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A fundamental theorem in [[Euclidean geometry|Euclidean geometry]] is the following result of F.S. Beckman and D.A. Quarles [[#References|[a1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b1102101.png" /> be a fixed positive real number and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b1102102.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b1102103.png" />) be a mapping satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b1102104.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b1102105.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b1102106.png" />. (Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b1102107.png" /> denotes the Euclidean distance
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b1102108.png" /></td> </tr></table>
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of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b1102109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021010.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021011.png" />.) The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021012.png" /> is then in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021013.png" />.
+
A fundamental theorem in [[Euclidean geometry|Euclidean geometry]] is the following result of F.S. Beckman and D.A. Quarles [[#References|[a1]]]. Let  $  k $
 +
be a fixed positive real number and let  $  f : {\mathbf R  ^ {n} } \rightarrow {\mathbf R  ^ {n} } $ ($  n \geq 2 $)
 +
be a mapping satisfying  $  d ( f ( x ) ,f ( y ) ) = k $
 +
for all  $  x,y \in \mathbf R  ^ {n} $
 +
with  $  d ( x,y ) = k $.
 +
(Here,  $  d ( x,y ) $
 +
denotes the Euclidean distance
 +
 
 +
$$
 +
d ( x,y ) = \sqrt {\sum _ {i = 1 } ^ { n }  ( x _ {i} - y _ {i} )  ^ {2} }
 +
$$
 +
 
 +
of  $  x = ( x _ {1}, \dots, x _ {n} ) $
 +
and  $  y = ( y _ {1}, \dots, y _ {n} ) $
 +
in  $  \mathbf R  ^ {n} $.)  
 +
The mapping $  x \rightarrow f ( x ) - f ( 0 ) $
 +
is then in $  O ( n, \mathbf R ) $.
  
 
It should be emphasized that no regularity assumption (like differentiability or continuity) is required in the theorem. For an analogue in [[Hyperbolic geometry|hyperbolic geometry]], see [[#References|[a4]]] and [[#References|[a5]]].
 
It should be emphasized that no regularity assumption (like differentiability or continuity) is required in the theorem. For an analogue in [[Hyperbolic geometry|hyperbolic geometry]], see [[#References|[a4]]] and [[#References|[a5]]].
  
