Toroidal circle planes that are not Minkowski planes

1998 ◽  
Vol 63 (1-2) ◽  
pp. 154-167 ◽  
Author(s):  
Burkard Polster
Keyword(s):  
Minkowski Planes

Covering Discs in Minkowski Planes

2009 ◽  
Vol 52 (3) ◽  
pp. 424-434 ◽  
Author(s):  
Horst Martini ◽  
Margarita Spirova
Keyword(s):  
Essential Role ◽  
Covering Problem ◽  
Minkowski Geometry ◽  
Minkowski Planes ◽  
Strictly Convex ◽  
Unit Circles ◽  
Circle Covering ◽  
Monotonicity Lemma

AbstractWe investigate the following version of the circle covering problem in strictly convex (normed or) Minkowski planes: to cover a circle of largest possible diameter by k unit circles. In particular, we study the cases k = 3, k = 4, and k = 7. For k = 3 and k = 4, the diameters under consideration are described in terms of side-lengths and circumradii of certain inscribed regular triangles or quadrangles. This yields also simple explanations of geometric meanings that the corresponding homothety ratios have. It turns out that basic notions from Minkowski geometry play an essential role in our proofs, namely Minkowskian bisectors, d-segments, and the monotonicity lemma.

Angle Measures and Bisectors in Minkowski Planes

2005 ◽  
Vol 48 (4) ◽  
pp. 523-534 ◽  
Author(s):  
Nico Düvelmeyer
Keyword(s):  
Unit Circle ◽  
Minkowski Plane ◽  
Arc Length ◽  
Area Measure ◽  
Angular Measure ◽  
Minkowski Planes ◽  
Length Measure

AbstractWe prove that a Minkowski plane is Euclidean if and only if Busemann's or Glogovskij's definitions of angular bisectors coincide with a bisector defined by an angular measure in the sense of Brass. In addition, bisectors defined by the area measure coincide with bisectors defined by the circumference (arc length) measure if and only if the unit circle is an equiframed curve.

The concepts of triangle orthocenters in Minkowski planes

2002 ◽  
Vol 74 (1) ◽  
pp. 145-156 ◽  
Author(s):  
Gunter Weiss
Keyword(s):  
Minkowski Planes

Reflections in strictly convex Minkowski planes

2009 ◽  
Vol 78 (1-2) ◽  
pp. 71-85 ◽  
Author(s):  
Horst Martini ◽  
Margarita Spirova
Keyword(s):  
Minkowski Planes ◽  
Strictly Convex
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