[hamdani yusuf (OP)]

Hypo Cycloid and its Parametric Equation

Hypo-Cycloid is the path traced by a point on a circle which is rotating on the inner circumference of other circle without sliding. This Short video illustrates what is Hypo-Cycloid and How to derive its parametric equation in terms of Radii and Angles. We also discussed about how different combinations of Radii generates Hypo-cycloids.

Chapters:
00:00 Hypo-Cycloid Introduction
00:08 Parametric Equation for Hypo-Cycloid
02:43 Hypo-Cycloid with different combinations of R & r : Tusi Couple
02:55 Deltoid
03:07 Astroid
03:19 Summary

[hamdani yusuf (OP)]

https://en.m.wikipedia.org/wiki/Hypocycloid
If the smaller circle has radius r, and the larger circle has radius R = kr, then the parametric equations for the curve can be given by either:


or



If k is an integer, then the curve is closed, and has k cusps (i.e., sharp corners, where the curve is not differentiable). Specially for k = 2 the curve is a straight line and the circles are called Tusi Couple. Nasir al-Din al-Tusi was the first to describe these hypocycloids and their applications to high-speed printing.[4][5]

If k is a rational number, say k = p/q expressed in simplest terms, then the curve has p cusps.

If k is an irrational number, then the curve never closes, and fills the space between the larger circle and a circle of radius R − 2r.

The parametric equations suggest that these polarization states can be generated in radio frequency using two dipole antennae perpendicular to each other.

Some examples.



If those polarization states can be successfully generated, some people might misidentify them as circularly polarized, or even unpolarized light.

[hamdani yusuf (OP)]

I asked chatGPT about it.
Quote
Cyclocycloid polarization is not a standard term in physics or optics, and it does not appear to describe any established scientific concept. It might be a term from a niche field, a hypothetical idea, or a misinterpretation of an existing concept.

It seems like this hypothetical polarization state hasn't been properly explored yet. Although the mathematical concept isn't particularly complicated, and the EM source to produce it shouldn't be that expensive nor hard to build. But I still have no clear idea what benefit from this type of polarization in practice. Perhaps it can increase information density for telecommunications. I'm still thinking how to detect it and distinguish it from other types of polarization.

[hamdani yusuf (OP)]

I've made a scatter chart in spreadsheet to visualize the hypocycloid polarization based on parametric equations.

[hamdani yusuf (OP)]

I asked chatGPT about it.
Quote
Cyclocycloid polarization is not a standard term in physics or optics, and it does not appear to describe any established scientific concept. It might be a term from a niche field, a hypothetical idea, or a misinterpretation of an existing concept.

It seems like this hypothetical polarization state hasn't been properly explored yet. Although the mathematical concept isn't particularly complicated, and the EM source to produce it shouldn't be that expensive nor hard to build. But I still have no clear idea what benefit from this type of polarization in practice. Perhaps it can increase information density for telecommunications. I'm still thinking how to detect it and distinguish it from other types of polarization.

I think this cyclocycloid polarization will be useful to improve our understanding of the physical mechanism of light propagation and its interaction, especially in interference of light. The positions and intensities of constructive and destructive interference may be different from linearly polarized light.

[hamdani yusuf (OP)]

Another way to generalize the oscillation is as cyclocycloid.

When light wave with linear, circular, or elliptical polarization passes through a slit, it will be diffracted and produces interference pattern on the screen behind the slit. The dark fringes are produced by destructive interference, where electromagnetic field on the screen produced by one of the light ray is cancelled out by the other light ray with 180 degrees phase difference.

But if the light is polarized like a deltoid, 180 degrees phase difference don't cancel out. If there's no attenuation during the propagation from the source to the screen, destructive interference requires three sources with 120 degrees phase shifts among one another.

[hamdani yusuf (OP)]

I've made a scatter chart in spreadsheet to visualize the hypocycloid polarization based on parametric equations.

I modified the formula slightly to make it more symmetrical and explore more patterns.


The last pattern seems to produce no cancellation when two sources with 180 degree phase difference are interfering.
And different set of parameters may produce similar patterns.

[hamdani yusuf (OP)]

Here's some more examples.
These patterns might look too mathematical/theoretical and don't seem practical. But if we look closer at the equations, they only involve two distinct frequencies on two orthogonal axis. I think it's possible to produce them using basic optical components like mirrors, beam splitters, linear polarizers, and quarter wave plates.

[hamdani yusuf (OP)]

Here's a basic setup to produce cycloid polarization.

[hamdani yusuf (OP)]

The last pattern seems to produce no cancellation when two sources with 180 degree phase difference are interfering.

I've made the plot for various phase difference. In general, patterns with odd number of lobes doesn't cancel out when interfering with pi radian phase difference. On the other hand, patterns with even number of lobes does cancel out when interfering with pi radian phase difference, and produce destructive interference.

[hamdani yusuf (OP)]

Here's my new video introducing cyclo-cycloid as a new type of polarization state of light, beyond the more familiar of linear, circular and elliptical polarization. This video describes the mathematical basic, as well as how to produce it.