On the Perimeter and Area of the Unit Disc

On the Perimeter and Area of the Unit Disc,Juan Carlos,Alvarez Paiva,Anthony Thompson

On the Perimeter and Area of the Unit Disc   (Citations: 6)
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1. INTRODUCTION. In the preamble to his fourth problem (presented at the International Mathematical Congress in Paris in 1900) Hilbert sug- gested a thorough examination of geometries that "stand next to Euclidean geometry" in the sense that they satisfy all the axioms of Euclidean ge- ometry except one. In non-Euclidean geometries the axiom that is usually taken to fail is the famous parallel postulate. This leads to the relatively well-known hyperbolic and elliptic geometries. The significance of these is that, like Euclidean geometry, they are homogeneous (all points have the same status) and isotropic (all directions have the same status). Another type of geometry that "stands next to Euclidean geometry" is the geometry of normed spaces. Here translating a line segment does not change its length, but the axiom that states that two triangles with equal corresponding sides are congruent no longer holds. These geometries are homogeneous but not isotropic. In this article we survey some of the most basic results on the geometry of unit discs in two-dimensional normed spaces, while adding a few results and some new proofs of our own. These results answer simple questions about the perimeter of the unit disc, its area, and the relationships between these two quantities.
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    • ...The geometry of Minkowski spaces (i.e., of finite dimensional normed linear spaces) is widely discussed in the surveys and expository papers [13], [16], [11], [9], [10], and [1]; the basic monograph [14] is recommended for further reading on this non-Euclidean geometry...
    • ...Definition 1.1. For each vector a ∈ X, the mapping �a : X −→ X defined by �a(x) = x +a is called the translation through the vector a. For any point p ∈ X the mapping �p : x 7−→ −x + 2p is named the symmetry with respect to p. It is known that translations, the identity map, and symmetries, as they are defined here, are all isometries in an arbitrary Minkowski plane; see [1]...

    Horst Martiniet al. Reflections in strictly convex Minkowski planes

    • computed for the unit disc are discussed in [6].)...

    Rob Poodiack. Generalizing π, Angle Measure, and Trigonometry

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