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One of the most famous 2D-curves are logarithmic spirals (or Spira mirabilis). They can be constructed by using a machinery that ensures a constant angle between the tangent and the radial lines all the time while plotting it.

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My question
I can see it in the picture that the spiral arms are getting bigger each turn and I see the math. What I don't understand intuitively is where the logarithm comes from - or put differently: Why does a geometric progression arise just by holding an angle constant. Could you give me some hints? Thank you,

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    $\begingroup$ You need to derive a relation between radius and rotating angle in a small differential form finite triangle. This sets up the basic differential equation for the mirabilis. Following the diagram small triangle trig gives $ \dfrac{r\cdot d \theta}{dr } = \tan \alpha $ where $\alpha$ is constant, which should be integrated. Transpose and recognize $dr/r$ gives log on integration. When variables are swapped exponetial relation between radius and rotation angle.$\theta $ ensues. $\endgroup$ – Narasimham Nov 9 '17 at 17:46
  • $\begingroup$ @Narasimham: Thank you! Could you add that to your answer? That would be awesome. $\endgroup$ – vonjd Nov 9 '17 at 18:56
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Much is explained by looking at the polar equation of the spiral:

$$r=\exp(\theta\cot\alpha)$$

Here, $\alpha$ is the constant angle any tangent to the curve makes with the radius vector (a line segment joining the origin and the point of tangency). This explains the adjective equiangular (the verification of this property from the defining equation is left as an exercise). As an aside, insects flying towards a point light source like a candle or a light bulb follow the path of an equiangular spiral, since the usual strategy of an insect flying at the daytime to get their bearing is to fly at a constant angle from the sun's rays, and this strategy works against them when encountering man-made light.

Now, suppose we have an arithmetic progression of angles $\theta,\theta+\Delta\theta,\theta+2\Delta\theta,\dots$; if we get the corresponding values of the radius vector using the defining equation for the logarithmic spiral (geometrically speaking, this corresponds to a clockwise rotation by $0,\Delta\theta,2\Delta\theta,\dots$ radians), we get

$$\exp(\theta\cot\alpha),\exp((\theta+\Delta\theta)\cot\alpha),\exp((\theta+2\Delta\theta)\cot\alpha),\dots$$

which can be re-expressed as

$$\exp(\theta\cot\alpha),\exp(\theta\cot\alpha)\cdot\exp(\Delta\theta\cot\alpha),\exp(\theta\cot\alpha)\cdot\exp(\Delta\theta\cot\alpha)^2,\dots$$

which as you can see is a geometric progression; that is to say, the logarithms of the members of this sequence form an arithmetic progression. This is where the logarithmic adjective arises from.

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  • $\begingroup$ M.: Thank you (also for the book tip!) To be honest with you: Now I get the part with the geometric progression - but I still don't see the connection between the const. angle and the logarithm :-( $\endgroup$ – vonjd Oct 25 '10 at 16:35
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    $\begingroup$ @vonjd: I'd chalk it up as a coincidence... the spiral happens to both have the constant angle property and radius vectors in a geometric progression. One way to proceed would be to derive the equation of the equiangular spiral from one property and then use it to derive the other. $\endgroup$ – J. M. is a poor mathematician Oct 25 '10 at 22:27
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If something increases at a rate proportionate to that same something on time or angle t basis, the thing is growing exponentially.

If $ dr/dt$ is proportional to $r$ with a proportionality constant cot($\alpha$), then $ r = {r_o} e^{ cot \alpha* t } $.

To appreciate where from cot(al) came, draw the differential right angled triangle where $ r dt / dr = CA/CB= tan(\alpha)$, treating $ rdt $ and $dr$ as finite lengths.

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So I shall draw that triangle after some three years..

For time being ignore the curvature of spiral infinitesmal segment $AB$ ( afterall any line zoomed in enough appears straight... like the Earth appears flat to us as nearby persons but not so to someone looking at us from the Moon..) and let $ACB$ be reckoned as a straight-sided right triangle... this is more accurately a curvilinear triangle:

enter image description here

$\alpha$ is constant in the integration of first line to result in the second line.

Constants in a differential equations are their own, eigen, or remain as their properties stay put that way when other variables change.

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