What is the name for the distance between two foci within an ellipse?

[Siri137]

The straight line between points F' and F.

I can't find the answer to this question anywhere, and yes, this is important because it represents a gap in my knowledge, and that's just not acceptable for a being that likes to think of herself as being omniscient. ALL the other characteristics have designations (ex. minor axis, semi-major axis). Within the eccentricity equation it is designated by the letter c. e = c/2a, with a the semi-major axis, but of course that's not a name, just a letter (sorry Elon, but X is just a letter. You should have kept the name 'Twitter').

Updates

12 d

Since it doesn't appear to have a name, I'm christening it now. I'm calling it the 'Bifocal Displacement'. Catchy name? Good. That's what it is. :)

[FatNeville]

This probably isn't the answer you are looking for, but this is the most accurate answer I can come up. The name is "the distance between the two foci". Like you just mentioned in the update, there isn't a universally-accepted name that is in common use. Any term used to refer to this will require it to be defined in order to be understood unambiguously by a general audience. The name you came up with is just as good as any I would be able to think up. I just wouldn't advise using the term in a proof or academic paper without also giving the definition if you aren't sure that all the readers know exactly what you mean.

[Siri137]

Don't worry, I'm not going to use the term in a test or proof or anything like that. I was just baffled I couldn't find a name for it, and it appears there isn't one.

[FatNeville]

It is a bit strange, considering that mathematics is filled with terms for the most random and arbitrary things that seem to come up from time to time. But I suppose in this case that particular distance hasn't been deemed commonly enough used to have a specialized jargon. Until today, of course.

[slatyb]

Let f be half the distance between the foci, so one is at (-f,0) and the other is at (0, f)

Let a be half the length of the semi-minor axis, and b be half the length of the semi-major axis.

Then consider the path from one focus to one end of the minor axis then to the other focus.
That path has length sqrt (f^2 + a^2).

Consider the path from one focus to one end of the major axis then to the other focus.
That path has length 2b. So we have f^2 + a^2 = b^2, or f = sqrt (b^2 - a^2)

[Siri137]

Err... I think you've misunderstood my question.

[slatyb]

Err... I think not. The distance between the foci is 2f, and f = sqrt (b^2 - a^2). I'm surprised you couldn't find this online. I didn't search for it, the geometry is simple enough.

[slatyb]

Notice that when a = b, the distance is zero.

[Siri137]

Yes, you did misunderstand the question. I want the NAME of the distance between the two foci, NOT the formula!

[slatyb]

🤦 So you did.

[SoJust]

It is called "major axis length of the ellipse" or simply "major axis".

[slatyb]

The ask was for the distance between the foci.

[FatNeville]

@slatyb Technically, it could be interpreted as "the length of the major axis" because the distance itself is a quantity, and these two are equal, but the connotation is different. The question seems to ask for a term that refers specifically to the distance in the context of the foci, though, rather than terms for the same quantity that is generally referring to something else.

[KostasKouvalis]

I got "axis major" as well

[Siri137]

@KostasKouvalis No, the major axis is the line between the two major vertices.

[KostasKouvalis]

Then it doesn't look like there's a special name for it, which is odd

[Siri137]

@KostasKouvalis Oh, there is now. I've given it one :)

[KostasKouvalis]

What did you name it?

[Siri137]

@KostasKouvalis See the update. I've called it the 'Bifocal Displacement'.

[KostasKouvalis]

You have to patent that though or someone will steal it. I'm having a hard time believing no mathematician has come up with an official term for that

[Siri137]

@KostasKouvalis I know. That's really, really surprising. They have names for so much else, so why not this?

[Siri137]

@KostasKouvalis I think you mean copyright. Patents apply to inventions.

[KostasKouvalis]

Yeah, you invented a new word😀

[Siri137]

@KostasKouvalis Okay, so how would I make it all mine? What's the procedure?

[KostasKouvalis]

You need to talk to a patent lawyer. Or a copyright lawyer

[RebelinSteel]

I think they are the vertices or verticals

[RebelinSteel]

Or vertexes

[Siri137]

Nope. Not vertices.

[NathanDavis]

it's called the fotaintci...

[Siri137]

Source? Link? I've never come across that name before.

[bowenw]

Is it focal length?

[Siri137]

Nope. That's in optics (lenses).

[bowenw]

Check this site that talks about the focal length of an ellipse:

www.khanacademy.org/.../ellipse-foci-review

[Siri137]

For some reason it won't load. I'll try again later.

[bowenw]

I will DM you a screen shot.

[Siri137]

Okay. Thanks :)

[FatNeville]

I have heard "focal distance" used to refer to the distance between either focus and a point on the circumference of the ellipse (example: www.vedantu.com/maths/focal-distance-of-ellipse ), or, alternatively, to the distance between a focus and the center (as it is used in the Khan Academy overview mentioned in the earlier reply). I haven't seen it used to refer to the distance between the foci.

[Siri137]

@FatNeville NOT the distance from the foci to the centre. The WHOLE distance between the two foci.

[FatNeville]

I have found one source that uses the term to refer to the distance between the two foci, but only because I specifically looked for one after reading this question: testbook.com/maths/focus-of-ellipse-formula