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Canadian Journal of Mathematics, 1982 An ovoid in an orthogonal vector space V of type Ω+(2n, q) or Ω(2n – 1, q) is a set Ω of qn–1 + 1 pairwise non-perpendicular singular points. Ovoids probably do not exist when n > 4 (cf. [12], [6]) and seem to be rare when n = 4. On the other hand, when n = 3 they correspond to affine translation planes of order q2, via the Klein correspondence ...
William M. Kantor
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Translation planes of order 27
Designs, Codes, and Cryptography, 1994Up to isomorphism there are exactly seven, already known translation planes of order 27. The author shows this with the help of a computer and describes the seven types by invariants that play an important role in the computer search. Independently of the computer proof it is shown which of these types occur in the case that there is an elation in the ...
U Dempwolff
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Doubly β-Derived Translation Planes
Designs, Codes and Cryptography, 2003 Let \(q>3\) be an odd prime power. A chain of circles in the Miquelian inversive plane \(M(q)\) is a collection of \({1\over 2}(q+3)\) circles such that every point covered by some circle in this set lies on exactly two circles from the collection. Using the correspondence between points and circles of \(M(q)\) and lines and reguli of a regular spread \
Abatangelo, V., Larato, Bambina
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Translation planes and configurations in Desarguesian planes
Archiv Der Mathematik, 1960T G Ostrom, Ostrom T G
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, 2003 This monograph is about symplectic translation planes, with particular regard to those of even order. A characteristic property is the existence of completely regular line-ovals, which relies on some properties of classical groups, especially the symplectic and orthogonal ones.
MASCHIETTI, Antonio
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Translating polygons in the plane
2005Let P = (p1,...,pn) and Q = (q1,...,qm) be two simple polygons with non-intersecting interiors in the plane specified by their cartesian coordinates in order. Given a direction d we can ask whether P can be translated an arbitrary distance in direction d without colliding with Q. It has been shown that this problem can be solved in time proportional to
Jörg-Rüdiger Sack +1 more
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The Translation Planes of Dempwolff
Canadian Journal of Mathematics, 1981In [2], Dempwolff constructs three translation planes of order 16 using sharply 2-transitive sets of permutations in S16. That is, if acting on Λ is a sharply 2-transitive set of permutations then an affine plane of order n may be defined as follows: The set of points = {(x, y)|x, y ∊ Λ} and the lines = {(x, y)|y = xg for fixed }, {(x, y)|x = c}, {(x,
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SOME CLASSES OF TRANSLATION PLANES
The Quarterly Journal of Mathematics, 1984This article considers the following. Let \(\pi\) be a finite translation plane of order \(p^ r\) with an autotopism group G which has an orbit of length \(p^ r\)-p on \(\ell_{\infty}\), the line at infinity. The authors make the following additional assumptions: (a) p is an odd prime and \(r=2\); (b) G acts faithfully on \(\ell_{\infty}\).
Cohen, Stephen D., Ganley, Michael J.
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A family of translation planes
Australas. J Comb., 2010Summary: An infinite family of non-Desarguesian translation planes of order \(q^4\) with kernel \(\text{GF}(q^2)\) is constructed, for any odd prime power \(q\). The collineation group of each plane has orbits of lengths 1, \(q^2\), and \(q^4- q^2\) on the translation line.
Andrew Hudson, Tim Penttila
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