New euclidean theorems by the use of Laguerre transformations ? Some geometry of Minkowski (2+1)-space

Fillmore, Jay P.; Springer, Arthur

doi:10.1007/bf01406828

Cited by 6 publications

(9 citation statements)

References 3 publications

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“…A good reference for affine Minkowski spaces is [7]. For more about the affine Minkowski space as a model for Laguerre geometry, the reader is referred to [4].…”

Section: Minkowski Space and Isotropic Projectionmentioning

confidence: 99%

“…We thus obtain a 1-1 correspondence between singular planes in M 3 and spears in E, called the isotropic projection of the singular planes of M 3 onto the spears of E (or of the spears of E onto the singular planes of M 3 ). For more on isotropic projection, the reader is referred to [4] and [5].…”

Section: Vol 90 (2008) a Euclidean Area Theorem Via Isotropic Projecmentioning

confidence: 99%

“…Let E be the Euclidean plane (embedded in M 3 ) containing the cycles c 1 , c 2 , c 3 , c 4 . Let e be a vector normal to E with e * e = −1, and set…”

Section: Proof Of the Area Theoremmentioning

confidence: 99%

“…The set of spears and cycles, the property touch, and the group of transformations of spears that preserve touch and map cycles to cycles, form classic Laguerre geometry (see, e.g., [2][3][4]6]). We introduce here only the elements of Laguerre geometry necessary for our exposition.…”

Section: Introductionmentioning

confidence: 99%

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A Euclidean Area Theorem via Isotropic Projection

Knight

2008

J. geom.

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We obtain a generalization of a property of the arbelos first stated as Proposition 4 in the Book of Lemmas by Archimedes, circa 250 BC. The new theorem relates the areas of a chain of four consecutively tangent circles to the area of a circle orthogonal to, and with a diameter tangent to, two of the original circles. The proof of the theorem uses isotropic projection of the 3-dimensional affine Minkowski space onto an embedded Euclidean plane. Proposition 14 of the Book of Lemmas (the Lemma of the Salinon) is then derived from the generalization. Mathematics Subject Classification (2000). 51B15, 51B20, 51A15, 51A30, 51M05.

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“…A good reference for affine Minkowski spaces is [7]. For more about the affine Minkowski space as a model for Laguerre geometry, the reader is referred to [4].…”

Section: Minkowski Space and Isotropic Projectionmentioning

confidence: 99%

Section: Vol 90 (2008) a Euclidean Area Theorem Via Isotropic Projecmentioning

confidence: 99%

“…Let E be the Euclidean plane (embedded in M 3 ) containing the cycles c 1 , c 2 , c 3 , c 4 . Let e be a vector normal to E with e * e = −1, and set…”

Section: Proof Of the Area Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Euclidean Area Theorem via Isotropic Projection

Knight

2008

J. geom.

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“…The latter model has been largely neglected, although its utility is demonstrated to good effect in both [7] and [8]. The interested reader may find a more thorough exposition of the classic and Minkowski models of Laguerre geometry in [8].…”

Section: Introductionmentioning

confidence: 99%

The Apollonius contact problem and Lie contact geometry

Knight

2005

J. geom.

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A simple classification of triples of Lie cycles is given. The class of each triad determines the number of solutions to the associated oriented Apollonius contact problem. The classification is derived via 2-dimensional Lie contact geometry in the form of two of its subgeometries-Laguerre geometry and oriented Möbius geometry. The method of proof illustrates interactions between the two subgeometries of Lie geometry. Two models of Laguerre geometry are used: the classic model and the 3-dimensional affine Minkowski space model. (2000): 50D45, 51B25, 51N10. Mathematics Subject Classification

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Dependencies of bundle forms in Laguerre planes

Knight

2014

J. Geom.

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New euclidean theorems by the use of Laguerre transformations ? Some geometry of Minkowski (2+1)-space

Cited by 6 publications

References 3 publications

A Euclidean Area Theorem via Isotropic Projection

A Euclidean Area Theorem via Isotropic Projection

The Apollonius contact problem and Lie contact geometry

Dependencies of bundle forms in Laguerre planes

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