# cone in $\mathbb{R}^{3}$

When a straight line moves in $\mathbb{R}^{3}$ 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 $\ell$ 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.

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 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 $\mathbb{R}^{3}$ 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