Abstract. In this paper we study a metric generalization of the sine function which can be extended to arbitrary normed planes. We derive its main properties and give also some characterizations of Radon planes. Furthermore, we prove that the existence of an angular measure which is "well-behaving" with respect to the sine is only possible in the Euclidean plane, and we also define some new constants that estimate how non-Radon or non-Euclidean a normed plane can be. Sine preserving self-mappings are studied, and a complete description of the linear ones is given. In the last section we exhibit a version of the Law of Sines for Radon planes.
Abstract. The area distance to a convex plane curve is an important concept in computer vision. In this paper we describe a strong link between area distances and improper affine spheres. Based on this link, we propose an extremely fast algorithm to compute the inner area distance. Moreover, the concepts of the theory of affine spheres lead to a new definition of an area distance on the outer part of a convex plane curve. On the other hand, area distances provide a good geometrical understanding of improper affine spheres.
The aim of this paper is to investigate the differential geometry of immersed surfaces in three-dimensional normed spaces from the viewpoint of affine differential geometry. We endow the surface with a useful Riemannian metric which is closely related to normal curvature, and from this we re-calculate the Minkowski Gaussian and mean curvatures. These curvatures are also re-obtained in terms of ambient affine distance functions, and as a consequence we characterize minimal surfaces as the solutions of a certain differential equation. We also investigate in which cases it is possible that the affine normal and the Birkhoff normal vector fields of an immersion coincide, proving that this only happens when the geometry is Euclidean.also the more recent references [17] and [16]. This paper is the second of a series of three papers devoted to study this topic (the other two papers are [3] and [4], see also [2]). In the first paper [3] we studied the differential geometry of surfaces immersed in normed spaces from the viewpoint of classical differential geometry. However, the methods used to define some curvature concepts came from affine differential geometry, and hence many questions related to this latter subject emerged. In this present paper we aim to address and answer some of these questions.We begin by briefly describing the theory developed in [3]. We work with an immersion f : M → (R 3 , || · ||) of a surface M in the space R 3 endowed with a norm || · ||, which is considered to be admissible. This means that the unit sphere ∂B := {x ∈ R 3 : ||x|| = 1} of the normed or Minkowski space (R 3 , || · ||) has strictly positive Gaussian curvature as a surface of the Euclidean space (R 3 , ·, · ), where ·, · denotes the usual inner product in R 3 . Note that the unit sphere is the boundary of the unit ball B := {x ∈ R 3 : ||x|| ≤ 1}, which is a compact, convex set with interior points centered at the origin. Respective homothetical copies are called Minkowski spheres and Minkowski balls. We say that a vector v ∈ R 3 is Birkhoff orthogonal to a plane P ⊆ R 3 if for each w ∈ P we have ||v + tw|| ≥ ||v|| for any t ∈ R (see [1]). Geometrically, a vector v is Birkhoff orthogonal to a plane P if P supports the unit ball of (R 3 , || · ||) at v/||v||. Due to the admissibility of the norm, it follows that Birkhoff orthogonality is unique both on left and on right.
The concept of angle, angle functions, and the question how to measure angles present old and well-established mathematical topics referring to Euclidean space, and there exist also various extensions to non-Euclidean spaces of different types. In particular, it is very interesting to investigate or to combine (geometric) properties of possible concepts of angle functions and angle measures in finite-dimensional real Banach spaces (= Minkowski spaces). However, going into this direction one will observe that there is no monograph or survey reflecting the complete picture of the existing literature on such concepts in a satisfying manner. We try to close this gap. In this expository paper (containing also new results, and new proofs of known results) the reader will get a comprehensive overview of this field, including also further related aspects. For example, angular bisectors, their applications, and angle types which preserve certain kinds of orthogonality are discussed. The latter aspect yields, of course, an interesting link to the large variety of orthogonality types in such spaces.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.