I have heard that certain conic sections are called “degenerate cases.” I have guessed that this means the point and the line would be the two degenerate conic sections, the plane passing through the point between the two cones that meet at their vertices, and making a point; and the plane tangent to the cone and thus just making a line. The question is 1) is this right? and
2) are there any other degenerate conic sections,
and 3) what else is called degenerate cases in mathematics? and while I’m thinking of conic sections, I don’t understand the parabola. I can visualize the plane going “straight across” the cone and making a circle; being a little tilted and making an ellipse; and being parallel to the central axis of the cone and so making the two parts (question 3 1/2 are these called cusps) of the hyperbola. The problem is if you take the plane and tilt it from the position it had to make the hyperbola, but at less than a certain angle of tilt so you do get the classic parabola, YOU WILL ALSO GET ANOTHER PARABOLA, BECAUSE THE PLANE GOES ON IN THE OTHER DIRECTION AWAY FROM THE CONE ON WHICH IT IS MAKING THE PARABOLA AND EVENTUALLY WILL CUT INTO THE OTHER CONE, IE. THE OTHER NAPE OF THE CONE, AND MAKE ANOTHER PARABOLA! So 4) what is a parabola then?
Signed All Mixed Up About Both Degeneracy and the Parabola!
A parabola is when the plane is parallel to one side of the cone, so you only get the one. (It will never intersect the other.)
tanstaafl is right, and that is why you only get one parabola. The equations have to be just right (=1) to make a parabola, whereas the ellipse is a short (<1) and the hyperbola is long (>1). You’ve heard of ellipsis and hyperbole? Same thing.
Another degenerate conic is a pair of crossed lines–which you get if the plane goes through the point and the cones as well.
As to other mathematical degeneracies, does Kari Mullis count?
If the plane intersects both halves of the cone, the curves are hyperbolas, whether the plane is parallel to the axis or not.
In general, a degeneracy is where two different answers give you the same thing. For instance, every quadratic equation has two roots: If my equation is x[sup]2[/sup] = 1, for instance, then my roots are x = 1 and x = -1. But what if I have the equation x[sup]2[/sup] = 0 ? Then my two roots are x = 0 and x = 0, a degenerate root.
Strictly speaking, I’m not sure how the “degenerate conic sections” match this definition, but the term “degenerate” is often also used in a looser sense of “uninteresting”, and things like a single point or a pair of intersecting lines are certainly rather uninteresting compared to the more usual conic sections.
A slightly tighter definition of a degenerate case is one where some of the complexity is lost (which usually does make it less interesting).
For instance, think about a quadratic equation,
ax^2 + bx + c = 0 where a, b and c are constants.
Now if you decide to set a = 0, you lose a lot of complexity - in fact end up with a linear equation – so this is a degenerate case. You can say that a linear equation is a degenerate case of the quadratic equation.
[Or, for another case, set b=-2a=-2c. Now you get a(x^2-2x+1)=0 or a(x-1)^2=0. This is less complex than the original equation, so it is also a degenerate case, though arguably less degenerate than a=0]
Same idea with a line as a conic section. By putting in the right constants, you can get a line from the equation for conic sections. It turns out that you end up with a much simpler equation when you cancel things out so you have a degenerate case. Obviously a point is pretty degenerate (which of course is why your mother told you never to point at the table. OK, I’ll stop now)