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Topic: How can I understand Euler's Equation as a 3-Dimensional Helix?
Replies: 5   Last Post: Apr 21, 2012 6:21 AM

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Andy Talmadge

Posts: 44
From: New Orleans
Registered: 2/2/05
Re: How can I understand Euler's Equation as a 3-Dimensional Helix?
Posted: Apr 9, 2011 8:37 PM
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> I'm quite interested to know how Euler's Formula can
> be both interpreted as a circle and as a helix inside
> the "Real" and "Imaginary" Plane.
>
> Wiki gives a "three dimensional visualization" of the
> curve. It would be helpful for me if anyone could
> explain how this "visualization" is derived with the
> use of real values. I'm having troubles with this
> formula; e^(i)(x)= (cos x + (i)sin x) and the 3
> dimensional interpretation of it.
>
> I'll explain a bit why I'm interested. I posted a
> question a while ago about how to formulate the
> growth of a spiral from the tail towards the inside.
> After not being able to complete this, I realized
> that the spiral I was looking at "face-on", was in
> fact a helix when looked at from the side. The
> "spiral-ity" of the face-on appearance was simply a
> perspective illusion.
>
> So, having understood that the growth of this spiral
> was in fact the same as with e^(i)(x) -(with some
> variation), I have since been trying to understand
> the 3-D interpretation of Euler's Formula. It would
> help me a lot with a certain project I'm looking in
> to.
>
> Another query I have is into the "Maclaurin Series"
> for the exponential function, and the sine and cosine
> functions. Would a better understanding of these
> series and their uses help in understanding the
> geometric 3-D form of Euler's Formula?
>
> Thanks a bunch. :)


The normal identification of complex plane with real plane is x + iy <-> (x,y). This would identify cos(t) + i sin(t) with (cos(t), sin(t)). You could build helix in R^3 using this as the xy-plane and get the helix:

(cos(t), sin(t), t)

Not sure this is what you are after.



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