cone in ℝ^3

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

ℓ

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

′

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

ℓ

,

m

through the apex such that

ℓ

∩

S

=

m

∩

S

=

{

O

}

.  In fact, it can be shown that there is a plane with the above

property

such that

ℓ

⟂

m

.

•

Given a conical surface

S

, if there is a plane

π

in

ℝ

3

such that

π

∩

S

=

∅

, 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

π

not passing through the apex

O

of a non-planar conical surface

S

, the intersection of

π

and

S

is non-empty, as guaranteed by the previous remark. Let

c

be the intersection of

π

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

π

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

=

A

⁢

h

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

π

⁢

r

⁢

s

.

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

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