Point on a archimedean spiral given distance from a known point.

Question

I have an archimedean spiral, and know these points:

angle and distance from center of the beginning of the spiral, angle and distance from center of the end of the spiral

I would need the angle and distance from center of a point that is X mm in distance along the line from the start...

all values are in mm.

for example (angle, distance to center):

P start: 2Pi , 20

P end: 6Pi, 60

Distance from P(start) to px: 110mm

how can this be caluclated? In my case semplicity is more important than precision, it is enough to have a approximation.

Answer 1

This spiral has equation $$r=10 \frac{t}{\pi}$$ Length of the spiral is $$L(x)=\int_0^x\sqrt{r^2+{r'}^2}\,dt$$ In this case we must have $$L(x)=\int_{2\pi}^x \frac{10 \sqrt{t^2+1}}{\pi } \, dt=110$$ which means $$\frac{5 x \sqrt{x^2+1}}{\pi }+\frac{5 \log \left(\sqrt{x^2+1}+x\right)}{\pi }-10 \sqrt{1+4 \pi ^2}-\frac{5 \log \left(\sqrt{1+4 \pi ^2}+2 \pi \right)}{\pi }=110$$ approximated $$x\approx 10.4\sim 360°+235.7°$$ Distance from origin is given by the spiral equation $$r=10 \frac{10.4}{\pi}\approx 33.1$$

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