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!
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"
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
Hi,
Am enjoying the posts so far so want to add a quick from 1st principles approach:
Hi,
Just could not resist it. A page of matrices and voila 3D plot in express.
Cheers
Terry
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
@AlanStevens wrote:
Alan
circle 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
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
Seems to have been a forum ban (Val sent me the following image):
@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 -> @admin , @AndrewK , @Jaime_Lee
I assume this was an automated system response. Perhaps an actual PTC person could overturn it?
Alan
@terryhendicott wrote:
Hi,
Your function uu() has two parameters so producing a 3D plot is simple with the matrix() function.
Cheers
Terry
I would strongly vote against using the matrix function to do the job! It may give a decent result here but you can't use it to provide intermediate values (non-integer arguments) or negative arguments.
The way to go sure is using the CreateMesh() function which is exactly made for this.
We could also plot the function uu directly but in this case it will fail because the function uu is written in a way so that it yields non-real values for negative input arguments. And because the standard interval for a 3D Quickplot is from -5 to 5, this makes the whole plot fail.
But if we force the argument values to be in the desired range, plotting uu directly works as well.
Nonetheless I sure would vote for using CreateMesh when creating 3D plots of functions in 2 variables.
BTW, here is a different approach allowing for negative input values, too (even they may not make much sense in the context of the ellipses here)
Can someone derive an equation for the contour lines ??
You should be able to open the xmcd file posted by Terry initially with your MC14!?
This isn't really a response to the challenge. The attached sheet poses a slightly different question.
I started out to answer the challenge; seemed easy to simply integrate along the curve and easier to do it in polar coordinates:
For the ellipse posed in the challenge:
Then I looked at the other responses. . . .
And went back and checked, did the same thing in cartesian coordinates:
So why won't polar work?
