hyperbolic rotation

hyperbolic rotation

Let

𝔼

be the

Euclidean plane

equipped with the

Cartesian coordinate system

. Recall that given a circle

C

centered at the origin

O

, one can define an “ordinary”

rotation

R

to be a

linear transformation

that takes any point on

C

to another point on

C

. In other words,

R

⁢

(

C

)

⊆

C

.

Similarly, given a

rectangular hyperbola

(the counterpart of a circle)

H

centered at the origin, we define a

hyperbolic rotation

(with respect to

H

) as a linear transformation

T

(on

𝔼

) such that

T

⁢

(

H

)

⊆

H

.

Since a hyperbolic rotation is defined as a linear transformation, let us see what it looks like in matrix form. We start with the simple case when a rectangular hyperbola

H

has the form

x

⁢

y

=

r

, where

r

is a non-negative real number.

Suppose

T

denotes a hyperbolic rotation such that

T

⁢

(

H

)

⊆

H

. Set

(

x

′

y

′

)

=

(

a

b

c

d

)

⁢

(

x

y

)

where

(

a

b

c

d

)

is the

matrix representation

of

T

, and

x

⁢

y

=

x

′

⁢

y

′

=

r

. Solving for

a

,

b

,

c

,

d

and we get

a

⁢

d

=

1

and

b

=

c

=

0

. In other words, with respect to rectangular hyperbolas of the form

x

⁢

y

=

r

, the matrix representation of a hyperbolic rotation looks like

(

a

0

0

a

-

1

)

Since the matrix is

non-singular

, we see that in fact

T

⁢

(

H

)

=

H

.

Now that we know the matrix form of a hyperbolic rotation when the rectangular hyperbolas have the form

x

⁢

y

=

r

, it is not hard to solve the general case. Since the two

asymptotes

of any rectangular hyperbola

H

are

perpendicular

, by an appropriate

change of bases

(ordinary rotation),

H

can be transformed into a rectangular hyperbola

H

′

whose asymptotes are the

x

and

y

axes, so that

H

′

has the algebraic form

x

⁢

y

=

r

. As a result, the matrix representation of a hyperbolic rotation

T

with respect to

H

has the form

P

⁢

(

a

0

0

a

-

1

)

⁢

P

-

1

for some

0

≠

a

∈

ℝ

and some

orthogonal matrix

P

. In other words,

T

is

diagonalizable

with

a

and

a

-

1

as

eigenvalues

(

T

is non-singular as a result).

Below are some simple

properties

:

•

Unlike an ordinary rotation

R

, where

R

fixes any circle centered at

O

, a hyperbolic rotation

T

fixing one rectangular hyperbola centered at

O

may not fix another

hyperbola

of the same kind (as implied by the discussion above).

•

Let

P

be the pencil of all rectangular hyperbolas centered at

O

. For each

H

∈

P

, let

[

H

]

be the subset of

P

containing all hyperbolas whose asymptotes are same as the asymptotes for

H

. If a hyperbolic rotation

T

fixing

H

, then

T

⁢

(

H

′

)

=

H

′

for any

H

′

∈

[

H

]

.

•

[

⋅

]

defined above

partitions

P

into disjoint subsets. Call each of these subset a sub-pencil. Let

A

be a sub-pencil of

P

. Call

T

fixes

A

if

T

fixes

any element of

A

. Let

A

≠

B

be sub-pencils of

P

. Then

T

fixes

A

iff

T

does not fix

B

.

•

Let

A

,

B

be sub-pencils of

P

. Let

T

,

S

be hyperbolic rotations such that

T

fixes

A

and

S

fixes

B

. Then

T

∘

S

is a hyperbolic rotation iff

A

=

B

.

•

In other words, the set of all hyperbolic rotations fixing a sub-pencil is closed under

composition

. In fact, it is a group.

•

Let

T

be a hyperbolic rotation fixing the hyperbola

x

⁢

y

=

r

. Then

T

fixes its branches (

connected components

) iff

T

has

positive

eigenvalues.

•

T

preserves

area.

•

Suppose

T

fixes the

unit hyperbola

H

. Let

P

,

Q

∈

H

. Then

T

fixes the (measure of) hyperbolic angle between

P

and

Q

. In other words, if

α

is the measure of the hyperbolic angle between

P

and

Q

and, by abuse of notation, let

T

⁢

(

α

)

be the measure of the hyperbolic angle between

T

⁢

(

P

)

and

T

⁢

(

Q

)

. Then

α

=

T

⁢

(

α

)

.

The

definition

of a hyperbolic rotation can be generalized into an arbitrary two-dimensional

vector space

: it is any diagonalizable linear transformation with a pair of eigenvalues

a

,

b

such that

a

⁢

b

=

1

.

Title

hyperbolic rotation

Canonical

name

HyperbolicRotation

Date of creation

2013-03-22 17:24:34

Last modified on

2013-03-22 17:24:34

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 53A04

Classification

msc 51N20

Classification

msc 51-00

Related topic

Hyperbola2

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