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This is a homework problem I am having trouble with:

Show that if a planar unit speed curve $q(s)$ satisfies $$\kappa_s = \frac{1}{es+f}$$ for constants $e, f >0$, then the curve is a logarithmic Spiral.


Here is what I have so far.

$\kappa_s$ is the signed curvature of $q(t)$ and is defined by

$$\kappa_s = N_s \cdot \frac{dT}{ds}$$

so my starting point is

$$N_s \cdot \frac{dT}{ds} = \frac{1}{es+f}$$

Here's what i know about Logarithmic spirals:

Parametric form: $x(t)=ae^{bt}cos(t)$ and $y(t) = ae^{bt}sin(t)$.

Polar form: $r= ae^{b\theta}$ or $\theta = \frac{1}{b} \ln{(\frac{r}{a})}$

I honestly don't know how to get started on this problem. I have a suspicion I need to integrate both sides of the equation in the problem with respect to s, because this will give me a logarithm on the right hand side. But I wouldn't know how to tackle that integral on the left hand side, or relate that expression to r. Any general nudging in the right direction would be very helpful.

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  • $\begingroup$ What, exactly, is the definition of $N_s$? Cheers! $\endgroup$ – Robert Lewis Oct 21 '15 at 3:59
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    $\begingroup$ I don't really have a textbook for my class, which can be problematic, but I understand that $ N_s = dT/d\theta$, correct? $\endgroup$ – Kyle Funk Oct 21 '15 at 4:13
  • $\begingroup$ Well, Kyle, I was sort of hoping you would know if it's correct or not, since before I saw the definition in your comment, I didn't know what $N_s$ is! Cheers! $\endgroup$ – Robert Lewis Oct 21 '15 at 4:34
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    $\begingroup$ $N_s$ is the signed normal, which my teacher defined the following way: if $T = (a,b)$, then $N_s = (-b,a)$. The other definition I found from wolfram math world, and it seems more useful, though i have trouble intuitively understanding what it means $\endgroup$ – Kyle Funk Oct 21 '15 at 4:38
  • $\begingroup$ So $N_s$ is just $T$ rotated counter-clockwise by $\pi /2$ . . . $\endgroup$ – Robert Lewis Oct 22 '15 at 18:53

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