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I have a parametric function for an ellipse:

$$f\left(t\right)=\left(a\cos\left(t\right),b\sin\left(t\right)\right)$$

As the function goes linearly through t from 0 to 2pi, the point speeds up near the minor axis and slows down near the major axis. I don't want this, I want it to have a constant speed.

I have tried taking the derivative, but then I don't know what to do with the derivative to make the parametric function output a constant speed. I have tried plugging the linearly increasing value into functions that I put inside the parametric function, trying to find a function that increases at a rate inverse to the speed. I have been using Desmos for my work.

In other words, how can I derive a function that creates the same shape over time but has a constant speed or derivative of magnitude of velocity? As well as this, can the same method be applied to any parametric function?

Kitty Craft0
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  • I think you'd have to use elliptic integrals, which are non-elementary functions. – mr_e_man Nov 21 '23 at 19:49
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    The arclength $s$ is given by $$\left(\frac{ds}{dt}\right)^2=\lVert f'(t)\rVert^2=a^2\sin^2t+b^2\cos^2t$$ $$s=\int\sqrt{a^2\sin^2t+b^2\cos^2t},dt,$$ and then you'd have to invert this function, solving for $t$ in terms of $s$. – mr_e_man Nov 21 '23 at 19:56
  • Unit speed parametrization is equivalent to parametrization by arclength. Please refer to my post here. – Ng Chung Tak Nov 23 '23 at 00:09

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