A distance space is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021014.png" />, a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021015.png" /> and a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021016.png" />. The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021017.png" /> is called the distance of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021019.png" /> (in this order). Beckman–Quarles-type theorems for distance spaces are, up to generalizations, statements claiming that a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021020.png" /> preserves all occurring distances <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021021.png" /> if it preserves one single distance. Beckman–Quarles-type theorems belong to a class of statements called characterizations of geometrical mappings under mild hypotheses.
+
A distance space is a set $  S \neq \emptyset $,  
 +
a set $  W $
 +
and a mapping $  d : {S \times S } \rightarrow W $.  
 +
The element $  d ( x,y ) $
 +
is called the distance of $  x $
 +
and $  y $ (in this order). Beckman–Quarles-type theorems for distance spaces are, up to generalizations, statements claiming that a mapping $  f : S \rightarrow S $
 +
preserves all occurring distances $  w $
 +
if it preserves one single distance. Beckman–Quarles-type theorems belong to a class of statements called characterizations of geometrical mappings under mild hypotheses.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021022.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021023.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021024.png" /> be a fixed real number and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021025.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021026.png" />) be a mapping satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021027.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021028.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021029.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021030.png" /> is a [[Lorentz transformation|Lorentz transformation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021031.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021033.png" />, this was proved by J. Lester; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021034.png" /> and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021036.png" /> this was proved by W. Benz (see [[#References|[a2]]] for all these results). The proofs are different for the three cases, and no common proof is known (1996).
+
Let $  D ( x,y ) = ( x _ {1} - y _ {1} )  ^ {2} + \dots + ( x _ {n - 1 }  - y _ {n - 1 }  )  ^ {2} - ( x _ {n} - y _ {n} )  ^ {2} $,  
 +
for $  x,y \in \mathbf R  ^ {n} $.  
 +
Let $  k \neq0 $
 +
be a fixed real number and let $  f : {\mathbf R  ^ {n} } \rightarrow {\mathbf R  ^ {n} } $ ($  n \geq 2 $)  
 +
be a mapping satisfying $  D ( f ( x ) ,f ( y ) ) = k $
 +
for all $  x,y \in \mathbf R  ^ {n} $
 +
with $  D ( x,y ) = k $.  
 +
Then $  x \rightarrow f ( x ) - f ( 0 ) $
 +
is a [[Lorentz transformation|Lorentz transformation]] of $  \mathbf R  ^ {n} $.  
 +
For $  n > 2 $
 +
and $  k > 0 $,  
 +
this was proved by J. Lester; for $  n = 2 $
 +
and for $  n > 2 $
 +
and $  k < 0 $
 +
this was proved by W. Benz (see [[#References|[a2]]] for all these results). The proofs are different for the three cases, and no common proof is known (1996).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021038.png" /> be normed real vector spaces such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021039.png" /> is strictly convex and the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021040.png" /> is at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021041.png" /> (cf. also [[Vector space|Vector space]]; [[Convex set|Convex set]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021042.png" /> be a fixed real number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021043.png" /> a fixed integer. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021044.png" /> is a mapping satisfying
+
Let $  X $
 +
and $  Y $
 +
be normed real vector spaces such that $  Y $
 +
is strictly convex and the dimension of $  X $
 +
is at least $  2 $ (cf. also [[Vector space|Vector space]]; [[Convex set|Convex set]]). Let $  k > 0 $
 +
be a fixed real number and $  N > 1 $
 +
a fixed integer. Suppose that $  f : X \rightarrow Y $
 +
is a mapping satisfying
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021045.png" /></td> </tr></table>
+
$$
 +
\left \| {a - b } \right \| = k \Rightarrow \left \| {f ( a ) - f ( b ) } \right \| \leq  k,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021046.png" /></td> </tr></table>
+
$$
 +
\left \| {a - b } \right \| = N k \Rightarrow \left \| {f ( a ) - f ( b ) } \right \| \geq  N k ,
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021047.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021048.png" /> is an [[Isometric operator|isometric operator]], and hence an [[Affine transformation|affine transformation]] (the Benz–Berens theorem, [[#References|[a2]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021050.png" /> are, in addition, pre-Hilbert spaces (cf. [[Pre-Hilbert space|Pre-Hilbert space]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021051.png" /> may be replaced by any real number greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021052.png" /> (Radó–Andreascu–Valcán theorem).
+
for all $  a,b \in X $.  
 +
Then $  f $
 +
is an [[Isometric operator|isometric operator]], and hence an [[Affine transformation|affine transformation]] (the Benz–Berens theorem, [[#References|[a2]]]). If $  X $
 +
and $  Y $
 +
are, in addition, pre-Hilbert spaces (cf. [[Pre-Hilbert space|Pre-Hilbert space]]), then $  N $
 +
may be replaced by any real number greater than $  \sqrt 3 $ (Radó–Andreascu–Valcán theorem).
  
