cone in ${ℝ}^3$

When a straight line moves in ${ℝ}^3$ passing constantly through a certain point $O$, the ruled surface it sweeps is called a conical surface or a generalized cone.  Formally and more generally, a conical surface $S$ is a ruled surface with the given condition:

there is a point $O$ on $S$, such that any ruling $\mathrm{\ell }$ on $S$ passes through $O$.

By definition, it is readily seen that this point $O$ is unique, for otherwise, all rulings of $S$ that pass through $O$ as well as another point $O^{\prime }$ must all coincide, concluding that $S$ must be nothing more than a straight line, contradicting the fact that $S$ is a surface.  The point $O$ is commonly known as the apex of the conical surface $S$.

Figure 1: A circular conical surface

Remarks.

• No plane can be a conical surface, because one can always find a line (ruling) on the plane not passing through $O$.

• A conical surface is, strictly speaking, not a (regular) surface in the sense of a differentiable manifold.  This is because of the differentiability at the apex breaks down.  In fact, no neighborhood of the apex is diffeomorphic to ${ℝ}^2$.  Nevertheless, it is easy to see that any two points on a conical surface can be joined by a simple continuous curve.

• Any plane passing through the apex $O$ of a conical surface $S$ must contain at least two lines $\mathrm{\ell },m$ through the apex such that $\mathrm{\ell }\cap S=m\cap S=\{O\}$.  In fact, it can be shown that there is a plane with the above property such that $\mathrm{\ell }⟂m$.

• Given a conical surface $S$, if there is a plane $\pi$ in ${ℝ}^3$ such that $\pi \cap S=\mathrm{\varnothing }$, then it can be shown that $S$ is planar (that is, $S$ lies on a plane).  This shows that if $S$ is a non-planar surface, it must have non-empty intersection with any plane in ${ℝ}^3$!

Given a plane $\pi$ not passing through the apex $O$ of a non-planar conical surface $S$, the intersection of $\pi$ and $S$ is non-empty, as guaranteed by the previous remark. Let $c$ be the intersection of $\pi$ and $S$. It is not hard to see that $c$ is necessarily a curve. The curve may be bounded or unbounded, and it may have disjoint components. If there is a plane not passing through the apex of a conical surface $S$, such that its intersection with $S$ is a bounded, connected closed loop, then we call this surface $S$ a closed cone, or cone for short. Intuitively, it can be pictured as a surface swept out by a moving straight line that returns to its starting position. Any plane not passing through the apex of a closed cone $S$ intersects each ruling of $S$ at exactly one point.

The solid bounded by a closed cone $S$, a plane $\pi$ not passing through the apex $O$ and the apex $O$ itself is called a solid cone.  The portion of the surface of the cone belonging to the conical surface is called the or the of the solid cone and the portion belonging to the plane is the base of the solid cone.

The intersections of a solid cone and the planes parallel to the base plane are similar.  The perpendicular of the apex and the base plane is the of the cone.  The volume ($V$) of the cone equals to the third of product of the base area ($A$) and the ($h$):

$$V=\frac{Ah}{3}.$$

The formula can be derived directly by observing that, if we were to take another parallel slice of the solid cone, the area $A(x)$ of the base from the slice is directly proportional to the square of the corresponding height $x$. Integrating $A(x)$ with respect to $x$, where $x$ is between $0$ and $h$ gives us the above formula.

If the base of a cone is a polygon, the cone is called a pyramid (which is a polyhedron).  The mantle of a pyramid consists of triangular faces having as common vertex the apex $O$.  A pyramid is if its base is a regular polygon and the of its base.  A tetrahedron is an example of a regular pyramid.  Note that, in a regular pyramid, all of the faces are isosceles.  Colloquially, “pyramid” typically refers to a regular square pyramid.

If the base is a circle, the cone is called circular.  If its of the base circle, the circular cone is .  Colloquially, “cone” typically refers to a right circular cone.

In any cone, the line segment of a ruling between the base plane and the apex is a of the cone.  All are equally long only in a right circular cone.  If in this case, the of the side line is $s$ and the radius of the base circle $r$, then the area of the mantle of the right circular cone equals $\pi rs$.

Title	cone in ${ℝ}^3$
Canonical name	ConeInmathbbR3
Date of creation	2013-03-22 15:29:54
Last modified on	2013-03-22 15:29:54
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	22
Author	pahio (2872)
Entry type	Definition
Classification	msc 51M20
Classification	msc 51M04
Synonym	generalized cone
Synonym	circular double cone
Synonym	right circular double cone
Related topic	SurfaceOfRevolution2
Defines	apex
Defines	base
Defines	circular cone
Defines	cone
Defines	conical surface
Defines	mantle
Defines	pyramid
Defines	regular pyramid
Defines	right circular cone
Defines	solid cone