How we can differentiate between the shapes of a parabola and a hyperbola?

Question

How we can differentiate between the shapes of the conics hyperbolas and parabolas?

If you just want to know the general characteristics of the curves, then see, for instance, Wikipedia's "Conic Section" entry. Otherwise, perhaps you should explain what you mean by "differentiate between their shapes". — Blue, Dec 01 '19 at 07:49
Locally the geometry in the plane is too similar for any visual metric to work, but globally we know hyperbola branches are asymptotically affine (i.e. they approach straight lines) while this fails spectacularly for parabolas — Brevan Ellefsen, Dec 01 '19 at 07:51
See https://math.stackexchange.com/questions/2353370/how-to-tell-the-difference-between-a-parabola-and-a-hyperbola-by-looking, where this has been asked already. — Cye Waldman, Dec 02 '19 at 21:31
If there's a second piece, it's a hyperbola. If there's only one piece, and it doesn't close up, it's a parabola. If there's only one piece, and it does close up, it's an ellipse. You have to look at the big picture. — Gerry Myerson, Dec 04 '19 at 23:00

Intelligenti pauca · Accepted Answer · 2019-12-10T22:57:44.447

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Draw two parallel chords of your conic, and let $a$ be the line through the midpoints of the chords. Then draw another pair of parallel chord (not parallel to the previous pair) and let $b$ be the line through their midpoints. If lines $a$ and $b$ are parallel, then the conic is a parabola, otherwise it is a hyperbola (and the lines meet at the center of the hyperbola).

An example can be seen in figure below: on the left a parabola and on the right a branch of hyperbola.

edited Dec 10 '19 at 22:57

answered Dec 01 '19 at 17:54

Intelligenti pauca

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+1 Nice. If you do this with an ellipse you find the center. So (speaking projectively) the center of the parabola is "at infinity". – Ethan Bolker Dec 04 '19 at 22:04
The figure shown in your answer is a parabola or a hyperbola? – DSP_CS Dec 09 '19 at 16:59
As per wording of your answer,your figure is hyperbola – DSP_CS Dec 09 '19 at 17:05
@Man The one on the left is a parabola, on the right a (branch of) hyperbola. – Intelligenti pauca Dec 09 '19 at 18:43
@Aretino please kindly include your this above comment in your answer – DSP_CS Dec 10 '19 at 17:07

score -3 · Answer 2 · answered Dec 01 '19 at 08:50

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A parabola is the intersection of a plane parallel to the surface of a cone (green) and a hyperbola the plane is parallel to the central axis of the cone (red).

answered Dec 01 '19 at 08:50

David G. Stork

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The hyperbola's plane doesn't have to be parallel to the cone axis, it just has to be "steeper" than the parabola's plane ... that is, it must meet both parts of a double-napped cone (only one of which is indicated in your figure), thereby creating the two branches of the conic. – Blue Dec 01 '19 at 12:06

How we can differentiate between the shapes of a parabola and a hyperbola?

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