(\frac{x(t)}{R_1+R_2} \right)^2 + \left(\frac{y(t)}{R_1-R_2} \right)^2 = \cos^2{\omega t} + \sin^2{\omega t} = 1.\tag{1}
\end{align*}
Equation (1) represents an ellipse with major axis \(R_1+R_2\) and minor axis \(R_1-R_2\) situated squarely in the coordinate system.

Setting the tilt angle to \(0°\) in the animation results in an ellipse squarely situated in the coordinate system, but an angle other than \(0°\) produces a tilted ellipse. This suggests an approach to the general proof. The equation of such a tilted ellipse is:
\begin{align*}
z(s) = e^{i\gamma} \left( R_1e^{i\omega s}+R_2e^{-i\omega s} \right)\\[0.7em]
\therefore z(s) = R_1e^{i(\gamma+\omega s)} +R_2e^{i(\gamma-\omega s)},\tag{2}
\end{align*}
where the parenthesized expression on the first line is a squarely situated ellipse as discussed above, \(R_1\) and \(R_2\) are real, and \(s\) runs through all real values. This is where the "Geometry of Complex Numbers" shows its value, considering that \(e^{i\gamma}\) rotates the ellipse counter-clockwise by angle \(\gamma\). Recall that the general equation for the locus being sketched out (the ellipse) is:
\begin{align*}
z(t) &= ae^{i\omega t} + be^{-i\omega t}\\[0.7em]
&= R_1e^{i\alpha} \cdot e^{i\omega t} + R_2e^{i\beta} \cdot e^{-i\omega t}\\[0.7em]
\therefore z(t) &= R_1e^{i(\alpha+\omega t)} + R_2e^{i(\beta-\omega t)},\tag{3}
\end{align*}
where \(a= R_1e^{i\alpha}\) and \(b= R_2e^{i\beta}\) are the initial fixed vectors. Identifying (2) and (3) results in:
\begin{align*}
\gamma + \omega s &= \alpha + \omega t\\[0.7em]
\gamma - \omega s &= \beta - \omega t.\\
\end{align*}
Solving this system for \(\gamma\) and \(\omega s\) leads to:
\begin{align*}
\gamma &= \frac{\alpha+\beta}{2}\\[1.0em]
\omega s &= \omega t + \frac{\alpha-\beta}{2},
\end{align*}
showing that the locus traced is indeed a tilted ellipse with tilt angle \(\gamma\). QED.

This theorem has a storied history going back to al-Tusi in the thirteenth century. Deaux's contribution is to show how complex numbers are a natural way to think about and prove it. Copernicus discussed a variant in de Revolutionibus. The names de La Hire (1706) and Fresnel (1820s) are connected with the theorem in more recent days, Fresnel applying it to explain light polarization.

Mike Bertrand

July 24, 2026

^ 1. Introduction to the Geometry of Complex Numbers by Roland Deaux, Dover Publications, Inc. (2008), ISBN 0-486-46629-9. This book was originally published in French in 1947. The Dover reprint is a translation into English by Howard Eves.

^ 2. I could've done this HTML5 / Canvas / Javascript animation given some time (weeks at my current level of productivity). In the event, I entered Deaux's statement of the theorem into Gemini and asked them to do it. It occurred to me when using early versions of the animation that if the two vectors pointed in different directions, then the ellipse would be tilted. I asked Gemini to add the third slider for the tilt angle. Amazing.