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I'm using Desmos, and have already combed through this site not finding anything close to what I need, nor have the equations and modifications I have tried been of help.

Desmos Trial by Combat

I need to either construct an Archimedes spiral from an adjustable amount of equal linear length segments, or use a slider to automatically mark a certain number of points that are each an equal predetermined length apart along the length of the spiral.

I do not understand calculus, and so would appreciate being taught like an idiot.

CryptoMynd
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  • The closest thing I found is https://math.stackexchange.com/questions/81636/how-to-place-objects-equidistantly-on-an-archimedean-spiral , but I'm using Desmos not python, and I don't understand enough about what they said to port it over. – CryptoMynd Jun 09 '24 at 23:25
  • I figured out that I somehow need to nullify the rate of change that happens due to the angle with increasing radius, but have not figured out how, nor how to apply that knowledge. – CryptoMynd Jun 10 '24 at 00:44

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As mentioned in this answer, there is no way to express the inverse of the arclength of the Archimedean spiral using elementary functions. However, one can numerically approximate the inverse using the Newton-Raphson method (by solving for $f(y)-x=0$) and a decent approximation. To find such an approximation, notice that the arclength of the spital can be written as $\frac{1}{2}(\operatorname{arcsinh}(x)+x\sqrt{x^2+1})$. $\operatorname{arcsinh}(x)$ grows logarithmically, while $x\sqrt{x^2+1}$ grows close to quadratically. Thus the arclength can be approximated as $\frac{1}{2}x|x|$, which has an inverse of $\operatorname{sign}(x)\sqrt{2|x|}$. An implementation of the above can be found here:

https://www.desmos.com/calculator/whbjmkijr8

Einsteinium
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