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### Definition of a Cone

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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

- 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?

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

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Definitions
High School Equations, Graphs, Translations
High School Geometry
High School Higher-Dimensional Geometry

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