Deriving the Nautilus shell spiral equation

Question

Suppose I need to estimate the cross section of a Nautilus shell, which is famously approximated by a logarithmic spiral, $r=ae^{b\theta}$.

The cross-section of this spiral could be found by integrating over the last revolution.

$$\frac{1}{2}\int_{6\pi}^{8\pi}r^2~d\theta$$

But I am struggling with finding $a$ and $b$. We can measure the width of the shell, $w$, along the horizontal axis, and the hight, $h$, along the vertical axis. How do we then find $a$ and $b$ via $w$ and $h$?

Answer 1

A better method is to convert the $\,(r,\theta)\,$ data points to do a linear fit of $\, \log(r) = \log(a)+b\theta.\,$

Answer 2

If the total number of rotations are 4, you have a set of 2 equations here: $$w = a(e^{8\pi b} - e^{7\pi b})$$ $$h = a(e^{7.5\pi b} - e^{6.5\pi b})$$ Further calculation of solving for $a$ and $b$ is trivial.