# Euler's relations & 'Epicycloids'

• saviourmachine
In summary, the conversation discusses the concept of epicycloids and their visualization through superimposed circles. The conversation also delves into the idea of using this technique to understand Euler's relations and complex Fourier series. The possibility of creating a java applet to demonstrate this concept is also mentioned.
saviourmachine
The idea
An epicycloid is a superimposed circle on another circle. http://www.math.dartmouth.edu/~dlittle/java/SpiroGraph/ can you find a java applet to show you. These epicycloids are tied to each other at their circumferences. But, what does change when using a slightly easier method, and superimpose the second circle (the referent) using it's centre!?

A point on the first circle moves along its circumference, and because it is the centre of the second circle, the whole second circle does move with along it. Now comes the clue: do the same with the second circle. Take a point at the circumference of the second circle and move in the opposite direction (with a negative frequency). Like you can see will this point move along the horizontal axis. Not much have to be imagined to realize that this traject will be equal to $$2 \cos{\omega}$$. Of course is this equals the sum of $$\exp{j \omega t}$$ and $$\exp{-j \omega t}$$, but it's cool that with rotating in the other direction a meaning is assigned to the concept "negative frequency".

http://www.annevanrossum.nl/pictures/science/Epicycloid.gif

Does anyone know of a clarification of Euler's relations using these drawing techniques?

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Edit: Changed math to Latex.

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Observation: Odd, that the image can't be displayed. That's for paying members?

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Amplitude modulation

Amplitude modulation
Now I'm thinking about it. The frequency $$f_1$$ of the point on the first circle can be the carrier frequency in AM modulation. The frequency $$f_2$$ of the point of the second circle can be a frequency that modulates the carrier. Projecting the second point on the horizontal x gives the resulting AM signal. If I had time I'd like to make a java applet to show that. The complex envelope is approximated a circle (radius equals sum of radius of circle 1 and 2)! $$g(t)=A_c \exp{j \theta(t)}$$

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Complex Fourier series

Another thought. Even the complex Fourier series can be visualized as superimposed circles with different frequencies isn't it?
$$f(t)=\sum_{n=-\infty}^{n=\infty}{A_n \exp{jn\omega_0t}}$$ with $$A_n$$ the radius of the circle n.

Does nobody know of this manner of visualization?

Can a moderator move this thread if it's not appropriate here? It's not homework...

## 1. What are Euler's relations?

Euler's relations refer to three mathematical equations discovered by Swiss mathematician Leonhard Euler. These equations describe the relationship between the trigonometric functions sine, cosine, and tangent, as well as the complex exponential function.

## 2. What are epicycloids?

Epicycloids are geometric curves created by tracing a point on the circumference of a circle as it rolls along the outside of another circle. They were first studied by Swiss mathematician Jacob Bernoulli in the 17th century.

## 3. How are Euler's relations related to epicycloids?

Euler's relations can be used to parametrize epicycloids, meaning they can be expressed in terms of sine and cosine functions. This allows for the calculation of various properties of epicycloids, such as arc length and curvature.

## 4. What are some real-world applications of Euler's relations and epicycloids?

Euler's relations and epicycloids have various applications in fields such as engineering, physics, and computer graphics. For example, they can be used to model the motion of planetary orbits, create gear designs for machinery, and generate visually appealing patterns in computer graphics.

## 5. Are there any other notable mathematicians who have studied Euler's relations and epicycloids?

Aside from Leonhard Euler and Jacob Bernoulli, other notable mathematicians who have made significant contributions to the study of Euler's relations and epicycloids include Pierre-Simon Laplace, Joseph-Louis Lagrange, and Augustin-Louis Cauchy.

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