Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



Re: How can I understand Euler's Equation as a 3Dimensional Helix?
Posted:
Apr 9, 2011 8:37 PM


> 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 "faceon", was in > fact a helix when looked at from the side. The > "spirality" of the faceon 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 3D 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 3D 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 xyplane and get the helix:
(cos(t), sin(t), t)
Not sure this is what you are after.



