2024/04/24/from-the-circle-to-epicycles

[utterances-bot]

From the Circle to Epicycles (Part 1) - An animated introduction to Fourier Series | andreinc

A visual introduction to Fourier Series

https://www.andreinc.net/2024/04/24/from-the-circle-to-epicycles

[NikhilSP]

Great work!!

[j2kun]

One thing I never quite understood is how you take an arbitrary 2d curve (not necessarily a function) and decompose it into Fourier components. The sort of "line drawing of Newton" thing they did in mathematica a while back

[yaakovgamliel]

Thank you for putting this work out there, excellent material

[jmgimeno]

"If we substitute 𝑎→𝑒 ..."

I suppose you mean a -> i.

[cypris75]

Great work!

I think there is an error within one of the visualizations:

[langley]

This is absolutely amazing! THANK YOU. As someone who can "do math", but lacks a deep understanding (I don't fully grok all the interconnections or fundamental reasons for some of the math I can "do", i.e. repeat like a monkey :^D ), this is an amazing resource! Thank you, I'll definitely be sharing it with other people!

[cwgreene]

One thing I never quite understood is how you take an arbitrary 2d curve (not necessarily a function) and decompose it into Fourier components. The sort of "line drawing of Newton" thing they did in mathematica a while back

@j2kun Can you clarify? An arbitrary curve can always be decomposed into a function $\langle f(t), g(t) \rangle$, the first can be decomposed into cosines, and the latter sines (and then can be paired off to form circle vectors). To obtain the initial $f(t)$ and $g(t)$ from an existing line drawing, there is room for creativity, is this what you're asking about?

One option is to transform the rasterized points into connected components, and then order them so that the hop between components is minimized. A greedy optimization algorithm would probably work well.

[j2kun]

One thing I never quite understood is how you take an arbitrary 2d curve (not necessarily a function) and decompose it into Fourier components. The sort of "line drawing of Newton" thing they did in mathematica a while back

@j2kun Can you clarify? An arbitrary curve can always be decomposed into a function $\langle f(t), g(t) \rangle$, the first can be decomposed into cosines, and the latter sines (and then can be paired off to form circle vectors). To obtain the initial $f(t)$ and $g(t)$ from an existing line drawing, there is room for creativity, is this what you're asking about?

One option is to transform the rasterized points into connected components, and then order them so that the hop between components is minimized. A greedy optimization algorithm would probably work well.

Yeah, it's how to get the parameterization from something that is not inherently parameterized, like a bitmap. I think you answered my question and then edited it.

[cwgreene]

Yeah, it's how to get the parameterization from something that is not inherently parameterized, like a bitmap. I think you answered my question and then edited it.

Yeah sorry about that, I made an editing change and nuked the entire post.

[AndreRouth]

Hi Andrei, Those plots are amazing! And helpful. The addition of the sinusoids is a great demo of how to create different waveforms. What can you do with modulation which involves multiplication of a carrier with modulating signal. Amplitude Modulation is quite easy to imagine with the carrier and sidebands. I've never seen a good demo of Frequency Modulation which produces a mindboggling splash of side frequencies. Please could you turn this creative streak in that direction? Many Thanks, Andre

[longluo]

Amazing, Thank you, Andrei