Parabola and Hyperbola Question

[DaveC426913]

Correct me if I'm wrong:
A parabola extends without limit toward parallel lines.
A hyperbola extends without limit toward diverging lines.

They have very different equations.

My question: is the former a specific instance of the latter?
Does a parabola = a hyperbola that happens to have parallel axes?

Some guy pointed me at this dumb video:


At 4:30 - 5:03 he says that is a parabola. It is not, since he literally shows the asymptotes at right angles.

 

[fresh_42]

Correct me if I'm wrong:
A parabola extends without limit toward parallel lines.

No. $y=x^2$ goes to $-\infty$ on the left and $+\infty$ on the right. But the tangents converge towards parallel lines - without ever reaching this status.

A hyperbola extends without limit toward diverging lines.

Not sure what this means. $y=1/x$ converges to $y=0$ on both branches, and also to $x=0$ for the other ends of the branches.

They have very different equations.

This is definitely wrong. Parabolas, hyperbolas, and ellipses are all sections of a cone. They only differ by the angle to which the cut by a plane is made. https://en.wikipedia.org/wiki/Conic_section

My question: is the former a specific instance of the latter?

They are all a specific case of a conic section.

Does a parabola = a hyperbola that happens to have parallel axes?

I don't know what you mean by axis (symmetry axis or coordinate axis) and what the equality sign should mean.

 

[DaveC426913]

No. $y=x^2$ goes to $-\infty$ on the left and $+\infty$ on the right. But the tangents converge towards parallel lines - without ever reaching this status.

Um. Is that not what I said?
Perhaps if I had specified 'asymptotically approaching parallel lines'?

This is definitely wrong. Parabolas, hyperbolas, and ellipses are all sections of a cone. They only differ by the angle to which the cut by a plane is made. https://en.wikipedia.org/wiki/Conic_section

I know that, but I asked about the equations.

Parabola: y^2=ax
Hyperbola: (y − k)^2 = a(x − h)
(I just quickly Googled these, rather than delving into my HS math).

I presume if you start with the equation of a hyperbola and set some value to zero (i.e. the conic slice is parallel to the slope), stuff will cancel out, and the equation for a parabola will be left. This is my question.

I don't know what you mean by axis (symmetry axis or coordinate axis) and what the equality sign should mean.

Sorry, I should have said asymptotes.

I'm asking if a parabola is a subset of the set of all hyperbolae.

 

[DaveC426913]

Of course. Now that I think of it, it must be so. A hyperbola would be formed by a cross-section through a cone that has a slope less than the slope of the cone. (Otherwise, you will get an ellipse.) So a parabola is simply a hyperbola where the slope of the cross-section exactly equals the slope of the cone.

(I realize I'm not using the correct terms. I'm intuiting.)

 

[fresh_42]

Here is the image from Wikipedia:

I used the German version, the picture looked nicer. Conic sections have axis, asymptotes and there is still the coordinate system, so quite a few "lines".

 

[DaveC426913]

Yes, I know what conic sections are.

(Okay, I said axes when I meant asymptotes. Mia Culpa. In the video, they start off as vertical and horizontal axes, before he rotates them and turns them into asymptotes with some trickery.)

It does make it easier to see how a parabola (where the cross-sectional plane is parallel to the slope) is simply one specific hyperbola.

 

[fresh_42]

I am not sure why you're showing these. I know what conic sections are.

Mostly because others might see it since we frequently have questions about them. And it contains all necessary information: one can imagine the coordinate system, or how complicate the equations will be if the coordinate system is anywhere in the image, one has all symmetries and axis, and even the asymptotes can be seen. I find that it tells more than descriptions could tell, and the more people have this picture in mind, the better.

 

[Mark44]

No. $y=x^2$ goes to $-\infty$ on the left and $+\infty$ on the right. But the tangents converge towards parallel line

I disagree. y goes to $\infty$ as $|x|$ goes to $\infty$. You might have been thinking about something else when you wrote this.

The tangent lines for the parabola become closer and closer to vertical lines as |x| gets large.

 

[Mark44]

A hyperbola extends without limit toward diverging lines.

A better way to say this, I believe, is that the graph of a hyperbola approaches two intersecting lines. Obviously, because the two lines intersect, they can't be parallel.

