The plots of $\sin(kx)$ over the real line are somehow boring and look essentially all the same:

enter image description here

For larger $k$ you cannot easily tell which $k$ it is (not only due to Moiré effects):

enter image description here

But when plotting $\sin(kx)$ over the unit circle by

$$x(t) = \cos(t) (1 + \sin(kt))$$ $$y(t) = \sin(t) (1 + \sin(kt))$$

interesting patterns emerge, e.g. for $k = 1,2,\dots,8$

enter image description here

Interlude: Note that these plots are the stream plots of the complex functions

$$f_k(z)=\frac{1}{2i}(z^k - \overline{z^k})z $$

on the unit circle (if I didn't make a mistake). Note that $f_k(z)$ is not a holomorphic function.

You may compare this with the stream plot of

$$g_k(z)=\frac{1}{2i}(z^k - \overline{z^k}) = f_k (z)/z$$

with $g_k(e^{i\varphi}) = \sin(k\varphi) $:

enter image description here

[End of the interlude.]

Even for larger $k$ one still could tell $k$:

enter image description here

Furthermore you can see specific effects of rational frequencies $k$ which are invisible in the linear plots. Here are the plots for $k=\frac{2n +1}{4}$ with $n = 1,2,\dots,8$:

enter image description here

The main advantage of the linear plot of $\sin(kx)$ is that it has a simple geometrical interpretation resp. construction: It's the plot of the y-coordinate of a point which rotates with constant speed $k$ on the fixed unit circle:

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Alternatively, you can look at the sine as the projection of a helix seen from the side. This was the idea behind one of the earliest depictions of the sine found at Dürer:

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Compare this to the cases of cycloids and epicycles. These also have a simple geometrical interpretation - being the plots of the x- and y-coordinates of a point on a circle that rolls on the line

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resp. moves on another circle with constant speed

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My question is:

By which geometrical interpretation resp. construction (involving circles or ellipses or whatsoever) can the polar plots of $\sin$ be seen resp. generated? Which construction relates to the construction of $\sin$ by a rotating point on a circle in the way that the construction of epicycles relates to the construction of cycloids?

Just musing: Might this question have to do with this other question on Hidden patterns in $\sin(kx^2)$? (Probably not because you cannot sensibly plot $\sin(kx^2)$ radially, since there is no well-defined period.)

  • $\begingroup$ Can you unabbreviate or clarify the meaning of "resp.", please? I'm not familiar with it. $\endgroup$ – timtfj Jan 30 at 11:08
  • $\begingroup$ (I still upvoted though) $\endgroup$ – timtfj Jan 30 at 11:16
  • 1
    $\begingroup$ "Resp." abbreviates "respectively" which just means "or" (with a slightly different connotation). $\endgroup$ – Hans-Peter Stricker Jan 30 at 11:22
  • $\begingroup$ @HansStricker I think it's more common to use "re" instead of "resp". $\endgroup$ – Jam Jan 30 at 12:23
  • $\begingroup$ @HansStricker I'm sure some people use "resp" but it's not very common. I'm a native English speaker and I've not seen it more than a couple of times in my life :) $\endgroup$ – Jam Jan 30 at 12:31

Thanks to Yves Danoust's hint to Grandi's roses I found this "answer without words" best fitting to my question:

enter image description here

– even though I'm not quite sure how to choose the parameters (radius of the circle, rotation speeds of line and circle) to exactly reproduce my plot:

enter image description here

– and even though I don't see clearly what happens. As Robert Ferréol describes it on his web page on Grandi's roses:

The roses can also be obtained as the trajectories of the second intersection point between a line and a circle in uniform rotation around one of their points.

The difference between Grandi's rose and my plot is the order, in which the curve is drawn: not lobe by lobe, but as a rose. This difference vanishes, when we plot $\sin(e^{ik\varphi})$ not over the circle (as the base line), but from the origin. Here for $0 \leq \varphi < \pi$ (left) and $\pi < \varphi < 2\pi$ (right):

enter image description hereenter image description here

  • $\begingroup$ better you take just a ray from the origin: then it is easier to controll the parameters $\endgroup$ – G Cab Jan 30 at 14:58
  • $\begingroup$ @GCab: Instead of what? Don't I need the whole line through the origin? And how would it help me to control the parameters? (Note that it was not me who created the animated gif.) $\endgroup$ – Hans-Peter Stricker Jan 30 at 15:01
  • $\begingroup$ judging by eye from the animation, that's constructed by crossing the entire line ($\pm \rho$) with the circle. If you take instead only the ray, then the ratio of the two angular speeds will govern (better) the number of lobes. $\endgroup$ – G Cab Jan 30 at 15:50

I did not grasp exactly what you are asking, however it might be of interest to know that in "old times" electrical engineers were used to visualize phase and frequency of a sinusoidal wave by feeding it to the $x$ axis of an oscilloscope in combination to a known and tunable signal (sinusoidal, triangular , ..) fed to the $y$ axis and produce a Lissajous figure.


  • $\begingroup$ No, the question is not related to the Lissajous curves. $\endgroup$ – Yves Daoust Jan 30 at 13:58
  • $\begingroup$ @YvesDaoust: now that the question has been edited, it is more clear. $\endgroup$ – G Cab Jan 30 at 15:00

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