M2023 - Fourier Series : 2 - a series of vectors

this post is meant for the tool : https://memo2007ultra.wixsite.com/m27u-l2016/foworier-sewies

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but before diving into the complexity of fourier series, we need to take a look at the simplest building block : a spinning unit vector, or a unit vector pointing at a certain angle in the complex plane :

this is one of the famous fomular : Euler's formula, there are many proofs available online, either tearing e into it's taylor series and then clump it into 2 taylor series which are later to be cos and sin, or looking it from a dynamic perspective suggested by 3B1B.

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here we can use time to describe θ

here

t refers to time,

f refers to the frequency (or how fast it spins)

φ refers ot it's initial direction (which direction the vector is pointing at when t=0)

<note>

both f and φ are in radiant not degrees

</note>

by combining these 2 equations, now we have a spinning unit vector rotating at a steady speed f starting with the direction φ

now we just need to scale this spinning unit vector so some length (amplitude), so that a single spinning vector (or in foworier sewies, we call it a wiper) can be written as :

the way that Fourier Series works is that we have some numbers of vectors w, each with a steady spinning speed f, each starting at a direction φ , and each has an amplitude/magnitude of a

so if we add all the vectors w tip to tail, the final vector v at time t will be something like this :

this is the notation / naming convention we will be using for the next few posts

below are some of the self made notations that are not available publicly :

the 1st one is almost like the sumation sign, but at least we can see the pattern easier, rather than trying unpack everything from the big sigma

the 2nd one is for compressing data, note that on the top right, we have the word "worldSpace", meaning the frequency and phases are in worldSpace, to make it localSpace, we just need to change "worldSpace" to "localSpace"