My question: is the former a specific instance of the latter?
Does a parabola = a hyperbola that happens to have parallel axes?

No and no.

 

[fresh_42]

I disagree. y goes to $\infty$ as $|x|$ goes to $\infty$. You might have been thinking about something else when you wrote this.

The tangent lines for the parabola become closer and closer to vertical lines as |x| gets large.

Right, I had left and right in mind, not up and down. The tangents approach vertical lines, but these drift apart.

 

[robphy]

At 4:30 - 5:03 he says that is a parabola. It is not, since he literally shows the asymptotes at right angles.

https://www.wikihow.com/Draw-a-Parabolic-Curve-(a-Curve-with-Straight-Lines)https://plus.maths.org/content/finding-intersection-envelope

 

[pbuk]

... I said axes when I meant asymptotes.

But a parabola does not have asymptotes. Yes the tangents to $y=x^2$ become more nearly vertical (and therefore more nearly parallel) as $x \to \pm \infty$, but in order for these tangents to be asymptotes they would have to be fixed i.e. $x = \pm k$ for some finite $k$. Clearly this is not the case.

It does make it easier to see how a parabola (where the cross-sectional plane is parallel to the slope) is simply one specific hyperbola.

This is not true because all hyperbolas are asymptotic; no parabola is.

Edit: corrected final sentence.

 

[pbuk]

A hyperbola would be formed by a cross-section through a cone that has a slope less than the slope of the cone. (Otherwise, you will get an ellipse.) So a parabola is simply a hyperbola where the slope of the cross-section exactly equals the slope of the cone.

If you want to take a more intuitive approach, the flaw in this argument can be demonstrated by restating it as follows:

An ellipse would be formed by a cross-section through a cone that has a slope greater than the slope of the cone (otherwise, you will get an hyperbola). So a parabola is simply an ellipse where the slope of the cross-section exactly equals the slope of the cone.

This is equally (but perhaps more obviously) incorrect - a parabola is no more a special kind of hyperbola than it is a special kind of ellipse.

 

[DaveC426913]

This is equally (but perhaps more obviously) incorrect - a parabola is no more a special kind of hyperbola than it is a special kind of ellipse.

Actually, it is. A parabola is effectively an ellipse with one focus at infinity.

And you can intuit this easily enough by decreasing the slop of the conic section closer and closer to parallel. The ellipse will get longer without limit until, when the slops becomes parallel, the ellispe will effectively be infinitely long.

 

[DaveC426913]

https://www.wikihow.com/Draw-a-Parabolic-Curve-(a-Curve-with-Straight-Lines)
View attachment 243344

Are you asserting that this diagram is a parabola?

 

[DaveC426913]

Does a parabola = a hyperbola that happens to have parallel axes?

No...

A hyperbola whose cross-section is within 0.0000000000001 degrees of parallel to the slope of the cone is still a hyperbola, but change that angle by 0.0000000000001 degrees and it is a completely different animal?That's like saying a circle (eccentricity = 0) is not a special case of an ellipse. No?

 

[pbuk]

That's like saying a circle (eccentricity = 0) is not a special case of an ellipse. No?

No it certainly isn't. A circle is a special case of an ellipse in the same sense that a right hyperbola is a special case of hyperbola. We even use the same term to describe both - we say the eccentricity is 0.

 

[DaveC426913]

OK, I guess I've digressed from my primary assertion. I wanted to assure myself that a parabola is a special case of hyperbola. I'll set that aside.

But the argument I'm making to this guy is that this:

is not a parabola. No matter how many lines you add to this, the limbs will never approach parallel.

It's a hyperbola, and the limbs are approaching right angles.

 

[pbuk]

It's a hyperbola (whose asymptotes are at right angles - definitely not parallel.)

Yes this is correct; this sketch approximates an hyperbola with eccentricity $\sqrt 2$ (i.e. a right hyperbola).

Edit: corrected value of eccentricity.

 

[DaveC426913]

Yes this is correct; this sketch approximates an hyperbola with eccentricity 0 (i.e. a right hyperbola).

Wait a minute. The eccentricity of a hyperbola must be greater than 1.
An eccentricity of 1 is a parabola.
(An eccentricity of less than one is an ellipse.
An eccentricity of 0 is a circle.)

https://en.wikipedia.org/wiki/Eccentricity_(mathematics)

 

[pbuk]

Yes indeed I am; an hyperbola with an eccentricity of $\infty$ is a line.

