Metric Geometry

Schröder, Eberhard M.

doi:10.1016/b978-044488355-1/50019-0

Cited by 10 publications

(10 citation statements)

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“…For results and references on other groups of Plücker transformations see, among others, [2], [3], [8], [9] and [16]. Finally, we refer to [5], [7], [11, p. 75], [12], [13], [14], [15] and [17] for an axiomatic descriptions of polarities, elliptic spaces and Clifford parallelism as well as a connection with quaternion skew fields.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…There is an infinite number of such unordered pairs, whence the image R γρ is an infinite subset on one side of the triangle {a γρ , b γρ , c γρ }; e.g., R L (c|a) γρ ⊂ (b γρ ∨ c γρ ). By (17) and Proposition 7, im ϕ L contains the κ L -self-polar triangle {a ϕγρ , b ϕγρ , c ϕγρ } as well as infinitely many points on each side of this triangle. We read off from [6, p. 4] that im ϕ L is either a projective subplane or a near-pencil (degenerate subplane) of E L .…”

Section: Propositionmentioning

confidence: 93%

“…If g ⊂ E L is a line, then (17) implies g ϕ L ⊂ g κ L ϕ L κ L . From this observation it is immediate that ϕ L is a lineation, that is, a collinearity-preserving mapping.…”

Section: Propositionmentioning

confidence: 99%

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A characteristic property of elliptic Pl�cker transformations

Havlicek

1997

J Geom

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Dedicated to Walter Benz on the occasion of his 65th birthdayWe discuss elliptic Plücker transformations of three-dimensional elliptic spaces. These are permutations on the set of lines such that any two related (orthogonally intersecting or identical) lines go over to related lines in both directions. It will be shown that for "classical" elliptic 3-spaces a bijection of its lines is already a Plücker transformation, if related lines go over to related lines. Moreover, if the ground field admits only surjective monomorphisms, then "bijection" can be replaced by "injection".

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Propositionmentioning

confidence: 93%

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A characteristic property of elliptic Pl�cker transformations

Havlicek

1997

J Geom

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“…[5] for a bibliography), including some (see [3,4]) that are constructive, in the following sense: the language contains only individual constants and operation symbols having clear geometric meaning, and all the axioms are universal sentences.…”

Section: )mentioning

confidence: 99%

Another Constructive Axiomatization of Euclidean Planes

Pambuccian

2000

MLQ - Math. Log. Quart.

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Abstract. H. Tietze has proved algebraically that the geometry of uniquely determined ruler and compass constructions coincides with the geometry of ruler and set square constructions. We provide a new proof of this result via new universal axiom systems for Euclidean planes of characteristic = 2 in languages containing only operation symbols.Mathematics Subject Classification: 03F65, 51M05, 51F15, 03B30.Keywords: Constructive axiomatization of Euclidean planes, Constructive geometry. [6 -8] the following question: Which points in the Euclidean plane can one construct with ruler and compass, if one starts with a certain finite set of points, and the ruler is to be used for drawing the line joining two already constructed points and finding the intersection point of two already constructed lines, and the compass is to be used to draw uniquely determined points of intersection of circles and lines or circles and circles ? In other words, one does not consider the two intersection points (if they exist) of a line determined by two points A and B with a circle C, with neither A nor B lying on C, to be constructed. The reason is twofold: (i) one does not know whether C will actually intersect the line joining A with B, as this depends on whether the distance from the centre of C to the line joining A with B is less than or equal to the radius of C; (ii) even if one knew that they do intersect, in case there are two distinct intersection points, one would be unable to separate them by metric properties alone (these being the only properties that pure ruler and compass constructions, which do not involve the choice of one point from a set of two, can discern), without taking recourse to betweenness considerations, so one could not say that any of the two points was uniquely determined. For the same reasons one does not consider the points of intersection of two circles to be constructed, unless one of them is an already constructed point (in which case the second point is uniquely determined even if it coincides with the first one). H. Tietze investigated inHe showed that the set of points uniquely constructible by ruler and compass from a given set of points coincide with those constructible by ruler and set square, where the set square is used to drop a perpendicular from a point to a line and to raise a perpendicular from a point on a line to that line. 1)

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“…See, for example, [96] and [97]. Blaschke (1885Blaschke ( -1962 showed that the set of spears is in one-one correspondence with the points of a circular cylinder of the Euclidean 3-space [17], now called the Blaschke [99], [129], [130], as well as the survey articles [67] and [102]. Note that in [129] the term inversive Galileian plane-named after Galileo Galilei (1564-1642)-is used instead.…”

Section: Chaptermentioning

confidence: 99%

Divisible designs, Laguerre geometry, and beyond

Havlicek

2012

J Math Sci

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Metric Geometry

Cited by 10 publications

References 154 publications

A characteristic property of elliptic Pl�cker transformations

A characteristic property of elliptic Pl�cker transformations

Another Constructive Axiomatization of Euclidean Planes

Divisible designs, Laguerre geometry, and beyond

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