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I'm in the need to draw (more like to use a wood router to carve a groove) a logarithmic spiral in a piece of wood. So, I got a router that is attached to a stick, I draw a circle by rotating the stick around a center.. but, I need to increase the radious (the router's distance from the center) in a logarithmic fashion to obtain the so elusive logarithmic spiral.

Now, the problem here is that I need to find a no-super-complicated way to increase the radius, do you know any method to achieve that?.

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In a mechanical way??? – copper.hat Aug 1 '14 at 5:00
Yeah, I mean, the mechanical part would be something like: - How much should I increase the radius by every degree I rotate the stick?, something like that. – Artemix Aug 1 '14 at 5:22

This may help - two properties of the logarithmic spiral:

  1. Every ray through the origin intersects the spiral at the same angle.
  2. The arc length from any point on the logarithmic spiral to the pole (center) is finite, although it takes infinitely many rotations to reach the pole.

Source - e: the Story of a Number

For a particular point on the curve and on a radius, draw the radius on which the next point will be marked. Then draw a line at the particular spiral angle for that curve from the first point to the next radius. The intersection will be the next point. If the radii are close enough together the linear segments will approximate the spiral.

There is probably a way of working out what angle to use for finite (as opposed to 'as small as possible') spacings of the radii. I think this is the critical equation to find.

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There's also a certain line whose length is the same as the arc length, though I can't remember it right now. – Akiva Weinberger Aug 1 '14 at 5:58

So this isn't an exact logarithmic spiral but it captures the idea

The equation for exponential-style spirals is

$$r = ae^{b\theta}$$

What that means is that if we consider the angles $\theta, \theta + k$ then

$$\frac{r(\theta + k)}{r(\theta)} = e^{bk}$$

Which means the ratio of the radiuses is a constant for fixed difference between the angle.

So as an example lets say you had your radius at 1 cm and you draw a point. Now if you tilt your router by a slight angle (say 1/3 of a degree or something else thats tiny) the appropriate radius would be just a multiple of 1 cm by a fixed constant. You can move like this plotting points and simply extending the length of the router by multiplying its current length by that fixed constant, then measuring it up with a ruler.

note that for any triplet of 3 points there exists a circle that passes through them so you can use circular arcs between sets of 3 points to connect them (or if the points are close enough just connect every pair with a plain old line made by ruler)

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From Wolfram MathWorld:

The logarithmic spiral can be constructed from equally spaced rays by starting at a point along one ray, and drawing the perpendicular to a neighboring ray.

Edit I searched for straightedge and compass constructions and found an old Popular Science article featuring the math behind the router. However, it doesn't show any spirals.

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Isn't that only one type of logarithmic spiral? – mistermarko Aug 1 '14 at 6:15
in the limit as the number of rays goes to infinity, though... – Semiclassical Aug 1 '14 at 6:17
Wont' be very smooth, but +1. The spacing between rays is a parameter you can vary. – Jyrki Lahtonen Aug 1 '14 at 6:20
The intersection is not necessarily a right angle - I'm just quoting the author of the book. – mistermarko Aug 1 '14 at 6:22
@mistermarko, I think the idea here is that the "non-right" angle at the other ray will remain constant. Anyway, your criticism is valid in the sense that (AFAICT) the separation from the right angle in this method equals the angular separation between adjacent rays, so we cannot "smoothen" the curve (by adding more rays) and simultaneously keep the angle. Do click the MathWorld link in Fred's answer! – Jyrki Lahtonen Aug 1 '14 at 17:55

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