# Thread: Arc Length of an Ellipse - derivatives

1. ## Arc Length of an Ellipse - derivatives

Hello,
Firstly I would like to say that I am new to this forum. I do not go to university yet, but I am sure this is university level mathematics. I have a quite specific question, and I am not sure it belongs to this sub-section of the forum. If it does not, then I apologize.

Calculation of Ellipse Arc Length

This website described the process of calculating the arc length of an ellipse. I understand everything up to the calculus part. Why exactly is the derivative of the arc length that specific value? It seems to somehow come from the Pythagoras theorem, but I do not quite see why the derivatives should be connected in such a way.

If you scroll down the webpage, a nice formula appears: (the ds/dx part)
Again, this seems to follow from the Pythagoras theorem, and again I do not know why. Why is dx/dx taken? Why are derivatives used at all?

And finally, once this is solved, how can I solve the integral to get the definite value of the arc length of part of the ellipse?

All credit from the website goes to the author of it.

2. ## Re: Arc Length of an Ellipse - derivatives

Originally Posted by Supernova008
Again, this seems to follow from the Pythagoras theorem, and again I do not know why. Why is dx/dx taken? Why are derivatives used at all?
There is no derivative. This is an integral. That 'dx' in the integral may represent nothing but a reminder of what variables you are using. It could also usefully mean a differential. "Slope" is defind a Rise/Run or dy/dx, or simply the change in y given the change in x. 'dx' may mean just the change in x.

Your Pythagorean theorem may make more sense if you would draw the hypotenuse.

As far as solving, ever hear of "Elliptical Integrals"? Not a trivial task. Let's get through the definition of the Riemann Integral, first.

3. ## Re: Arc Length of an Ellipse - derivatives

Well, let's not worry about solving it, I can just type it into any maths program and it should solve it.

However, you say these are not derivatives. So the dy/dx and dx/dx in the triangle are not derivatives?

I still don't quite get why the Pythagorean theorem is used. Could it be because it is an infinitely small x and an infinitely small y, and using the pythagoras theorem, they make an infinitely small s? But why are they all over dx?

4. ## Re: Arc Length of an Ellipse - derivatives

Look at the drawing. Must usefully, they are differentials. In the sense of the graph, they symbolize a small change in one direction or the other.

Did you draw in the hypotenuse where dx/dx and dy/dx are the legs of a right triangle?

5. ## Re: Arc Length of an Ellipse - derivatives

Yes, I did try to draw the hypotenuse. Yes, I realise it is an approximation of the arc length.

What I am trying to find out is why dx/dx and dy/dx are used to produce ds/dx. Why not for instance dy/dy, dx/dx and ds/dx (just an example).

And how could I mathematically show that (ds/dx)^2 = (dy/dx)^2 + (dx/dx)^2?

Also, knowing one point on the ellipse and the arc length, is it possible to extract the second point?

6. ## Re: Arc Length of an Ellipse - derivatives

It's a simplification. It was dx and dy, but that takes two differentials.

7. ## Re: Arc Length of an Ellipse - derivatives

Originally Posted by TKHunny
It's a simplification. It was dx and dy, but that takes two differentials.
I don't quite get it. Could you just explain it fully? I would really appreciate it

8. ## Re: Arc Length of an Ellipse - derivatives

$(dx)^{2} + (dy)^{2} = (ds)^{2}$

Divide by $(dx)^{2}$

9. ## Re: Arc Length of an Ellipse - derivatives

And if I have one point and the arc length, how can I extract the second point?

10. ## Re: Arc Length of an Ellipse - derivatives

?? What point? The drawing is representative of ALL points. dx is not a point.

11. ## Re: Arc Length of an Ellipse - derivatives

Originally Posted by TKHunny
?? What point? The drawing is representative of ALL points. dx is not a point.
I know. But the formula is used to calculate the arc length of an ellipse between two points, ie. the arc length of the ellipse is a definite integral. Now, say I have the point (x,y) on an ellipse, and want to find what the end point (a,b) of an arc of length m, how can I do that?

I would assume that I would subtract the value of the integral at (x,y) from the arc length to find the integral at (a,b). How can I then extract a and b from that?

12. ## Re: Arc Length of an Ellipse - derivatives

1) Such an answer is not unique. From (x,y), one can proceed two diretions to obtain the desired arc length. You should decide which way is "positive". Usually, the left-hand rule is a good convention.

2) The search has limitations. Can your integral defintion go around corners? A cartesian implementation may have more trouble with this. Polar representation may be more useful.

13. ## Re: Arc Length of an Ellipse - derivatives

Fine, let's say I decide on the left-hand rule. How can I extract the second point?