There are generalizations for finite planes [[#References|[a9]]], rational or constructible curves (B. Farrahi), non-Euclidean spaces [[#References|[a11]]], and planes over fields [[#References|[a10]]]. F. Radó [[#References|[a8]]] has proved the following theorem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021053.png" /> be a non-singular metric vector space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021054.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021057.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021058.png" /> be a fixed element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021059.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021060.png" /> is a bijection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021061.png" /> preserving distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021062.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021063.png" /> is a semi-affine mapping if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021064.png" />.
+
There are generalizations for finite planes [[#References|[a9]]], rational or constructible curves (B. Farrahi), non-Euclidean spaces [[#References|[a11]]], and planes over fields [[#References|[a10]]]. F. Radó [[#References|[a8]]] has proved the following theorem. Let $  V $
 +
be a non-singular metric vector space of dimension $  \geq 3 $
 +
over $  { \mathop{\rm GF} } ( p  ^ {m} ) $,  
 +
$  p \neq2 $,  
 +
$  m \geq 3 $,  
 +
and let $  k \neq0 $
 +
be a fixed element of $  { \mathop{\rm GF} } ( p  ^ {m} ) $.  
 +
If $  f $
 +
is a bijection of $  V $
 +
preserving distance $  k $,  
 +
then $  f $
 +
is a semi-affine mapping if $  n \not\equiv0, -1, - 2 ( { \mathop{\rm mod} } p ) $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021066.png" />, be a mapping satisfying
+
Let $  f : {\mathbf R  ^ {n} } \rightarrow {\mathbf R  ^ {n} } $,  
 +
$  n \geq 3 $,  
 +
be a mapping satisfying
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021067.png" /></td> </tr></table>
+
$$
 +
\forall a,b,c \in \mathbf R  ^ {n} : \Delta ( a,b,c ) = 1 \Rightarrow
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021068.png" /></td> </tr></table>
+
$$
 +
\Rightarrow
 +
\Delta ( f ( a ) ,f ( b ) ,f ( c ) ) = 1,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021069.png" /> denotes the area of the triangle with vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021070.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021071.png" /> is a Euclidean motion, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021072.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021073.png" /> (Lester's theorem), [[#References|[a3]]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021074.png" /> the equi-affine mappings are characterized similarly.
+
where $  \Delta ( a,b,c ) $
 +
denotes the area of the triangle with vertices $  a,b,c $.  
 +
Then $  f $
 +
is a Euclidean motion, i.e., $  x \rightarrow f ( x ) - f ( 0 ) $
 +
is in $  O ( n, \mathbf R ) $ (Lester's theorem), [[#References|[a3]]]. For $  n = 2 $
 +
the equi-affine mappings are characterized similarly.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021075.png" /> be a mapping of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021076.png" /> of lines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021078.png" />, into itself such that whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021079.png" /> are the lines making up the sides of a triangle of area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021080.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021081.png" /> are also the sides of a triangle of area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021082.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021083.png" /> is induced by a Euclidean motion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021084.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021085.png" /> and by an equi-affine mapping if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021086.png" /> (the Wen-ling Huang theorem, [[#References|[a3]]]).
+
Let $  \pi : {M  ^ {n} } \rightarrow {M  ^ {n} } $
 +
be a mapping of the set $  M  ^ {n} $
 +
of lines in $  \mathbf R  ^ {n} $,  
 +
$  n \geq 2 $,  
 +
into itself such that whenever $  a,b,c \in M  ^ {n} $
 +
are the lines making up the sides of a triangle of area $  1 $,  
 +
then $  \pi ( a ) , \pi ( b ) , \pi ( c ) $
 +
are also the sides of a triangle of area $  1 $.  
 +
Then $  \pi $
 +
is induced by a Euclidean motion of $  \mathbf R  ^ {n} $
 +
for $  n \geq 3 $
 +
and by an equi-affine mapping if $  n = 2 $ (the Wen-ling Huang theorem, [[#References|[a3]]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021087.png" /> be a fixed real number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021088.png" /> a normed real vector space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021089.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021090.png" /> be a function satisfying
+
Let $  k > 0 $
 +
be a fixed real number and $  X $
 +
a normed real vector space of dimension $  \geq 3 $.  
 +
Let $  f : X \rightarrow X $
 +
be a function satisfying
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021091.png" /></td> </tr></table>
+
$$
 +
\forall x,y \in X: \left \| {x - y } \right \| = k \Rightarrow
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021092.png" /></td> </tr></table>
+
$$
 +
\Rightarrow
 +
x - y,f ( x ) - f ( y )  \textrm{ linearly  independent  } .
 +
$$
  
Then there are elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021094.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021095.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110210/b11021096.png" />. This theorem was proved by Benz [[#References|[a2]]] and, anew, by D. Laugwitz [[#References|[a6]]].
+
Then there are elements $  \lambda \in \mathbf R $
 +
and $  t \in X $
 +
such that $  f ( x ) = \lambda x + t $
 +
for all $  x \in X $.  
 +
This theorem was proved by Benz [[#References|[a2]]] and, anew, by D. Laugwitz [[#References|[a6]]].
  
 
General references for this area are [[#References|[a2]]], [[#References|[a3]]] and [[#References|[a7]]].
 
General references for this area are [[#References|[a2]]], [[#References|[a3]]] and [[#References|[a7]]].

