### But can't you just use integration on the arc length?

I recently came across the following video

It made me think surly this can be derived in the same way as one can derive the formula for a circle. I'd gone through that derivation a number of times at school and since the formula for an ellipse is so close thought the same approach would surly work for an ellipse.

I decided to give it a go believing that I surely wouldn't get the answer I expected but it would be interesting to see why the arc-length method wouldn't work.

At this point, if I had to guess I would guess that the impossibility lay in the issue with integration.

The formula for a circle can be represented parametrically as:

To adapt this to an ellipse one simply needs to add constants in front of each trigonometric function:

The arc length of a line can be calculated by the following identity which follows from Pythagoras:

Using this identity we can get a formula for s in terms of x and y. However, this is where the signs of the problem begin as to find s from ds we need to perform an integration.

## Comments

## Post a Comment