Can Euler's Formula Be Modified to Create a Logarithmic Spiral?

[Ahmidahn]

Is there a way to take Euler's formula "e^(i∅)" -which gives a circle; and change it into a logarithmic spiral?

Does a simple modification like " e^-(i∅/n) " make any sense mathematically?

If it actually does, my other question would be; supposing that such a logarithmic spiral is in fact just a head on view of a space curve with parameters like:

y=tsint
x=tcost
z=e^(-t/12)

how does one go about finding the arc length of such a curve? I've attempted to use the general arc length formula on this one, but continually end up with something like:

s = ∫ √e^-t/12 + t^2 + 1

which is, apparently, impossible to solve.

Is there a way to use the complex variable in a formula like Euler's formula to create a logarithmic spiral? And if so, how does one go about calculating the arc length of such a thing?

 

[Simon Bridge]

Yeah - you change the amplitude with angle.

remember - euler's formula is for a circle in the complex plane.
the logarithmic spiral formula is for the real plane.
to change it to complex - just multiply the y parameterization by the square-root of minus one.

OR: you could just set Aexp(it) so that A=the radius of the spiral at angle t.

note: exp(it/n) just changes the frequency of the rotation.

 

[Ahmidahn]

Maybe something like:

(cos $\theta$ + i sin$\theta$)/w

would translate into:

[exp(i$\theta$)] / w where w is some decreasing function related to $\theta$? I don't know. I've been stuck on this one for almost two years.

Thanks for your help.

 

[Ahmidahn]

The Wikipedia entry for "Circular Polarization" has the "classical sinusoidal plane wave solution of the electromagnetic wave equation" - and the math behind it as well.

It's similar, in general, to what I'm looking for. The math is beyond me. I actually know what some of those things are, but I think I would need to enroll in school again to understand usefully.

Tough stuff. Maybe I should ask some physics guys how to transform Circular Polarization into Spiral Polarization.

 

[robert2734]

what if we take e^i$\Theta$ and turn it into e^-a+i$\Theta$? Now we have a logrithmic spiral no?.

 

[Ahmidahn]

Yes. That would seem to make sense. Correct me if I'm wrong reinterpreting the equation.

e^{-a + i$\theta$} = (e^-a)(e^i$\theta$)=

(e^-a)(cos$\theta$ + i sin$\theta$)

What kind of variable is "a" in this situation?

Is this a three dimensional curve, or rather, can it be visualized in 3 dimensions?

Thanks for your help there.

 

[Ahmidahn]

I'm going to have to suss that out for the arc length...hmm...getting closer.

 

[Simon Bridge]

What kind of variable is "a" in this situation?

a is an angle. Anything inside the exponential function must be dimensionless.

Is this a three dimensional curve, or rather, can it be visualized in 3 dimensions?

You can represent/visualize the curve how you like.

Taken as a locus of points in the complex plane, it is two dimensional.
If a is a constant, then the locus is a circle.

But you can also make the angle, any of them, a function of time - in which case, $e^{i\theta(t)}$ is rotating. You can make $a$ a function of time, or even of a third space dimension if you want ... and plot a locus in 3D or an evolving spiral path in 3+1.

Consider:
$$Ae^{\alpha t}e^{i\beta t} = Ae^{(\alpha+i\beta)t} = Ae^{zt}$$
if $\alpha=\beta=1$, what is the shape mapped out in the complex plane?

You could also look at: $at[\cos(bt)+i\sin(bt)]$, where a and b are arbitrary constants.

Have fun.

 

[tommyli]

Replace $$i$$ in Euler's formula with $$i -1$$ and you get a logarithmic spiral:
$$e^{(i - 1)t}$$ parametrically describes a logarithmic spiral in the complex plane

 

[Simon Bridge]

In other words, in OP notation, a=-t \theta = t.
But you can have fun experimenting with lots of different spirals besides the golden one :)

 

[Dickfore]

Is there a way to take Euler's formula "e^(i∅)" -which gives a circle; and change it into a logarithmic spiral?

Does a simple modification like " e^-(i∅/n) " make any sense mathematically?

Please define a "logarithmic spiral".

 

[Ahmidahn]

"Logarithmic spiral"= r = a(exp)-b(theta) in polar coordinates

Parametric form:

x(t) = r(t) \cos(t) = ae^{bt} \cos(t)\,
y(t) = r(t) \sin(t) = ae^{bt} \sin(t)\,

 

[Ahmidahn]

Are you wondering why I write the power in the exponent (theta) over "n"? Or what?

 

[Ahmidahn]

Thanks again for all the replies. I will be pondering this one for a while.