Latest revision as of 09:58, 8 May 2022


A fundamental theorem in Euclidean geometry is the following result of F.S. Beckman and D.A. Quarles [a1]. Let $ k $ be a fixed positive real number and let $ f : {\mathbf R ^ {n} } \rightarrow {\mathbf R ^ {n} } $ ($ n \geq 2 $) be a mapping satisfying $ d ( f ( x ) ,f ( y ) ) = k $ for all $ x,y \in \mathbf R ^ {n} $ with $ d ( x,y ) = k $. (Here, $ d ( x,y ) $ denotes the Euclidean distance

$$ d ( x,y ) = \sqrt {\sum _ {i = 1 } ^ { n } ( x _ {i} - y _ {i} ) ^ {2} } $$

of $ x = ( x _ {1}, \dots, x _ {n} ) $ and $ y = ( y _ {1}, \dots, y _ {n} ) $ in $ \mathbf R ^ {n} $.) The mapping $ x \rightarrow f ( x ) - f ( 0 ) $ is then in $ O ( n, \mathbf R ) $.

It should be emphasized that no regularity assumption (like differentiability or continuity) is required in the theorem. For an analogue in hyperbolic geometry, see [a4] and [a5].

A distance space is a set $ S \neq \emptyset $, a set $ W $ and a mapping $ d : {S \times S } \rightarrow W $. The element $ d ( x,y ) $ is called the distance of $ x $ and $ y $ (in this order). Beckman–Quarles-type theorems for distance spaces are, up to generalizations, statements claiming that a mapping $ f : S \rightarrow S $ preserves all occurring distances $ w $ if it preserves one single distance. Beckman–Quarles-type theorems belong to a class of statements called characterizations of geometrical mappings under mild hypotheses.

Let $ D ( x,y ) = ( x _ {1} - y _ {1} ) ^ {2} + \dots + ( x _ {n - 1 } - y _ {n - 1 } ) ^ {2} - ( x _ {n} - y _ {n} ) ^ {2} $, for $ x,y \in \mathbf R ^ {n} $. Let $ k \neq0 $ be a fixed real number and let $ f : {\mathbf R ^ {n} } \rightarrow {\mathbf R ^ {n} } $ ($ n \geq 2 $) be a mapping satisfying $ D ( f ( x ) ,f ( y ) ) = k $ for all $ x,y \in \mathbf R ^ {n} $ with $ D ( x,y ) = k $. Then $ x \rightarrow f ( x ) - f ( 0 ) $ is a Lorentz transformation of $ \mathbf R ^ {n} $. For $ n > 2 $ and $ k > 0 $, this was proved by J. Lester; for $ n = 2 $ and for $ n > 2 $ and $ k < 0 $ this was proved by W. Benz (see [a2] for all these results). The proofs are different for the three cases, and no common proof is known (1996).

Let $ X $ and $ Y $ be normed real vector spaces such that $ Y $ is strictly convex and the dimension of $ X $ is at least $ 2 $ (cf. also Vector space; Convex set). Let $ k > 0 $ be a fixed real number and $ N > 1 $ a fixed integer. Suppose that $ f : X \rightarrow Y $ is a mapping satisfying

$$ \left \| {a - b } \right \| = k \Rightarrow \left \| {f ( a ) - f ( b ) } \right \| \leq k, $$

$$ \left \| {a - b } \right \| = N k \Rightarrow \left \| {f ( a ) - f ( b ) } \right \| \geq N k , $$

for all $ a,b \in X $. Then $ f $ is an isometric operator, and hence an affine transformation (the Benz–Berens theorem, [a2]). If $ X $ and $ Y $ are, in addition, pre-Hilbert spaces (cf. Pre-Hilbert space), then $ N $ may be replaced by any real number greater than $ \sqrt 3 $ (Radó–Andreascu–Valcán theorem).

There are generalizations for finite planes [a9], rational or constructible curves (B. Farrahi), non-Euclidean spaces [a11], and planes over fields [a10]. F. Radó [a8] has proved the following theorem. Let $ V $ be a non-singular metric vector space of dimension $ \geq 3 $ over $ { \mathop{\rm GF} } ( p ^ {m} ) $, $ p \neq2 $, $ m \geq 3 $, and let $ k \neq0 $ be a fixed element of $ { \mathop{\rm GF} } ( p ^ {m} ) $. If $ f $ is a bijection of $ V $ preserving distance $ k $, then $ f $ is a semi-affine mapping if $ n \not\equiv0, -1, - 2 ( { \mathop{\rm mod} } p ) $.

