quadratic surfaces

quadratic surfaces

The common equation of all

quadratic surfaces

is

A

⁢

x

2

+

B

⁢

y

2

+

C

⁢

z

2

+

2

⁢

A

′

⁢

y

⁢

z

+

2

⁢

B

′

⁢

z

⁢

x

+

2

⁢

C

′

⁢

x

⁢

y

+

2

⁢

A

′′

⁢

x

+

2

⁢

B

′′

⁢

y

+

2

⁢

C

′′

⁢

z

+

D

=

0

where

A

,

B

,

C

,

A

′

,

B

′

,

C

′

,

A

′′

,

B

′′

,

C

′′

,

D

are

constants

and at least one of the six first does not vanish.  The different

non-degenerate

kinds are as follows; we give also the simplest equation.

This classification is based on examining the

signature

(

http://planetmath.org/SylvestersLaw

) of the

quadratic form

A

⁢

x

2

+

B

⁢

y

2

+

C

⁢

z

2

+

2

⁢

A

′

⁢

y

⁢

z

+

2

⁢

B

′

⁢

z

⁢

x

+

2

⁢

C

′

⁢

x

⁢

y

and the signature of the form

A

⁢

x

2

+

B

⁢

y

2

+

C

⁢

z

2

+

2

⁢

A

′

⁢

y

⁢

z

+

2

⁢

B

′

⁢

z

⁢

x

+

2

⁢

C

′

⁢

x

⁢

y

+

2

⁢

A

′′

⁢

x

⁢

w

+

2

⁢

B

′′

⁢

y

⁢

w

+

2

⁢

C

′′

⁢

z

⁢

w

+

D

⁢

w

2

Note that, because of the fact that the equation describes the same

surface

if we simultaneously change the signs of all the coefficients, we obtain the same type of surface if we change all the signs in both signatures.

Surfaces without

midpoints

(

http://planetmath.org/Midpoint3

):

\includegraphics

plotA.png

a) Elliptic paraboloid,

x

2

a

2

+

y

2

b

2

=

2

⁢

z

Signatures:

[

+

+

0

]

,

[

+

+

+

-

]

(or

[

-

-

0

]

,

[

+

-

-

-

]

)

\includegraphics

plotB.png

b)

Hyperbolic paraboloid

,

x

2

a

2

-

y

2

b

2

=

2

⁢

z

;  it is a

doubly ruled surface

.

Signatures:

[

+

-

0

]

,

[

+

+

-

-

]

\includegraphics

plotC.png

c) Parabolic cylinder,

x

2

=

2

⁢

p

⁢

z

;  it is a

developable surface

.

Signatures:

[

+

00

]

,

[

+

+

-

0

]

(or

[

-

00

]

,

[

+

-

-

0

]

)

Surfaces with one midpoint:

\includegraphics

plotD

a)

Ellipsoid

,

x

2

a

2

+

y

2

b

2

+

z

2

c

2

=

1

Signature:

[

+

+

+

]

,

[

+

+

+

-

]

(or

[

-

-

-

]

,

[

+

-

-

-

]

)

\includegraphics

plotE

b)

One-sheeted hyperboloid

,

x

2

a

2

+

y

2

b

2

-

z

2

c

2

=

1

;  it is a doubly ruled surface.

Signatures:

[

+

+

-

]

,

[

+

+

-

-

]

(or

[

+

-

-

]

,

[

+

+

-

-

]

)

\includegraphics

plotF

c)

Two-sheeted hyperboloid

,

x

2

a

2

-

y

2

b

2

-

z

2

c

2

=

1

Signature:

[

+

+

-

]

,

[

+

+

+

-

]

(or

[

+

-

-

]

,

[

+

-

-

-

]

)

Δ

<

0

\includegraphics

plotG

d) ,

x

2

a

2

+

y

2

b

2

-

z

2

c

2

=

0

;  it is a developable surface.

Signatures:

[

+

+

-

]

,

[

+

+

-

0

]

(or

[

+

-

-

]

,

[

+

-

-

0

]

)

Surfaces with infinitely many midpoints

\includegraphics

plotH

a)

Hyperbolic cylinder

,

x

2

a

2

-

y

2

b

2

=

1

;  it is a developable surface.

Signatures:

[

+

-

0

]

,

[

+

-

-

0

]

(or

[

+

-

0

]

,

[

+

+

-

0

]

)

\includegraphics

plotI

c)

Elliptic cylinder

,

x

2

a

2

+

y

2

b

2

=

1

;  it is a developable surface.

Signatures:

[

+

+

0

]

,

[

+

+

-

0

]

(or

[

-

-

0

]

,

[

+

-

-

0

]

)

b) Two intersecting planes,

x

2

a

2

-

y

2

b

2

=

0

Signatures:

[

+

-

0

]

,

[

+

-

00

]

d) Two

parallel planes

,

x

2

=

a

2

Signatures:

[

+

00

]

,

[

+

-

00

]

(or

[

-

00

]

,

[

+

-

00

]

)

e) Double plane,

x

2

=

0

Signatures:

[

+

00

]

,

[

+

000

]

(or

[

-

00

]

,

[

-

000

]

)

Algebraically, there are other possibilities for the signatures, such as

[

+

+

+

]

and

[

+

+

+

+

]

.  However, these give rise to equations which have no real solutions, hence they have been ignored.

Title

quadratic surfaces

Canonical name

QuadraticSurfaces

Date of creation

2013-03-22 14:59:40

Last modified on

2013-03-22 14:59:40

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

53

Author

pahio (2872)

Entry type

Topic

Classification

msc 51N20

Synonym

surfaces of second degree

Related topic

TangentPlaneOfQuadraticSurface

Related topic

Ellipsoid

Related topic

SurfaceOfRevolution2

Related topic

GeneratricesOfOneSheetedHyperboloid

Related topic

GeneratricesOfHyperbolicParaboloid

Related topic

AnalyticGeometry

Related topic

IntersectionOfQuadraticSurfaceAndPlane

Defines

elliptic paraboloid

Defines

hyperbolic paraboloid

Defines

parabolic cylinder

Defines

ellipsoid

Defines

one-sheeted hyperboloid

Defines

two-sheeted hyperboloid

Defines

cone surface

Defines

hyperbolic cylinder

Defines

elliptic cylinder

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