Talk:Logarithmic spiral

From Wikipedia, the free encyclopedia

What type of spirals do subatomic particles track in cloud chambers?

a log spiral -- if the drag force is proportional to speed (which I doubt). (If a uniform magnetic field runs perpendicular to the complex plane, with Im(z) the strength of the field, -Re(z) the drag coefficient, and

x'

the velocity, then

x'' = z x'

. Integrating twice is trivial.) 142.177.124.178 21:05, 20 Jul 2004 (UTC)

Is it "spiralis mirabilis" or "spira mirabilis"? I've seen both, which is correct?

I urge caution with the pedal curve comment (I merely preserved it); I suspect it's only so if the pedal point is the center/origin of the spiral. 142.177.124.178 20:27, 20 Jul 2004 (UTC)

D'oh, it's so obvious. If the pedal point is not the spiral's center then the pedal curve passes through the pedal point an infinite number of times. No log spiral passes through any point more than once. 142.177.124.178 06:58, 23 Jul 2004 (UTC)

I fixed my mistake in the differential geometric definition of the logarithmic spiral.MathMartin 21:50, 20 Jul 2004 (UTC)

Presumably some log spirals are exactly their own evolute (no rotation needed) ... anyone know which? They're the only such curves, ya? 142.177.124.178 07:33, 23 Jul 2004 (UTC)

'Cording to my ciphering, the evolute of $t\mapsto e^{zt}$ is $t\mapsto {\bar z+z\over\bar z-z}e^{zt}$ ; if that's right, then the spiral is exactly its evolute iff ${\Re z\over\Im z}\left(2n+\frac12\right)\pi=\ln {\Re z\over\Im z}$ for some integer n. 142.177.124.178 18:05, 23 Jul 2004 (UTC)

[edit] z^t vs e^wt

I'd prefer it if

The spiral is nicely parametrized in the complex plane:

z t

, given a z with Im(z)≠0 and |z|≠1.

were changed to use the exponential function directly, rather than relying on complex exponentiation .. there are difficulties there that we need not deal with in this article, plus the exponential function arises more directly from solving certain differential equations.

Comments?

RandomP 12:35, 19 May 2006 (UTC)

Go for it. Melchoir 21:09, 19 May 2006 (UTC)

[edit] Broken explanation of a and b

This bit of text is internally inconsistent:

with positive real numbers a and b. a is a scale factor which determines the size of the spiral, while b controls how tightly and in which direction it is wrapped. For b >0 the spiral expands with increasing θ, and for b <0 it contracts; the only difference between positive and negative b being that one spirals to the left and the other to the right.

If b is a positive real number it can't be < 0. "Positive real number" is a left-over from the original formula $r = a b θ$ . We could let b be a real number (and note that for b = 0 you get a circle; b = 1 in the original formula), or we could go back to the original formula.

Negative b won't cause it to spiral to the left. I think the confusion here stems from the word "direction" which I think was meant to indicate inward/outward. I therefore suggest to change it to "direction (inward/outward)". By the way, negative a would make the spiral turn to the left (in essence it mirrors the spiral over the X axis), but we've posited that a is a positive real number. We could drop the "positive".

24.6.122.223 13:40, 24 May 2006 (UTC)

And that's not all that's broken: "scale factor which determines the size of the spiral"? That's just nonsense. Anyway, a negative a in r=a exp(rt) wouldn't mirror over the X axis; remember we're in polar coordinates. I'll have a go at fixing it... Melchoir 14:21, 24 May 2006 (UTC)

[edit] Merge spira mirabilis

someone suggested we merge Spira mirabilis into here, but didn't start the discussion.

Support. That article is just a stub on the same topic as this one, and includes misinfo e.g. about divine proportion. Dicklyon 00:10, 12 October 2006 (UTC)

Support. We don't need multiple articles on the same thing just because there are multiple names for it. —David Eppstein 19:03, 15 October 2006 (UTC)

I merged in a few bits already. All we need to do now is remove the merge tag and convert spira mirabilis to a redirect. And maybe integrate better that what I've done. Any objections? Dicklyon 19:50, 15 October 2006 (UTC)

Retrieved from "http://en.wikipedia.org../../../l/o/g/Talk%7ELogarithmic_spiral_9712.html"