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I am studying the (somewhat old-fashioned) "Analytical Conics" by D.M.Y. Sommerville (3rd edition, 1933). (It's fascinating, but the order of presentation of material makes it come across as muddled sometimes).

I have found this definition in Chapter V: The Hyperbola, section 2.

"... chords through the center are diameters."

Okay, straightforward enough: he does not actually define "chord", but by extrapolating from its definition for the circle I assume he means a line segment joining 2 points, whether they be on the same branch or not. Hence this definition is simple and straightforward.

However, when I seek to confirm this online, I get the following mumbo-jumbo, usually accompanied by a diagram in which the line so constructed is anything but the locus of midpoints:

"The locus of the mid point of a system of parallel chords of a hyperbola is called a diameter of the hyperbola and the point where the diameter intersects the hyperbola is called the vertex of the diameter."

Now I have run it up on GeoGebra, constructed a system of parallel chords, constructed a line through its midpoints, and indeed, it passes neatly through the centre:

enter image description here

... and indeed, it is definitely the case that the locus of the midpoints do in fact form a diameter.

The question I have is: if it is indeed blindingly simple to define a diameter as (the infinite production in both directions of) a chord which passes through the centre, why use the complicated definition?

And, having used this definition, why accompany it with a horribly misleading diagram? Examples of such are ubiquitous:

enter image description here

Is there a subtlety here by which it is not always the case that "a chord through the centre (produced appropriately) is the locus of a system of midpoints of parallel chords"?

In due course I will work out the algebra to prove the above, and satisfy myself that it is indeed true (I can't find it in Sommerville -- but then its presentational style is dense and I haven't studied the chapter on hyperbolas line by line yet). I will no doubt also find it is true for all conic sections, as I believe it is a projective property: it's obviously true for circles, and I expect there is a form for the parabola that allows it to make sense for it there as well.

Now I understand, from the language in which many of such pages are couched, that this is something which is often pitched at elementary-school level or perhaps high-school level. Here, for example, is a particularly patronising example:

https://www.toppr.com/ask/content/story/amp/diameter-of-hyperbola-52033/

... so it is obviously something which is considered pretty basic, mathematically speaking (although I can't remember anything about this in my own long-ago formal studies of geometry via the Kleinian approach).

Is anyone able to shed any light on this from the point of view of a coherent approach? That is, the equivalence of the definitions of a diameter being straightforward to demonstrate and (presumably) fairly straightforward to prove (although I haven't done this formally myself to my own satisfaction), why is it (in general) not even mentioned in the literature?

EDIT: I have checked for a parabola, and it appears (as you'd expect by a projective argument) that the "diameters" of a parabola (as defined by the locus of the midpoints of parallel chords) are in fact parallel to the axis. This is consistent with the "centre" of a parabola being the point at infinity of the axis of the parabola.