 

[pbuk]

(Although you can have a hyperbola with an eccentricity of ∞-1)

?

 

[DaveC426913]

Yes indeed I am; an hyperbola with an eccentricity of $\infty$ is a line.

Ah. I see. Because the cone has an angle of zero.

OK, that leads me to an obvious question then. You can slice a cone anywhere along its side, from the apex toward the base without limit. There must be an offset variable that defines how far from the apex the cut is.

 

[pbuk]

You can slice a cone anywhere along its side, from the apex toward the base without limit.

The base is the limit; once you reach that limit the conic section is said to be degenerate, but that is another story...

You can slice a cone anywhere along its side, from the apex toward the base without limit. There must be an offset variable that defines how far from the apex the cut is.

That distance does not change the eccentricity of the hyperbola, it just makes it "bigger" in the sense that the major and minor axes are multiplied by the same factor; you can visualise creating a family of ellipses by moving the plane of intersection in a similar way.

 

[DaveC426913]

Here are two cross-sections through a cone:
Blue is parallel to the cone's slope; red is at an infinitesimally small angle.

The red cross-section forms a hyperbola, but the blue does not? It forms a parabola instead? They are two different animals?

 

[pbuk]

The red cross-section forms a hyperbola, but the blue does not? It forms a parabola instead? They are two different animals?

The red section forms an ellipse!

 

[DaveC426913]

The red section forms an ellipse!

You were too quick! I saw the error right away and corrected the pic.

Here it is again:

The blue line forms a parabola. The red line forms a hyperbola.
The angle between them could be infinitesimally small.

I would have thought that a hyperbola with E=∞ would be the same animal as a hyperbola with E=∞-1. Just like a circle with E=0 is the same animal as an ellipse with E=0.0000000000000000001.

But OK., I'll let that go.

 

[kith]

No parabola is simultaneously a hyperbola simply because it doesn't fit the definition. But parabolas can be seen as limiting cases of both ellipses and hyperbolas. Examples:
https://math.stackexchange.com/ques...ola-which-is-similar-to-a-branch-of-hyperbolahttps://math.stackexchange.com/ques...othly-between-cartesian-ellipse-and-hyperbola

 

[Mark44]

I would have thought that a hyperbola with E=∞ would be the same animal as a hyperbola with E=∞-1.

∞-1 is not defined. You can't use infinity in arithmetic computations.

 

[DaveC426913]

You can't use infinity in arithmetic computations.

Nor am I doing so. I am conceptualizing an arbitrarily large number.

You must grant that a hyperbola can have an eccentricity of an arbitrarily large value - there is no value on the real number line greater than 1 that a hyperbola cannot have as its eccentricity. Literally, anything shy of infinity.

 

[Mark44]

Nor am I doing so. I am conceptualizing an arbitrarily large number.

Actually, you did use ∞ in an arithmetic expression. You wrote E=∞-1 in post #27, to which I replied that ∞ - 1 is meaningless, as is any arithmetic expression involving the symbol ∞.

Having said that, some expressions using this symbol are meaningful, such as $\infty + \infty$ or $\infty \times \infty$, which should really be thought of as the sum or product, respectively, of functions whose limits are $\infty$.

Other arithmetic expressions, such as $\infty - \infty$, $\infty - n$, where n is any finite number, and $\frac \infty \infty$ are not defined, and are therefore meaningless.

You must grant that a hyperbola can have an eccentricity of an arbitrarily large value - there is no value on the real number line greater than 1 that a hyperbola cannot have as its eccentricity. Literally, anything shy of infinity.

But all you need to say is that the eccentricity is greater than 1.

 

[Charles Link]

The hyperbola of $xy=1$ was introduced above. I don't think this was previously mentioned, and it may be worth mentioning, that it will take the standard form of a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ if you introduce a rotation of axes by $\theta=45^{\circ}$ using $\\$ $x'=x \cos{\theta}+y \sin{\theta}$ $\\$ and $\\$ $y'=-x \sin{\theta}+y \cos{\theta}$. $\\$ You then solve for $x$ and $y$ and substitute into $xy=1$. $\\$ The result is $\frac{x'^2}{2}-\frac{y'^2}{2}=1$.