I have to place a series of line segments along an $r=k\theta$ spiral. All the line segments are the same length (which I will call $L$) and the starting point of each is the finish point of the previous.
For my task I decided it will be easiest to do if I can determine the next radius $r_2$ from the previous radius $r_1$ (or the angle $\theta$ from the previous radius $r_1$).
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So, I have the equations:
$r_2 = r_1 - \Delta r$
$\Delta r = k \theta$
and of course all the triangle relations from $\theta$ being between $r_1$ and $r_2$ and opposite $L$.
I plugged them together to get $r_1-k \theta=r_1\cos(\theta)\pm\sqrt{L^2-r_1^2\sin^2(\theta)}$
So, I need to solve for $\theta$ to get my $\theta$ and $r_2$. But, I don't know how to do this and the online solvers can't manage it (although they can solve very close ones...) Can anyone help with it?
Thank you!
It might be easiest simply to do it numerically -- write a parametric equation for your spiral and use bisection (or possibly Newton-Raphson) to find the parameter for the next point that has the desired distance from your current one.