This has tripped me up in the past, so let’s write it down.

Imagine we have an ellipse

Let’s say we have some parametrization (x(t), y(t)) of the ellipse and we want to convert it into a unit-speed parametrization. We can do this by composing with the inverse of the function s(t) that tells us how much arclength our parametrization has traced out on [0, t], which is given by

Differentiating the equation cutting out our ellipse, we have

Let’s solve for y’ so that we can eliminate it from the integral:

Squaring,

Plugging this in, we have

Let’s eliminate y from the equation as well. From the defining equation of the ellipse, we have

Plugging this in, we get

Now let’s assume that our parametrization has and for small values of — for instance, we could start the parametrization from the top of the ellipse and go counter-clockwise. Then, for in the second quadrant, we can write

Putting , this becomes

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