Minkowski plane

In mathematics, a Minkowski plane (named after Hermann Minkowski) is one of the Benz planes (the others being Möbius plane and Laguerre plane).

Classical real Minkowski plane

Applying the pseudo-euclidean distance

d

(

P

1

,

P

2

)

=

(

x

1

′

−

x

2

′

)

2

−

(

y

1

′

−

y

2

′

)

2

{\displaystyle d(P_{1},P_{2})=(x'_{1}-x'_{2})^{2}-(y'_{1}-y'_{2})^{2}}

on two points

P

i

=

(

x

i

′

,

y

i

′

)

{\displaystyle P_{i}=(x'_{i},y'_{i})}

(instead of the euclidean distance) we get the geometry of hyperbolas, because a pseudo-euclidean circle

{

P

∈

R

2

∣

d

(

P

,

M

)

=

r

}

{\displaystyle \{P\in \mathbb {R} ^{2}\mid d(P,M)=r\}}

is a hyperbola with midpoint ⁠

M

{\displaystyle M}

⁠.

By a transformation of coordinates ⁠

x

i

=

x

i

′

+

y

i

′

{\displaystyle x_{i}=x'_{i}+y'_{i}}

⁠, ⁠

y

i

=

x

i

′

−

y

i

′

{\displaystyle y_{i}=x'_{i}-y'_{i}}

⁠, the pseudo-euclidean distance can be rewritten as ⁠

d

(

P

1

,

P

2

)

=

(

x

1

−

x

2

)

(

y

1

−

y

2

)

{\displaystyle d(P_{1},P_{2})=(x_{1}-x_{2})(y_{1}-y_{2})}

⁠. The hyperbolas then have asymptotes parallel to the non-primed coordinate axes.

The following completion (see Möbius and Laguerre planes) homogenizes the geometry of hyperbolas:

the set of points:

P

:=

(

R

∪

{

∞

}

)

2

=

R

2

∪

(

{

∞

}

×

R

)

∪

(

R

×

{

∞

}

)

∪

{

(

∞

,

∞

)

}

,

∞

∉

R

,

{\displaystyle {\mathcal {P}}:=\left(\mathbb {R} \cup \left\{\infty \right\}\right)^{2}=\mathbb {R} ^{2}\cup \left(\left\{\infty \right\}\times \mathbb {R} \right)\cup \left(\mathbb {R} \times \left\{\infty \right\}\right)\ \cup \left\{\left(\infty ,\infty \right)\right\}\ ,\ \infty \notin \mathbb {R} ,}

the set of cycles

Z

:=

{

{

(

x

,

y

)

∈

R

2

∣

y

=

a

x

+

b

}

∪

{

(

∞

,

∞

)

}

∣

a

,

b

∈

R

,

a

≠

0

}

∪

{

{

(

x

,

y

)

∈

R

2

∣

y

=

a

x

−

b

+

c

,

x

≠

b

}

∪

{

(

b

,

∞

)

,

(

∞

,

c

)

}

∣

a

,

b

,

c

∈

R

,

a

≠

0

}

.

{\displaystyle {\begin{aligned}{\mathcal {Z}}:={}&\left\{\left\{\left(x,y\right)\in \mathbb {R} ^{2}\mid y=ax+b\right\}\cup \left\{\left(\infty ,\infty \right)\right\}\mid a,b\in \mathbb {R} ,a\neq 0\right\}\\&\quad \cup \left\{\left\{\left(x,y\right)\in \mathbb {R} ^{2}\mid y={\frac {a}{x-b}}+c,x\neq b\right\}\cup \left\{\left(b,\infty \right),\left(\infty ,c\right)\right\}\mid a,b,c\in \mathbb {R} ,a\neq 0\right\}.\end{aligned}}}

The incidence structure

(

P

,

Z

,

∈

)

{\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}

is called the classical real Minkowski plane.

The set of points consists of ⁠

R

2

{\displaystyle \mathbb {R} ^{2}}

⁠, two copies of

R

{\displaystyle \mathbb {R} }

and the point ⁠

(

∞

,

∞

)

{\displaystyle (\infty ,\infty )}

⁠.

Any line

y

=

a

x

+

b

,

a

≠

0

{\displaystyle y=ax+b,a\neq 0}

is completed by point ⁠

(

∞

,

∞

)

{\displaystyle (\infty ,\infty )}

⁠, any hyperbola

y

=

a

x

−

b

+

c

,

a

≠

0

{\displaystyle y={\frac {a}{x-b}}+c,a\neq 0}

by the two points

(

b

,

∞

)

,

(

∞

,

c

)

{\displaystyle (b,\infty ),(\infty ,c)}

(see figure).

Two points

(

x

1

,

y

1

)

≠

(

x

2

,

y

2

)

{\displaystyle (x_{1},y_{1})\neq (x_{2},y_{2})}

can not be connected by a cycle if and only if

x

1

=

x

2

{\displaystyle x_{1}=x_{2}}

or ⁠

y

1

=

y

2

{\displaystyle y_{1}=y_{2}}

⁠.

We define:

Two points

P

1

{\displaystyle P_{1}}

,

P

2

{\displaystyle P_{2}}

are (+)-parallel (⁠

P

1

∥

+

P

2

{\displaystyle P_{1}\parallel _{+}P_{2}}

⁠) if

x

1

=

x

2

{\displaystyle x_{1}=x_{2}}

and (−)-parallel (⁠

P

1

∥

−

P

2

{\displaystyle P_{1}\parallel _{-}P_{2}}

⁠) if ⁠

y

1

=

y

2

{\displaystyle y_{1}=y_{2}}

⁠.