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}} . Both these relations are equivalence relations on the set of points. Two points P 1 , P 2 {\displaystyle P_{1},P_{2}} are called parallel ( P 1 ∥ P 2 {\displaystyle P_{1}\parallel P_{2}} ) if
P 1 ∥ + P 2 {\displaystyle P_{1}\parallel _{+}P_{2}} or P 1 ∥ − P 2 {\displaystyle P_{1}\parallel _{-}P_{2}} . From the definition above we find: Lemma:
For any pair of non parallel points A {\displaystyle A} , B {\displaystyle B} there is exactly one point C {\displaystyle C} with A ∥ + C ∥ − B {\displaystyle A\parallel _{+}C\parallel _{-}B} . For any point P {\displaystyle P} and any cycle z {\displaystyle z} there are exactly two points A , B ∈ z {\displaystyle A,B\in z} with A ∥ + P ∥ − B {\displaystyle A\parallel _{+}P\parallel _{-}B} . For any three points A {\displaystyle A} , B {\displaystyle B} , C {\displaystyle C} , pairwise non parallel, there is exactly one cycle z {\displaystyle z} that contains A , B , C {\displaystyle A,B,C} . For any cycle z {\displaystyle z} , any point P ∈ z {\displaystyle P\in z} and any point Q , P ∦ Q {\displaystyle Q,P\not \parallel Q} and Q ∉ z {\displaystyle Q\notin z} there exists exactly one cycle z ′ {\displaystyle z'} such that z ∩ z ′ = { P } {\displaystyle z\cap z'=\{P\}} , i.e. z {\displaystyle z} touches z ′ {\displaystyle z'} at point P {\displaystyle P} . Like the classical Möbius and Laguerre planes Minkowski planes can be described as the geometry of plane sections of a suitable quadric. But in this case the quadric lives in projective 3-space: The classical real Minkowski plane is isomorphic to the geometry of plane sections of a hyperboloid of one sheet (not degenerated quadric of index 2).
Axioms of a Minkowski plane Let ( P , Z ; ∥ + , ∥ − , ∈ ) {\displaystyle \left({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in \right)} be an incidence structure with the set P {\displaystyle {\mathcal {P}}} of points, the set Z {\displaystyle {\mathcal {Z}}} of cycles and two equivalence relations ∥ + {\displaystyle \parallel _{+}} ((+)-parallel) and ∥ − {\displaystyle \parallel _{-}} ((−)-parallel) on set P {\displaystyle {\mathcal {P}}} . For P ∈ P {\displaystyle P\in {\mathcal {P}}} we define:
P ¯ + := { Q ∈ P ∣ Q ∥ + P } {\displaystyle {\overline {P}}_{+}:=\left\{Q\in {\mathcal {P}}\mid Q\parallel _{+}P\right\}} and P ¯ − := { Q ∈ P ∣ Q ∥ − P } {\displaystyle {\overline {P}}_{-}:=\left\{Q\in {\mathcal {P}}\mid Q\parallel _{-}P\right\}} . An equivalence class P ¯ + {\displaystyle {\overline {P}}_{+}} or P ¯ − {\displaystyle {\overline {P}}_{-}} is called (+)-generator and (−)-generator, respectively. (For the space model of the classical Minkowski plane a generator is a line on the hyperboloid.) Two points A , B {\displaystyle A,B} are called parallel ( A ∥ B {\displaystyle A\parallel B} ) if A ∥ + B {\displaystyle A\parallel _{+}B} or A ∥ − B {\displaystyle A\parallel _{-}B} . An incidence structure M := ( P , Z ; ∥ + , ∥ − , ∈ ) {\displaystyle {\mathfrak {M}}:=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )} is called Minkowski plane if the following axioms hold:
C1: For any pair of non parallel points A , B {\displaystyle A,B} there is exactly one point C {\displaystyle C} with A ∥ + C ∥ − B {\displaystyle A\parallel _{+}C\parallel _{-}B} . C2: For any point P {\displaystyle P} and any cycle z {\displaystyle z} there are exactly two points A , B ∈ z {\displaystyle A,B\in z} with A ∥ + P ∥ − B {\displaystyle A\parallel _{+}P\parallel _{-}B} . C3: For any three points A , B , C {\displaystyle A,B,C} , pairwise non parallel, there is exactly one cycle z {\displaystyle z} which contains A , B , C {\displaystyle A,B,C} . C4: For any cycle z {\displaystyle z} , any point P ∈ z {\displaystyle P\in z} and any point Q , P ∦ Q {\displaystyle Q,P\not \parallel Q} and Q ∉ z {\displaystyle Q\notin z} there exists exactly one cycle z ′ {\displaystyle z'} such that z ∩ z ′ = { P } {\displaystyle z\cap z'=\{P\}} , i.e., z {\displaystyle z} touches z ′ {\displaystyle z'} at point P {\displaystyle P} . C5: Any cycle contains at least 3 points. There is at least one cycle z {\displaystyle z} and a point P {\displaystyle P} not in z {\displaystyle z} . For investigations the following statements on parallel classes (equivalent to C1, C2 respectively) are advantageous.
