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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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| + | $#A+1 = 96 n = 0 |
| + | $#C+1 = 96 : ~/encyclopedia/old_files/data/B110/B.1100210 Beckman\ANDQuarles\AAhtype theorems |
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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]]]. |
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 |