surface of revolution

surface of revolution

If a curve in

ℝ

3

rotates about a line, it generates a

surface of revolution

. The line is called the

axis of revolution

.  Every point of the curve generates a

circle of latitude

. If the surface is intersected by a

half-plane

beginning from the axis of revolution, the intersection curve is a

meridian curve

. One can always think that the surface of revolution is generated by the

rotation

of a certain

meridian

, which may be called the

0-meridian

.

Let

y

=

f

⁢

(

x

)

be a curve of the

x

⁢

y

-plane rotating about the

x

-axis. Then any point

(

x

,

y

)

of this 0-meridian draws a circle of latitude,

parallel

to the

y

⁢

z

-plane, with centre on the

x

-axis and with the radius

|

f

⁢

(

x

)

|

. So the

y

- and

z

-

coordinates

of each point on this circle satisfy the equation

y

2

+

z

2

=

[

f

⁢

(

x

)

]

2

.

This equation is thus satisfied by all points

(

x

,

y

,

z

)

of the surface of revolution and therefore it is the equation of the whole surface of revolution.

More generally, if the equation of the meridian curve in the

x

⁢

y

-plane is given in the implicit form

F

⁢

(

x

,

y

)

=

0

,  then the equation of the surface of revolution may be written

F

⁢

(

x

,

y

2

+

z

2

)

=

0

.

Examples.

When the

catenary

y

=

a

⁢

cosh

⁡

x

a

rotates about the

x

-axis, it generates the

catenoid

y

2

+

z

2

=

a

2

⁢

cosh

2

⁡

x

a

.

The catenoid is the only surface of revolution being also a

minimal surface

.

The

quadratic surfaces

of revolution:

•

When the

ellipse

x

2

a

2

+

y

2

b

2

=

1

rotates about the

x

-axis, we get the

ellipsoid

x

2

a

2

+

y

2

+

z

2

b

2

=

1

.

This is a

stretched ellipsoid

, if

a

>

b

,  and a

flattened ellipsoid

, if

a

<

b

, and a sphere of radius

a

, if

a

=

b

.

•

When the

parabola

y

2

=

2

⁢

p

⁢

x

(with

p

the

latus rectum

or the

parameter of parabola

) rotates about the

x

-axis, we get the

paraboloid

of revolution

y

2

+

z

2

=

2

⁢

p

⁢

x

.

•

When we let the

conjugate hyperbolas

and their common

asymptotes

x

2

a

2

-

y

2

b

2

=

s

(with

s

=

1

,

-

1

,

0

) rotate about the

x

-axis, we obtain the

two-sheeted hyperboloid

x

2

a

2

-

y

2

+

z

2

b

2

=

1

,

the

one-sheeted hyperboloid

x

2

a

2

-

y

2

+

z

2

b

2

=

-

1

and the

cone of revolution

x

2

a

2

-

y

2

+

z

2

b

2

=

0

,

which apparently is the common

asymptote cone

of both hyperboloids.

References

1

Lauri Pimiä

:

Analyyttinen geometria

.  Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1958).

Title

surface of revolution

Canonical

name

SurfaceOfRevolution

Date of creation

2013-03-22 17:17:08

Last modified on

2013-03-22 17:17:08

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Topic

Classification

msc 57M20

Classification

msc 51M04

Related topic

SurfaceOfRevolution

Related topic

PappussTheoremForSurfacesOfRevolution

Related topic

QuadraticSurfaces

Related topic

ConicalSurface

Related topic

Torus

Related topic

SolidOfRevolution

Related topic

LeastSurfaceOfRevolution

Related topic

ConeInMathbbR3

Defines

surface of revolution

Defines

axis of revolution

Defines

circle of latitude

Defines

meridian curve

Defines

0-meridian

Defines

cone of revolution

Defines

asymptote cone

Defines

catenoid

Generated on Sat Feb 10 11:47:19 2018 by

LaTeXML