Date: Apr 9, 2011 3:42 PM
Subject: How can I understand Euler's Equation as a 3-Dimensional Helix?
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. :)