Logarithmic spiral transform function

I've got a rectangular image with x and y in uniform space, and I want to twist it round a logarithmic spiral, so that the centreline of the x follows the spiral and all the space is filled. We can assume that the width of the image is a lot greater than the height.

So I need to derive a transform function from each pixel in x, y to a pixel in logarithmic space. Note the space-filling criterion is important, the top pixel at turn theta should be adjacent to the bottom pixel at theta - 2PI.

The filling occurs automatically in a parametric plot by incrementing $\beta$ parameter.

$$r \rightarrow r\cdot ( \cos(\theta+ \beta), \sin(\theta + \beta))$$

A single log spiral is given:

Say you wish to reach same $\theta$ after one rotation when radius $r$ increases from starting $r_1$

Log spiral has differential equation

$$\frac{dr}{r d\theta}= \cot \alpha$$

and by integration polar equation is

$$r = r_1\cdot e ^ {\cot\alpha \, \theta } \tag1$$

$$\alpha =\tan^{-1}[2 \pi/\log( r_2/r_1)]$$

If for example $r_1= 1$, $r_2= 16$, $\alpha$ calculates to $\approx 1.15522$

Choose a small increment like $\beta=\pi/30.$ This parameter rotates the entire log spiral by this increment.

$$(x,y) = r_1\cdot e ^ {\cot\alpha \, \theta } ( \cos(\theta+ \beta), \sin(\theta + \beta)) \tag2$$

• See if I can understand this. I need to start at r1 = 1.0, so the image is 1pixel high. I can then give a free parameter which indicates the multiplication factor for a full turn. That gives me alpha, or the spiral tightness, using your equation. However I'm a bit shaky on beta. – Malcolm McLean Apr 27 '17 at 14:03
• Does equn (2) makes it clear ? $\beta$ rotates each log spiral. – Narasimham Apr 27 '17 at 14:27
• So I just need to rotate theta to correct for pixels off the x axis? – Malcolm McLean Apr 27 '17 at 14:30
• Right, but all around the origin, not just x- or y- axis. Can you show how you work with pixels? I cannot figure out much here although input is only geometrical. – Narasimham Apr 27 '17 at 17:05
• Essentially I have an image which is rectangular and a lot wider than it is high, say 25 pixels by 1000 pixels. It might contain text. I want the text to be scaled, in a space filling manner, round the log curve. So I imagine that y = 12.5 would be the line on the curve itself. But y= 0 should touch y = 26 on the previous loop. – Malcolm McLean Apr 28 '17 at 8:41