Prime Mover
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    To me, a chord is a segment joining two pts, and a diameter is a chord through the center. Defining (the line of) a diameter as the locus of mdpts of parallel chords is weird to me, but consider ... As seen in the "misleading" diagrams, a particular diameter-line isn't the locus of mdpts of just any family of parallel chords; rather, there's a specific family, parallel to the given diameter's so-called "conjugate diameter" (which in turn has the given diameter as its conjugate). The author may be paving the way for this discussion. – Blue Apr 09 '21 at 07:44
  • @Blue Yes, that second diagram of the pair does indeed come from a discussion about the conjugate. But the point is, yes indeed: a particular diameter line is the midpoint of a specific family of parallel chords, parallel to the conjugate diameter, but the diagram does not show this. It just shows a random parallel chord system which has no relation to either diameter or conjugate. And the same applies to every single illustration of the concept that I can find online. This puzzles me. There has to be a reason why, unless all such websites are missing the mark. They can't all be. – Prime Mover Apr 09 '21 at 07:54
  • "... unless all such websites are missing the mark. They can't all be." Not all are. Wikipedia's "Diameter" entry notes "a diameter of a conic section is typically defined as any chord which passes through the conic's centre". Cut-the-Knot's "Conjugate Diameters" says similar (albeit about ellipses). ... Anyway, using the mdpt property as the definition puzzles me, too; then again, I prefer to introduce trig functions via triangles instead of the unit circle, so... shrug :) – Blue Apr 09 '21 at 08:19
  • BTW: That toppr lesson is awfully patronizing! :) For what it's worth, a misleading toppr solution was referenced in this recent question, and a commenter suggested that toppr content may not be exactly authoritative. Whether the commenter's opinion is authoritative, I can't say; yours is my second exposure to the site. – Blue Apr 09 '21 at 08:31
  • @Blue "All such websites" applies to all the sites I've seen presenting a construction of a diameter based on the midpoints of chords. I have not found one with a diagram that even pretends to be accurate. Yes, I see a few sites defining a diameter as a chord thru the centre but universally this is in relation to circle and possibly ellipse. Yes, even Wikipedia. The latter has a page on "conjugate diameters" but then does not actually define the diameter of a conic section. – Prime Mover Apr 09 '21 at 09:02
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    A non-mathematical comment: congratulations for your writing style ; though English isn't my mother language, I appreciate very much the nuanced language you use. – Jean Marie Apr 09 '21 at 09:20
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    @JeanMarie Thank you for the compliment. I (usually) take great care to construct my communications so as to be both accurate and understandable. I am gratified that my attention to such detail is appreciated. – Prime Mover Apr 09 '21 at 09:25
  • For those who want to "leaf through" Sommerville's book: https://archive.org/details/dli.ernet.247333/mode/2up – Jean Marie Apr 09 '21 at 09:59
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    @JeanMarie Interesting comparing it to the 3rd edition, which appears to be a hasty revision in which extra material was squeezed in as an afterthought. But what's confusing about this online edition is the fact that it has the odd pages on the left and the even pages on the right! :-) – Prime Mover Apr 09 '21 at 10:05

1 Answers1

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I've spent some time over the last few days working on this.

There is good reason to define a diameter as the locus of the midpoints of parallel chords.

This is because the definition as "chord which passes through the center" is inadequate as a definition for the following reasons:

a) It does not encompass the case of the parabola, where it is straightforward to demonstrate that the locus of the midpoints of parallel chords is a straight line parallel to the axis of the parabola. Thus, and because there is no center of a parabola (unless you go down the rocky road of defining the center as the point at infinity, and there is no real need to do that here) it is meaningless (or at best cumbersome and complicated) to define a diameter in relation to such a point.

b) In the case of the hyperbola, it does not take account of the diameters which are the loci of parallel chords either ends of which are on opposite branches of the hyperbola.

Hyperbola in blue, chords black, diameter red

Hence the "chord through center" definition misses these important diameters which do not intersect the hyperbola at all.

So the question is resolved. The reason for defining a diameter as the locus of the midpoints of a system of parallel chords is because it is universally applicable.

The fact that a diameter passes through the center of the (central) conic section is a direct consequence of the above definition.

It would be so nice, though, if more texts (and websites) took the trouble to explain the above.

It would also be nice if the websites would construct adequate and accurate diagrams illustrating the concept without forcing the readers to scratch their heads and say: "But those are not the midpoints ..."

Prime Mover
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  • +1. Nice resolution to this. ... I apologize for not giving enough thought or credit to "exterior" diameters. (I agree that sources should do a better job of explaining the situation.) The locus-of-midpoints definition does neatly grant them peer status with "interior" diameters, but now I'm wondering: What geometric results require nuanced formulation for exterior diameters? That is, apart from whether or not we can consider a diameter a(n extended) chord, when might the interior-vs-exterior nature of a diameter matter? Hmmm ... – Blue Apr 11 '21 at 13:50
  • @Blue Sommerville (my source work) goes on about imaginary elements of conics, i.e. such that fulfil the equations of the various bits and pieces only with imaginary or complex numbers. But during the course of discussion of the hyperbola, he talks about the asymptotes partitioning the diameters into those which intersect the hyperbola and those which do not. I get it, but it's an ill-designed book structure. In fact, it's such a muddled and confusing text I'm considering casting it aside and seeing what else I've got on my shelf which may cover the same ground. – Prime Mover Apr 11 '21 at 19:25