Why isn't a V-shape a conic section?

Factual Questions
1

We’re doing conic sections in math and my teacher says there are only four types of conic sections: circles, ellipses, parabolas, and hyperbolas. Someone in the class mentally cut a cone right through the middle, vertically, leaving a cross section that was a V-shape (or an X-shape if you use a double-cone). He asked why that wasn’t considered a conic section. The teacher explained that even though it looks like a pair of straight lines, they’re really curved, because the sides of a cone are curved. I tried to argue that they were both curved and straight, but I didn’t know how to express that idea clearly. In any case, I’m pretty sure she was wrong, and that a V-shape is a cross section of a cone.

So why isn’t it a conic section?

2

You’re right, it is a conic section, but it’s a so-called degenerate conic section, along with a line and a point. It’s a hyperbola with the constant equal to 0, eg:

x[sup]2[/sup] - y[sup]2[/sup] = 0

Similarly, a point can be thought of as a circle with radius 0:

x[sup]2[/sup] + y[sup]2[/sup] = 0

So, while these things are conic sections, they’re just special cases of ellipses, parabolae, and hyperbolae.

3

Wouldn’t that just be an absolute value function?

4

It’s an absolute value function combined with its negative. The equation x[sup]2[/sup] - y[sup]2[/sup] = 0 is satisfied by all points on the lines y = x and y = -x.

5

Here’s a good site on the subject:

http://www.math2.org/math/algebra/conics.htm

6

So, it’s actually correct to say that there are four curved conic sections and two non-curved conic intersections (the ‘V’ [or, rather, the ‘X’ when you consider the attached inverted cone] and the point).

Peace.

I thought this was going to be about ice cream cones on a plane.

7

It depends on how you count them. There’s no real need to count a circle as separate from an ellipse.

8

Note that there are other degenerate cases:

Take the parabola case (parallel to the sides of the cones) and have it intersect the point where the cones meet: you have a line.

Twist it down some and you get a point.

Note that there are also intersections that don’t give true hyperbolas or parabolas by just tilting the cut a bit.

Just because you can generate certain figures by cuts doesn’t always mean that are the only ones possible.

9

I definitely agree with you on the point and the line, but care to explain this exception?

10

I think it’s the misconception that a true hyperbola is only generated when the intersecting plane is parallel to the axis of the cone. Of course, true hyperbolas are generated by any plane intersecting at any angle between parallel to the the axis of the cone and parallel to the side of the cone.

I’d bet this error arises because so many text books show the plane generating the hyperbola as parallel to the cone’s axis and students then think that this is the only orientation that produces a symmetric hyperbola. I know I thought that for some time.

11

Looking at ccwaterback’s site, isn’t example 3 (the parabola) actually just an extended ellipse? If you extend the cone out far enough, it should curve back on itself. It seems to me that a parabola really can only tilt away from the center of the cone – if it tilts at all toward the center, it will eventually reach the center, and curve back on itself – an ellipse.

12

I believe the “sides” of a parabola are forever expanding, by definition. So it’s not a U-shape or a U-shape that’s coming back together, more like a U-shape that’s beeing pulled apart.

13

Isn’t glass just a hardened form of sand? Isn’t cheese just congealed milk with added flavoring? Isn’t an adult just an older form of child?

Sorry for the sarcasm. (Well, not that sorry. It’s one of life’s great joys.) But obviously the reason these figures — ellipses and parabolas — are given different names is that their differences are considered significant enough to warrant it.

There’s a sense in which an ellipse can “evolve” into a parabola, if that’s how you wish to think of it. In astronomy they measure orbits by their eccentricity — a single value which fully specifies the shape of conic section. An orbit with eccentricity 0 is a perfect circle; values greater than 0 and less than 1 are ellipses; a value of exactly 1 is a parabola; and values greater than 1 are hyperbolae.

So if you imagine an “eccentricity” knob on a magic box that draws conic sections, then twisting the knob from 0.1 to 1 would show an ellipse that stretches longer and longer until finally, at 1, it instantly “pops open” into a parabola. In astronomical terms, e = 1 is the threshhold separating objects that are gravitationally bound from those that aren’t.