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!

16-Pearl

Question

Mathcad Community Challenge March 2024 - Perimeter of an Ellipse

Forum|Forum|2 years ago
March 1, 2024
13 replies
19059 views

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!

T

terryhendicott

21-Topaz II

Hi,

File enclosed Prime Express 9.0

Cheers

Terry

1_20240302-Elliptical-Perimeter.zip

V

ValeryOchkov

24-Ruby IV

@terryhendicott wrote:

Hi,

File enclosed Prime Express 9.0

Cheers

Terry

1_Ellipse.zip

V

ValeryOchkov

24-Ruby IV

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)!

PS

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

Leo Tolstoy. "Anna Karenina"

V

ValeryOchkov

24-Ruby IV

1_Ellipse-Perimeter.zip

V

ValeryOchkov

24-Ruby IV

More interest plot

зз.png

A

AlanStevens

19-Tanzanite

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

1_March2024Challenge_AS.zip

2_March2024Challenge_AS.pdf

D

DJF

16-Pearl

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.

1_Ellipse-Perimeter-DJF1.pdf

2_Ellipse-Perimeter-DJF1.zip

T

terryhendicott

21-Topaz II

Hi,

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

1_240305-EllipseFirstPrinciples.zip

T

terryhendicott

21-Topaz II

Hi,

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

.

Cheers

Terry

1_240306-Elliptical-Perimeter.zip

B

BertBeirinckx

1-Visitor

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

1_PerimeterEllipse_march2024.zip

A

AlanStevens

19-Tanzanite

@Bert

Alan

W

Werner_E

25-Diamond I

@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

A

AlanStevens

19-Tanzanite

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

W

Werner_E

25-Diamond I

I guess and hope that it is not a ban on the part of the forum operators, but on the part of the Russian rulers.
If this is true, the use of a VPN could probably help Val to overcome this hurdle.
If not, maybe the admins can say something about it -> @unknown , @AndrewK , @Jaime_Lee

A

AlanStevens

19-Tanzanite

@Werner_E wrote:

I guess and hope that it is not a ban on the part of the forum operators, but on the part of the Russian rulers.
If this is true, the use of a VPN could probably help Val to overcome this hurdle.
If not, maybe the admins can say something about it -> @unknown , @AndrewK , @Jaime_Lee

Seems to have been a forum ban (Val sent me the following image):

I assume this was an automated system response. Perhaps an actual PTC person could overturn it?

Alan

A

AlfredFlaßhaar

15-Moonstone

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.

1_Ellipsenumfang-Ramanujan.zip

T

terryhendicott

21-Topaz II

Hi,

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

Cheers

Terry

1_240310--3D-Plot.zip

A

AlfredFlaßhaar

15-Moonstone

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.