C1′: For any two points A {\displaystyle A} , B {\displaystyle B} we have | A ¯ + ∩ B ¯ − | = 1 {\displaystyle \left|{\overline {A}}_{+}\cap {\overline {B}}_{-}\right|=1} . C2′: For any point P {\displaystyle P} and any cycle z {\displaystyle z} we have: | P ¯ + ∩ z | = 1 = | P ¯ − ∩ z | {\displaystyle \left|{\overline {P}}_{+}\cap z\right|=1=\left|{\overline {P}}_{-}\cap z\right|} . First consequences of the axioms are
Analogously to Möbius and Laguerre planes we get the connection to the linear geometry via the residues. For a Minkowski plane M = ( P , Z ; ∥ + , ∥ − , ∈ ) {\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )} and P ∈ P {\displaystyle P\in {\mathcal {P}}} we define the local structure
A P := ( P ∖ P ¯ , { z ∖ { P ¯ } ∣ P ∈ z ∈ Z } ∪ { E ∖ P ¯ ∣ E ∈ E ∖ { P ¯ + , P ¯ − } } , ∈ ) {\displaystyle {\mathfrak {A}}_{P}:=({\mathcal {P}}\setminus {\overline {P}},\{z\setminus \{{\overline {P}}\}\mid P\in z\in {\mathcal {Z}}\}\cup \{E\setminus {\overline {P}}\mid E\in {\mathcal {E}}\setminus \{{\overline {P}}_{+},{\overline {P}}_{-}\}\},\in )}
and call it the residue at point P. For the classical Minkowski plane A ( ∞ , ∞ ) {\displaystyle {\mathfrak {A}}_{(\infty ,\infty )}} is the real affine plane R 2 {\displaystyle \mathbb {R} ^{2}} . An immediate consequence of axioms C1 to C4 and C1′, C2′ are the following two theorems.
Minimal model
The minimal model of a Minkowski plane can be established over the set K ¯ := { 0 , 1 , ∞ } {\displaystyle {\overline {K}}:=\{0,1,\infty \}} of three elements:
P := K ¯ 2 {\displaystyle {\mathcal {P}}:={\overline {K}}^{2}}
Z : = { { ( a 1 , b 1 ) , ( a 2 , b 2 ) , ( a 3 , b 3 ) } ∣ { a 1 , a 2 , a 3 } = { b 1 , b 2 , b 3 } = K ¯ } = { { ( 0 , 0 ) , ( 1 , 1 ) , ( ∞ , ∞ ) } , { ( 0 , 0 ) , ( 1 , ∞ ) , ( ∞ , 1 ) } , { ( 0 , 1 ) , ( 1 , 0 ) , ( ∞ , ∞ ) } , { ( 0 , 1 ) , ( 1 , ∞ ) , ( ∞ , 0 ) } , { ( 0 , ∞ ) , ( 1 , 1 ) , ( ∞ , 0 ) } , { ( 0 , ∞ ) , ( 1 , 0 ) , ( ∞ , 1 ) } } {\displaystyle {\begin{aligned}{\mathcal {Z}}:\!&=\left\{\{(a_{1},b_{1}),(a_{2},b_{2}),(a_{3},b_{3})\}\mid \{a_{1},a_{2},a_{3}\}=\{b_{1},b_{2},b_{3}\}={\overline {K}}\right\}\\&=\{\{(0,0),(1,1),(\infty ,\infty )\},\;\{(0,0),(1,\infty ),(\infty ,1)\},\\&\qquad \{(0,1),(1,0),(\infty ,\infty )\},\;\{(0,1),(1,\infty ),(\infty ,0)\},\\&\qquad \{(0,\infty ),(1,1),(\infty ,0)\},\;\{(0,\infty ),(1,0),(\infty ,1)\}\}\end{aligned}}} Parallel points:
( x 1 , y 1 ) ∥ + ( x 2 , y 2 ) {\displaystyle (x_{1},y_{1})\parallel _{+}(x_{2},y_{2})} if and only if x 1 = x 2 {\displaystyle x_{1}=x_{2}}
( x 1 , y 1 ) ∥ − ( x 2 , y 2 ) {\displaystyle (x_{1},y_{1})\parallel _{-}(x_{2},y_{2})} if and only if y 1 = y 2 {\displaystyle y_{1}=y_{2}} . Hence | P | = 9 {\displaystyle \left|{\mathcal {P}}\right|=9} and | Z | = 6 {\displaystyle \left|{\mathcal {Z}}\right|=6} .
Finite Minkowski-planes For finite Minkowski-planes we get from C1′, C2′:
This gives rise of the definition: For a finite Minkowski plane M {\displaystyle {\mathfrak {M}}} and a cycle z {\displaystyle z} of M {\displaystyle {\mathfrak {M}}} we call the integer n = | z | − 1 {\displaystyle n=\left|z\right|-1} the order of M {\displaystyle {\mathfrak {M}}} . Simple combinatorial considerations yield
Miquelian Minkowski planes We get the most important examples of Minkowski planes by generalizing the classical real model: Just replace R {\displaystyle \mathbb {R} } by an arbitrary field K {\displaystyle K} then we get in any case a Minkowski plane M ( K ) = ( P , Z ; ∥ + , ∥ − , ∈ ) {\displaystyle {\mathfrak {M}}(K)=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )} . Analogously to Möbius and Laguerre planes the Theorem of Miquel is a characteristic property of a Minkowski plane M ( K ) {\displaystyle {\mathfrak {M}}(K)} .
Theorem (Miquel): For the Minkowski plane M ( K ) {\displaystyle {\mathfrak {M}}(K)} the following is true:
If for any 8 pairwise not parallel points P 1 , . . . , P 8 {\displaystyle P_{1},...,P_{8}} which can be assigned to the vertices of a cube such that the points in 5 faces correspond to concyclical quadruples, then the sixth quadruple of points is concyclical, too. (For a better overview in the figure there are circles drawn instead of hyperbolas.) Theorem (Chen): Only a Minkowski plane M