# Polar plots of $\sin(kx)$

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

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

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 $$k = 1,2,\dots,8$$

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

[End of the interlude.]

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

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 $$n = 1,2,\dots,8$$:

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:

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:

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

resp. moves on another circle with constant speed

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.)

• Can you unabbreviate or clarify the meaning of "resp.", please? I'm not familiar with it. – timtfj Jan 30 at 11:08
• (I still upvoted though) – timtfj Jan 30 at 11:16
• "Resp." abbreviates "respectively" which just means "or" (with a slightly different connotation). – Hans-Peter Stricker Jan 30 at 11:22
• @HansStricker I think it's more common to use "re" instead of "resp". – Jam Jan 30 at 12:23
• @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 :) – 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:

– 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:

– 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):

• better you take just a ray from the origin: then it is easier to controll the parameters – G Cab Jan 30 at 14:58
• @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.) – Hans-Peter Stricker Jan 30 at 15:01
• 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. – 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.

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