Definition of a Cone

Date: 07/13/2000 at 22:15:00
From: Joshua Romanowski
Subject: Parallel planes in cones

I'm having trouble seeing how a right circular cone cut parallel to 
the axis of symmetry reveals a hyperbola. Shouldn't it be a parabola?

Thanks.

Date: 07/13/2000 at 23:00:43
From: Doctor Peterson
Subject: Re: Parallel planes in cones

Hi, Joshua.

One thing that may be confusing you is that the "cone" we have in mind 
when we talk about conics is a double cone made of two identical cones 
end to end, such as you would form if you held a long thin stick at 
the middle and moved one end in a circle. A "vertical" slice parallel 
to its axis will go through both parts of the cone, producing the two 
parts of a hyperbola. It's only when you cut parallel to the slant of 
the cone that you get a parabola, a single infinite curve rather than 
one finite curve (the ellipse) or two infinite curves (the hyperbola.)

A discussion and illustrations can be found in Xah Lee's Special Plane 
Curves pages:

http://xahlee.org/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.html   

Any cut steeper than the slope of the cone will give a hyperbola, not 
just a cut parallel to the axis; what amazes me is that such a 
hyperbola is symmetrical, even though the two parts seem so different 
due to the slant.

If I've missed your objection to the curve being a hyperbola, tell me 
more about your thinking, so we can clarify the subject.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 07/14/2000 at 22:53:59
From: maximus_gold
Subject: Re: Parallel planes in cones

Doctor Peterson,

Thank you very much for the reply. It makes much more sense now. 
However, what is bugging me is the assumption that the term implies 
two cones end to end. As you recall, the question stated a "right 
circular cone." Doesn't this imply just one side? What am I missing?

Thanks in advance.
Josh

Date: 07/15/2000 at 21:05:39
From: Doctor Peterson
Subject: Re: Parallel planes in cones

Hi, Josh.

There are a couple of ways to answer. First, even if all you get is 
one half of the hyperbola, it's still (part of) a hyperbola, not a 
parabola; so this aspect of the definition doesn't really make a lot 
of difference. Second, however, the term "right circular cone," or 
"cone" in general, is used in a couple of different ways. When we talk 
about volumes, we refer to a finite object bounded by a cone and a 
plane perpendicular to the cone's axis. That's what you usually 
picture as a cone at an elementary level. But the cone we refer to 
when we discuss conics is the surface generated by constructing all 
lines that pass through a circle and a point not in its plane; that's 
the "double cone" I described. In other words, what you think of as a 
cone is actually only part of one half of a full cone.

Math terminology, I'm afraid, is not entirely consistent, especially 
where a term is used in different fields, or in both elementary and 
advanced math.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/