Journal of Formalized Mathematics
Volume 4, 1992
University of Bialystok
Copyright (c) 1992
Association of Mizar Users
Reper Algebras
-
Michal Muzalewski
-
Warsaw University, Bialystok
Summary.
-
We shall describe $n$-dimensional spaces with the reper
operation [7, pages 72-79]. An inspiration to such
approach comes from the monograph [9] and so-called
Leibniz program. Let us recall that the Leibniz program is a
program of algebraization of geometry using purely geometric
notions. Leibniz formulated his program in opposition to
algebraization method developed by Descartes.
The Euclidean geometry in Szmielew's approach [0]
{SZMIELEW:1} is a theory of structures $\langle S$; $\parallel,
\oplus, O \rangle$, where $\langle S$; $\parallel, \oplus, O
\rangle$ is Desarguesian midpoint plane and $O \subseteq S\times S\times S$
is the relation of equi-orthogonal basis. Points $o, p, q$ are in relation
$O$
if they form an isosceles triangle with the right angle in vertex $a$.
If we fix vertices $a, p$, then there exist exactly two points $q, q'$ such
that $O(apq)$, $O(apq')$. Moreover $q \oplus q' = a$.
In accordance with the Leibniz program we replace the relation
of equi-orthogonal basis by a binary operation $\ast : S\times S
\rightarrow S$, called the reper operation. A standard model for the
Euclidean geometry in the above sense is the oriented plane over the
field of real numbers with the reper operations $\ast$ defined by the
condition: $a \ast b = q$ iff the point $q$ is the result of rotating of $p$
about right angle around the center $a$.
MML Identifier:
MIDSP_3
The terminology and notation used in this paper have been
introduced in the following articles
[10]
[12]
[3]
[4]
[2]
[5]
[1]
[11]
[6]
[8]
-
Substitutions in tuples
-
Reper Algebra Structure and its Properties
-
Reper Algebra and its Atlas
Bibliography
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\em Foundations of Metric-Affine Geometry.
Dzial Wydawnictw Filii UW w Bialymstoku, Filia UW w
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- [8]
Michal Muzalewski.
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Journal of Formalized Mathematics,
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Wanda Szmielew.
\em From Affine to Euclidean Geometry, volume 27.
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- [10]
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Received May 28, 1992
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