Circle w/ circumference equals to that of an ellipse

[Francis Ocoma]

Hi. Unfortunately, it looks like my first ever PhysicsForums post isn't even about Physics. I'll think of a Physics-related question later. :)

Anyway, I know that each ellipse has an inscribed circle and a circumscribed circle, but I was wondering about a third circle associated with an ellipse. Specifically, I'm thinking of the circle whose circumference is equal to that of the given ellipse. Is there a name for this circle?

I swear I've been Googling this for hours now, but all I could find are pages showing how difficult it is to calculate the circumference of an ellipse. Thank you guys in advance!

 

[Noctisdark]

Circumference of an ellipse is C = 2*pi*sqrt((a^2 + b^2)/2), that of a circlr is 2*pi*a, assuming a > b , theythey have the same circumference means a= b (solve the eq), which implies that the original ellipse is actually a circle, made you mean something else, please try to.make a pic describing what you mean in that case !

 

[Francis Ocoma]

Circumference of an ellipse is C = 2*pi*sqrt((a^2 + b^2)/2), that of a circlr is 2*pi*a, assuming a > b , theythey have the same circumference means a= b (solve the eq), which implies that the original ellipse is actually a circle, made you mean something else, please try to.make a pic describing what you mean in that case !

Let's see if I can rephrase my question better. Suppose there is a ellipse with an eccentricity of 0.5 and circumference of 100. A circle with radius 50/pi would also have a circumference of 100. If you drew the ellipse's incircle and circumcircle, the circle with 50/pi radius would fit somewhere in between. Is there a name for that circle?

By the way, your math there is wrong. The circumference of the circle is not 2*pi*a (where a is the semi-major axis of the ellipse), but is rather 2*pi*whatever its radius is. You are trying to equate the semi-major axis of the ellipse to the circle's radius; if that were the case, you'd get the circumcircle of the ellipse, which isn't what I was looking for at all.

 

[onomatomanic]

I doubt it. Coinages are elevated to the status of "name" by being widely used, generally speaking, and to some extent that holds even in a field like this. The reason we have "incircle", and not just the descriptive compound "inscribed circle", is ultimately simply that they're frequently referred to because they're frequently useful. For constructions for which that is not the case, which I'm guessing includes what you're describing, you're typically going to have to create ad-hoc labels, which can be either sufficiently descriptive to be readily understood, or not, in which case you explain them the first time you use the term. After all, language is little more than a bunch of conventions, and where there isn't one, you're free to come up with your own. ;)

 

[Francis Ocoma]

I doubt it. Coinages are elevated to the status of "name" by being widely used, generally speaking, and to some extent that holds even in a field like this. The reason we have "incircle", and not just the descriptive compound "inscribed circle", is ultimately simply that they're frequently referred to because they're frequently useful. For constructions for which that is not the case, which I'm guessing includes what you're describing, you're typically going to have to create ad-hoc labels, which can be either sufficiently descriptive to be readily understood, or not, in which case you explain them the first time you use the term. After all, language is little more than a bunch of conventions, and where there isn't one, you're free to come up with your own. ;)

Ah, I was afraid that was the case. I'm thinking of calling it "the equicircumferenced circle of the ellipse" or the "equicircle of the ellipse". Would you care to suggest a better name?

 

[onomatomanic]

The (at least somewhat) established adverbial form is "circumferential". And while "equi-" isn't as specific as one might wish in general (an "equipotential surface" is one which is defined by equal potentials across it, rather than with reference to something else), it should serve well enough in this specific context.

Shortening that to "equicircle" may not be the best idea, though, because that term no longer signifies how the circle is "equal" to the ellipse - it could as easily refer to one having the same area, say. That's not as much the case with the two established terms, because the figurative "scribing" is largely implied by "in-" and "circum-" in and of themselves, which justifies the omission.

Does that help at all? :)

 

[Francis Ocoma]

Hmmm... I wonder if "perimeter circle" would be a better term. It could then be generalized beyond ellipses, e.g. "the perimeter circle of a square". The shortened form could be "pericircle".

Thanks for the input, by the way!

 

[onomatomanic]

"Circumference" is superior to "perimeter" in that it denotes or at least connotes the idea of length of a boundary, and not just the boundary as such.

Personally, I usually try to pick the option whose etymology best fits the desired meaning, all else being equal - mainly just for the heck of it, but also because it gives people who come across the term without the benefit of already being familiar with what it's supposed to mean the best chance of puzzling it out for themselves. That would argue against "peri-", which emphasizes the radial rather than the tangential (which is to say, length-wise, for these purposes) direction.

The principal, and not-to-be-underestimated, argument in favour or "pericircle" is that it has an undeniably nice ring to it... :P