topic Re: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse in Mathcad

Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

DaveMartin — Tue, 04 Feb 2025 21:49:22 GMT

I am shocked that I have walked the surface of this planet for over 5 decades without realizing that there is no exact closed-form solution for the perimeter of an ellipse. In school, we learn the perimeter of shapes like triangles, squares, rectangles, circles, parallelograms, and so on, but I feel like schools and teachers conveniently skipped over ellipses.

(An ellipse, in case it’s been a while, is the set of points where the sum of the distances from two points – the foci – is a constant.)

Your challenge, should you choose to accept it, is to correct that oversight. Create a Mathcad worksheet that:

Derives, depicts, or shows one (or more) of the various approximation formulas / methods for the perimeter of an ellipse.
Create a calculator whereby someone can change the values of the semi-major and semi-minor axis lengths in order to find the perimeter.
Create a 3D plot of the perimeter as a function of the ellipse semi-major and semi-minor axis lengths.

Although there is no exact solution for the perimeter of an ellipse, this is a fairly well-documented problem. Therefore, your documentation in your worksheet is key! This worksheet should be able to stand on its own and be understood by someone with a basic knowledge of calculus (since integrals and infinite series are involved).

Good luck and have fun!

Find the Mathcad Community Challenge Guidelines here!

Re: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

terryhendicott — Sat, 02 Mar 2024 07:16:08 GMT

Hi,

File enclosed Prime Express 9.0

Cheers

Terry

Re: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

ValeryOchkov — Sat, 02 Mar 2024 07:58:45 GMT

I once won such an argument a long time ago - I showed two ellipses (Mean) for which there is an exact formula for the perimeter. One of them is a circle, and the second - see the picture.
A more challenging and interesting problem is the Cassini Oval (gmean) and the Cayley Oval (hmean)!

“The race course was a large three-mile ring of the form of an ellipse..”

Leo Tolstoy. "Anna Karenina"

Re: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

ValeryOchkov — Sat, 02 Mar 2024 08:21:02 GMT

@terryhendicott wrote:

Hi,

File enclosed Prime Express 9.0

Cheers

Terry

Re: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

ValeryOchkov — Sat, 02 Mar 2024 09:16:05 GMT

Re: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

ValeryOchkov — Sat, 02 Mar 2024 10:58:11 GMT

More interest plot

Re: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

AlanStevens — Mon, 04 Mar 2024 21:07:52 GMT

My attempt attached (excludes part 3 as 3D -plotting not available in Express).

(As I couldn't do the 3D plot in Prime Express, I decided to do it in M15 - see below.

The plot is really boring!).

Alan

Re: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

DJF — Mon, 04 Mar 2024 16:32:04 GMT

Every problem deserves a numeric solution. I'll leave the complicated derivations for the experts. This was good enough for Archimedes so good enough for me too. Bit too much to make the 3D plot at this time.

Prime 7 attached.

Re: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

terryhendicott — Mon, 04 Mar 2024 16:46:14 GMT

Hi,

Am enjoying the posts so far so want to add a quick from 1st principles approach:

Re: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

terryhendicott — Wed, 06 Mar 2024 03:03:59 GMT

Hi,

Just could not resist it. A page of matrices and voila 3D plot in express.

Cheers

Terry

Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

BertBeirinckx — Fri, 08 Mar 2024 20:32:00 GMT

This is one I couldn't let go. This question is easily solved using polar coordinates:

Calculation speed can be increased using the ellipse's symmetry by limiting the integral between 0 and pi/2 and multiply that value by 4. With a similar technique surface, moment's of inertia and so on are easily calculated.

All this is pretty basic, really. The fun starts when adding parameters to the equation, for instance making Superellipses and even (far) beyond. In the following article I made use of Mathcad Prime to illustrate the above: https://www.athena-publishing.com/series/atmps/issbg-22/articles/260/view

Enjoy!

Bert Beirinckx

Re: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

AlanStevens — Sat, 09 Mar 2024 10:24:40 GMT

@Bert

Alan

Re: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

Werner_E — Sat, 09 Mar 2024 10:55:03 GMT

@AlanStevens wrote:

@Bert

Alan

"... which is exactly ..."

exactly?? The perimeter of a circle can't be calculated exactly, either. 😞

The closed form for the circle circumference is actually a deception, because the infinite series is hidden in the constant pi, which we have invented for the ratio perimeter: (2*r) for the ellipse with a=b=r.

So there is no real difference between the perimeter of a general ellipse and a circle other than the pi approximation key being available on every pocket calculator...

I, too, enjoyed Bert's article.

BTW, I nice video on the topic of the ellipse perimeter can be seen here on Stand-Up Maths channel: Why is there no equation for the perimeter of an ellipse

Re: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

AlanStevens — Sat, 09 Mar 2024 11:15:46 GMT

I would say the circumference of a circle is exactly 2pi.r. Of course, that doesn't mean we can get an exact numerical value. However, let's not worry too much about semantic interpretations! Perhaps it would have been better if I'd replaced the words "is exactly" by " has the closed form solution".

I also liked the Stand-Up Maths video.

Alan

Re: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

Werner_E — Sat, 09 Mar 2024 19:39:39 GMT

@AlanStevens wrote:

I would say the circumference of a circle is exactly 2pi.r. Of course, that doesn't mean we can get an exact numerical value. However, let's not worry too much about semantic interpretations! Perhaps it would have been better if I'd replaced the words "is exactly" by " has the closed form solution".

I also liked the Stand-Up Maths video.

Alan

What I meant was that 2* pi *r just looks like a closed form solution but it actually isn't. It just looks that way because we decided to choose a simple looking greek letter as an abbreviation for he involved infinite sum. That way I guess we could also derive "closed form" solutions for the perimeter of an ellipse with any other ratio a:b as well.

Re: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

AlanStevens — Sun, 10 Mar 2024 08:43:07 GMT

Val Ochkov has asked me to post this picture:

Val says he has just been banned from the forum! I hope that's just a temporary glitch as he's been a great supporter of Mathcad and the forum for years.

Alan

Betreff: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

AlfredFlaßhaar — Sun, 10 Mar 2024 09:42:26 GMT

A friendly hello to the puzzle group,
I only have MC14 available. Therefore I couldn't read all the posts. Therefore, here is just a reference to a very good approximation function for the circumference of the ellipse - it is Ramanujan's formula. I don't know of a better one from the literature. I have attached them, but I cannot provide the desired 3D plot. I'm asking for help here.

Betreff: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

terryhendicott — Sun, 10 Mar 2024 11:53:11 GMT

Hi,

Your function uu() has two parameters so producing a 3D plot is simple with the matrix() function.

Cheers

Terry

Betreff: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

AlfredFlaßhaar — Sun, 10 Mar 2024 12:15:46 GMT

I use MC14 and therefore cannot load your file. Would you please reply in the appropriate file format? I haven't created a 3D plot yet. They were mostly vicious equations that I solved using equivalent minimum problems.

Betreff: Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

terryhendicott — Sun, 10 Mar 2024 12:19:47 GMT

Mathcad 12 version you should be able to open in Mathcad 14