Parabolas and Infinity: Why Do Parabolas Close at Infinity?

[Math Is Hard]

Why does a parabola close at infinity?

A science teacher made this statement and I didn't quite get it.

Thanks!

 

[cookiemonster]

Parabolas blow up to infinity (or negative infinity, depending on the sign of the x^2 coefficient) when x = infinity.

cookiemonster

 

[Math Is Hard]

well certainly - but do they close?? I am thinking "close" like an ellipse.
thanks, cm.

 

[matt grime]

The idea he's getting at is that they 'meet' at infinity: there is a model for the plane that adds a 'point at infinity', imagine the plane in space, and a ball such that the plane cuts through the equator. There is a projection map from the north pole to the plane obtained by taking lines from the north pole - these pass through exactly one point in the plane and one point in the sphere. so you can identify the plane with the surface of the sphere minus the north pole. we call teh north pole the point at infinity, as it has the property that every curve in the plane that 'goes to infinity' tends towards the north pole on the sphere.

 

[HallsofIvy]

FIRST- "infinity" is not a single, well-defined point. Standard geometry does not include a "point at infinity" but there are several models that do. How you think about infinity depends upon which model you use- matt grime gave the most common 2-dimensional model.

What your teacher was thinking about is probably this:

Start with an ellipse with one focus at (0,0), the other on the y-axis (at (0,y), say). Imagine "stretching" the ellipse so that, while the first focus remains at (0,0), the second focus moves along the y-axis with y getting larger and larger. You will find that the eccentricity of the ellipse increases- getting closer and closer to 1.

If you imagine stretching that ellipse out "to infinity"- so that the second focus "goes to infinity", then the eccentricity goes to 1 (technically, "in the limit") so that the ellipse "becomes" a parabola. That is the sense in which a parabola "closes at infinity"- a parabola is like an infinitely long ellipse.

Of course, if stretch "to infinity and beyond" (!) the eccentricity becomes larger than 1- a hyperbola. The focus that went out to infinity along the positive y-axis now reappears on the negative y axis! That happens in precisely the same sense that the hyperbola y= 1/x has y going to infinity as x approaches 0 (from above) and then reappears with negative y on the other side of 0.

In order to make that precise, you have to use the "spherical" model for the plane that matt grime talked about.

 

[Hurkyl]

Polar coordinates gives another interpretation of that statement; the further along the parabola you go, the closer theta gets to vertical.

 

[Math Is Hard]

What your teacher was thinking about is probably this:

Start with an ellipse with one focus at (0,0), the other on the y-axis (at (0,y), say). Imagine "stretching" the ellipse so that, while the first focus remains at (0,0), the second focus moves along the y-axis with y getting larger and larger. You will find that the eccentricity of the ellipse increases- getting closer and closer to 1.

I believe that's exactly what he was saying, because we were talking about unbound orbits being hyperbolic rather than parabolic. But this was quite shocking to me because I had never heard a math teacher state that "parabolas close at infinity."

I can see a "parabolic orbit" existing only as transition phase between the bound elliptical orbit and the unbound hyperbolic orbit, but it was confusing to try to work with his statement outside of that context.

Thanks for the responses all. This helps.

 

[matt grime]

you must have heard of people saying parallel lines meet at infinity? it's the same thing.

 

[Math Is Hard]

you must have heard of people saying parallel lines meet at infinity? it's the same thing.

Isn't that dependent on the geometry? For parallel lines on a flat surface that shouldn't be true. On a curved surface - sure - I'll buy that.

 

[matt grime]

infinity isn't part of the plane. it's an additional point to the plane, so the first objection doesn't hold. note that the plane isn't some phyiscal flat object, just a set of ordered pairs of real numbers, and the 'lines' are defined algebraically really.

 

[fourier jr]

Yeah imagine you're God (or something) and you "zoom out" enough to see the real line in its entirety. Then the whole real line would look like a circle with a single point missing, the "point at infinity". So there isn't a positive infinity and a negative infinity; there's just one infinity and you get there by going in the positive direction or negative direction. So yeah I guess parabolas close at infinity, but you've got to think about it a bit. A picture of a horocycle would be really helpful right about now...

there's a pic of a Poincare disc in the top right here:
http://mcs.open.ac.uk/tcl2/nonE/nonE.html
That disc is what "God" would see if he looked down on our whole universe, each curve that intersects at 90 degree angle in that disc is a "line" (so there can be many parallel lines that don't touch instead of just one btw). If a curve touches the outer circle ("the absolute" is the math lingo) in one & only one point instead of 2 as with a "line" that's a horocycle, which is like the real line and its single point at infinity. I wish I could find a better pic.
Here's some more info:
http://mathworld.wolfram.com/PoincareHyperbolicDisk.html

 

[Math Is Hard]

Thank you for both of those links, FJ!
I thought quite a bit about parallel lines intersecting at infinity yesterday, and here's how I have reconciled it. First, if we have two straight vertical lines situated next to each other and we take them and lean them just slightly in toward each other by the same degree, (like / \) they will intersect somewhere before infinity. The more they are leaned into toward each other, the quicker we'll get to their intersection. If we take the two lines and lean them away from each other (like \ / ) they will never intersect.
OK, so back to the inward leaning lines, if we start gradually "straightening them out" until they are getting closer and closer to becoming parallel lines then we'll see the intersection point get farther and farther away. Eventually they will become parallel lines. They will intersect at an infinite point because if they were tilted toward each other by even the tiniest degree they would intersect somewhere BEFORE infinity and if they were tilted away from each other by even the tiniest degree they would NEVER intersect.
Considering a parabola, I went back and looked at how the slope of each side of the curve behaves. (I was just considering a simple function of y=x^2 and it's derivative y=2x for example. I hope that wasn't a bad choice!)
As we go "up" the parabola on the right side the slope becomes a steeper and steeper positive slope. If we go up the left side we get a steeper and steeper negative slope.
The higher we go up, the closer we get to approaching parallel lines.
So at infinity we'll have parallel lines, and by virtue of the above logic, they will intersect.