Types of Cones

Date: 01/19/99 at 15:57:04
From: Jamie Rowan
Subject: Edges of cones

Does a cone have an edge?

Date: 01/20/99 at 08:55:54
From: Doctor Rob
Subject: Re: Edges of cones

There are several meanings to the word "cone." In some, it does not 
have an edge; in others, it does. I will try to describe the various 
meanings precisely and perhaps then you will see what I mean.

Set up a Cartesian rectangular xyz-coordinate system as follows. Take 
the origin to be the vertex of the cone, and the z-axis to be the axis 
of the cone. Then a right circular cone can be defined by an equation 
of the form

   a^2*z^2 = x^2 + y^2

for some nonzero constant a. The words "right circular" mean that a 
cross-section perpendicular (at a right angle) to the axis is a circle. 
Often these words are omitted, since it is quite uncommon to see an 
oblique circular cone or a right elliptical cone, or other types. An 
"infinite right circular cone of two sheets" consists of all points 
whose coordinates satisfy this equation. In this form, you can see that 
there is no limit on the size of z, so it extends infinitely in both 
positive and negative z-directions; hence the word "infinite." The 
words "of two sheets" mean that if you remove the vertex, the surface 
is split into two connected parts disjoint from each other.

If you restrict z to z >= 0, then you get an "infinite right circular 
cone of one sheet."

If you restrict z to 0 <= z <= b, for some positive b, then you get 
part of a right circular cone of one sheet. I would call this a "finite 
right circular cone of one sheet."

If you add the set of points z = b, x^2 + y^2 <= a^2*b^2 (the base of 
the cone), you get a closed surface having two faces, with an interior 
region. I would call this a "closed right circular cone of one sheet."

If you consider the region of xyz-space enclosed by that surface, that 
is, points whose coordinates satisfy 0 <= z <= b, x^2 + y^2 <= a^2*z^2, 
you get a "right circular conical region," or "solid right circular 
cone."

Any of the above can be loosely called a "cone."  Only the last two 
have an edge, the circle z = b, x^2 + y^2 = a^2*b^2, with center 
(0,0,b) and radius a*b.

The moral of this is that you may need to be precise about which of the 
above "cones" you mean, then the answer as to whether or not there is 
an edge can be easily determined.

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