Let $ f : {\mathbf R ^ {n} } \rightarrow {\mathbf R ^ {n} } $, $ n \geq 3 $, be a mapping satisfying

$$ \forall a,b,c \in \mathbf R ^ {n} : \Delta ( a,b,c ) = 1 \Rightarrow $$

$$ \Rightarrow \Delta ( f ( a ) ,f ( b ) ,f ( c ) ) = 1, $$

where $ \Delta ( a,b,c ) $ denotes the area of the triangle with vertices $ a,b,c $. Then $ f $ is a Euclidean motion, i.e., $ x \rightarrow f ( x ) - f ( 0 ) $ is in $ O ( n, \mathbf R ) $ (Lester's theorem), [a3]. For $ n = 2 $ the equi-affine mappings are characterized similarly.

Let $ \pi : {M ^ {n} } \rightarrow {M ^ {n} } $ be a mapping of the set $ M ^ {n} $ of lines in $ \mathbf R ^ {n} $, $ n \geq 2 $, into itself such that whenever $ a,b,c \in M ^ {n} $ are the lines making up the sides of a triangle of area $ 1 $, then $ \pi ( a ) , \pi ( b ) , \pi ( c ) $ are also the sides of a triangle of area $ 1 $. Then $ \pi $ is induced by a Euclidean motion of $ \mathbf R ^ {n} $ for $ n \geq 3 $ and by an equi-affine mapping if $ n = 2 $ (the Wen-ling Huang theorem, [a3]).

Let $ k > 0 $ be a fixed real number and $ X $ a normed real vector space of dimension $ \geq 3 $. Let $ f : X \rightarrow X $ be a function satisfying

$$ \forall x,y \in X: \left \| {x - y } \right \| = k \Rightarrow $$

$$ \Rightarrow x - y,f ( x ) - f ( y ) \textrm{ linearly independent } . $$

Then there are elements $ \lambda \in \mathbf R $ and $ t \in X $ such that $ f ( x ) = \lambda x + t $ for all $ x \in X $. This theorem was proved by Benz [a2] and, anew, by D. Laugwitz [a6].

General references for this area are [a2], [a3] and [a7].

References

[a1] F.S. Beckman, D.A. Quarles, jr., "On isometries of Euclidean spaces" Proc. Amer. Math. Soc. , 4 (1953) pp. 810–815
[a2] W. benz, "Geometrische Transformationen (unter besonderer Berücksichtingung der Lorentztransformationen)" , BI Wissenschaftsverlag (1992)
[a3] W. Benz, "Real geometries" , BI Wissenschaftsverlag (1994)
[a4] B. Farrahi, "A characerization of isometries of absolute planes" Resultate Math. , 4 (1981) pp. 34–38
[a5] A.V. Kuz'minykh, "Mappings preserving a unit distance" Sibirsk. Mat. Zh. , 20 (1979) pp. 597–602 (In Russian)
[a6] D. Laugwitz, "Regular hexagons in normed spaces and a theorem of Walter Benz" Aequat. Math. , 45 (1993) pp. 163–166
[a7] J. Lester, "Distance preserving transformations" F. Buekenhout (ed.) , Handbook of Incidence geometry , Elsevier (1995)
[a8] F. Radó, "On mappings of the Galois space" Israel J. Math. , 53 (1986) pp. 217–230
[a9] H.-J. Samaga, "Zur Kennzeichnung von Lorentztransformationen in endlichen Ebenen" J. Geom. , 18 (1982) pp. 169–184
[a10] H. Schaeffer, "Der Satz von Benz–Radó" Aequat. Math. , 31 (1986) pp. 300–309
[a11] E.M. Schröder, "Zur Kennzeichnung distanztreuer Abbildungen in nichteuklidischen Räumen" J. Geom. , 15 (1980) pp. 108–118
How to Cite This Entry:
Beckman-Quarles-type theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Beckman-Quarles-type_theorems&oldid=22075
This article was adapted from an original article by W. Benz (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article