When they present the conic sections in mathematics books, they generally leave out the degenerate cases. They have the ellipse, the circle, the parabola, and the hyperbola, but the degenerate cases happen when the plane is passed through the vertex of the double cone, resulting in the degenerate conic section of a point. I’m wondering if they count a line as another degenerate conic section, the plane passing so that it goes through the vertex again but also touches or is tangent to the two cones, which would make a straight line. And are there any other possible degeneracies with the plane and the cone? Also, what are some of the other degenerate cases in mathematics? Shouldn’t this be worth a whole book?
You also have two intersecting lines, when the plane passes through the vertex, and also through the double cones.
Bill
I knew I forgot one, so there are 3 degenerate cases?
A point, a line, and a pair of lines that intersect?
Also I forgot before to wonder if the shapes of the conic sections change if the cone still has a circle base but instead of the vertex being over the center of the circle, it is over a point on the circle? I mean straight up over it so that the line connecting the point on the circle to the vertex is perpendicular to the plane that the circle is in?
I have a cone like this that I got at a flea market, a nice wood one. It only has one nappe, as it is really a drum, but other than the missing nappe, it is a perfect cone as I describe. Also, are you, Groundskeeper Willie, Scottish, the Scot in the Simpsons? Or an admirer of this character?
I also like the ship’s captain that goes, “Arrrrrrgh.”
Well, my high school calculus teacher was rumored to be unusually fond of sheep…
Oh, wait, that isn’t what you meant is it? Never mind.
don –
The additional case you mentioned (vertex over a point not the center) is equivalent to the centered vertex case. The cones extend to infinity – there really is no plane that establishes a base. The centered vertex case can be rotated so that any point is directly above/below the vertex.
If you’re describing a case where the cross-section of the cone is no longer circular – well, then they are no longer conic sections in the traditional sense.
Well, he certainly is honest! Although, if he were a Teeming Mass I think that he would spend all his time in the Pit.
As for why I chose that name, the truth is rather uninspiring: when I first posted to the SDMB, and had to chose a User Name, the Simpsons was on the television.
Bill