Mappings preserving the area equality of hyperbolic triangles are motions |
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Authors: | Victor Pambuccian |
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Institution: | (1) Department of Integrative Studies, Arizona State University, West Phoenix, AZ, 85069-7100, U.S.A. |
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Abstract: | It is shown that a mapping ${\varphi: \mathfrak{A}\rightarrow \mathfrak{B}}It is shown that a mapping
j: \mathfrakA? \mathfrakB{\varphi: \mathfrak{A}\rightarrow \mathfrak{B}} between models
\mathfrakA{\mathfrak{A}} and
\mathfrakB{\mathfrak{B}} of elementary plane hyperbolic geometry, coordinatized by Euclidean ordered fields, that maps triangles having the same area
and sharing a side into triangles that have the same property, must be a hyperbolic motion onto
j(\mathfrakA){\varphi(\mathfrak{A})}. The relations that Tarski and Szmielew used as primitives for geometry, the equidistance relation ≡ and the betweenness
relation B are shown to be positively existentially definable in terms of the quaternary relation Δ, with Δ(abcd) standing for “the triangles abc and abd have the same area.” |
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Keywords: | |
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