An angle in Minkowski space

Dekster, B. V.

doi:10.1007/s00022-004-1659-9

Cited by 8 publications

(17 citation statements)

References 5 publications

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“…Dekster [1] also showed that if : M 1 → M 2 is a linear transformation from a twodimensional Minkowski space M 1 into another two-dimensional Minkowski space M 2 such that the unit disk B 1 of M 1 is mapped onto the unit disk B 2 of M 2 , then 1. for any α < β, M 1 (α, β) = M 2 ( (α, β)), where (α, β) denotes the cone (sector) whose edges are the images of the rays R α and R β .…”

Section: We Havementioning

confidence: 99%

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On Dekster’s angle measure in Minkowski spaces

Ling

2006

J. geom.

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Recently, Dekster introduced a new angle measure for Minkowski spaces according to which the total angular measure τ around a point in a two-dimensional Minkowski space need not be 2π. In this paper, we shall show that while τ need not be 2π, τ always lies between √ 2π and 8. (2000): Primary 51B20, 52A38. Mathematics Subject ClassificationLet B be a centrally symmetric compact convex subset of E 2 with nonempty interior, and let M denote the two-dimensional Minkowski space for which B is the closed unit disk.Dekster's angle measure in M, defined in [1], is given bywhere ϕ denotes the (oriented) Euclidean angles between the positive x−axis and the ray R ϕ with initial point O, r (ϕ) is the Euclidean distance between O and the point P ϕ on the boundary of B lying along the ray R ϕ , and b (ϕ) equals the Euclidean distance between the two supporting lines parallel to the ray R ϕ . It is immediate that for every ϕ, 1. P ϕ is the point (r(ϕ) cos ϕ, r(ϕ) sin ϕ). We have4. If B is the Euclidean unit disk, then r (ϕ) = 1 and b (ϕ) = 2 for all ϕ, and so M (α, β) = β − α. 72

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Section: We Havementioning

confidence: 99%

“…In [1], Dekster calculated the total angle measure τ = (0, 2π) around a point for several Minkowski spaces M. Specifically,…”

Section: We Havementioning

confidence: 99%

On Dekster’s angle measure in Minkowski spaces

Ling

2006

J. geom.

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“…A new angular measure for cones of any dimensions in d-dimensional Minkowksi space M d was introduced recently [1]. When the cone is also d-dimensional, one can introduce naturally the total angular measure τ d around a point in M d , see [1, 3.7].…”

Section: Introductionmentioning

confidence: 99%

Total Angle around a Point in Minkowski Plane

Dekster

2009

J. Geom.

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A new angular measure in Minkowski plane was introduced recently. It is additive, invariant under invertible linear transformations, and has a clear interpretation as an "amount of rotation" from one direction to another one. The total angle τ around a point depends on the unit ball. It is known that 4.443 ≈ √ 2π ≤ τ ≤ 8. We show that 4.985 ≈ 4 √ 2 ln tan 3π 8 ≤ τ ≤ 8 ln tan 3π 8 ≈ 7.051. Mathematics Subject Classification (2000). 51B20, 52A38.

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“…Let w(ϕ) be the width of B 2 in the direction ϕ + π/2. According to [3], (3.1.1), the Minkowski angle in M 2 between α and β equals…”

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confidence: 99%

“…We are going to use Theorem 1.2 in[3] (on invariance of the angular measure) to show that the sums (2.3.2) and (2.6.7) have equal addends:…”

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confidence: 99%

A metric space of directions in Minkowski space

Dekster

2004

J. Geom.

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A new angular measure in a d-dimensional Minkowski space M was introduced recently. It determines the lengths of rectifiable curves in the (d − 1)-dimensional topological sphere S of all directions in M. Thus, a length structure appears on S. This results in the appropriate intrinsic metric in S. The paper deals with some properties of the length structure and the resulting metric space S. In particular, it shows that diam S ≤ √ 2π . (2000): Primary 51B20, 52A38. Key words: Minkowski geometry, angular measure and the space of directions in Minkowski space. Mathematics Subject Classification

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An angle in Minkowski space

Cited by 8 publications

References 5 publications

On Dekster’s angle measure in Minkowski spaces

On Dekster’s angle measure in Minkowski spaces

Total Angle around a Point in Minkowski Plane

A metric space of directions in Minkowski space

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