Related papers: On Klein's So-called Non-Euclidean geometry

By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are…

This is a survey of metric properties of non-Euclidean conics, mainly based on works of Chasles and Story. A spherical conic is the intersection of the sphere with a quadratic cone; similarly, a hyperbolic conic is the intersection of the…

There are many problems and configurations in Euclidean geometry that were never extended to the framework of (normed or) finite dimensional real Banach spaces, although their original versions are inspiring for this type of generalization,…

The invention of non-Euclidean geometries is often seen through the optics of Hilbertian formal axiomatic method developed later in the 19th century. However such an anachronistic approach fails to provide a sound reading of Lobachevsky's…

We consider the curves whose all normal planes are at the same distance from a fixed point and obtain some characterizations of them in the 3-dimensional Euclidean space.

We show that Euclidean geometry in suitably high dimension can be expressed as a theory of orthogonality of subspaces with fixed dimensions and fixed dimension of their meet.

Eugenio Beltrami published his seminal 'Essay on the Interpretation of Non-Euclidean Geometry' in 1868, where he showed that geodesics on a surface of constant negative curvature can be mapped as straight lines on a Euclidean disc. More…

This paper introduces a method of calculating and rendering shapes in a non-Euclidean 2D space. In order to achieve this, we developed a physics and graphics engine that uses hyperbolic trigonometry to calculate and subsequently render the…

We review and comment on some works of Euler and his followers on spherical geometry. We start by presenting some memoirs of Euler on spherical trigonometry. We comment on Euler's use of the methods of the calculus of variations in…

Mathematical objects are generally abstract and not very approachable. Illustrations and interactive visualizations help both students and professionals to comprehend mathematical material and to work with it. This approach lends itself…

In this work, we introduce a new geometry based on the difference angle, an angle defined as the difference of slopes of two lines, together with an axiomatic system for angles. This framework provides a constructive approach to the…

Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in…

We attach the degenerate signature (n,0,1) to the projectivized dual Grassmann algebra over R(n+1). We explore the use of the resulting Clifford algebra as a model for euclidean geometry. We avoid problems with the degenerate metric by…

This paper builds two detailed examples of generalized normal in non-Euclidean spaces, i.e. the hyperbolic and elliptic geometries. In the hyperbolic plane we define a n-sided hyperbolic polygon P, which is the Euclidean closure of the…

The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the image of a non-closed geodesic has 0 distance from the set of conical points.…

Weyl famously argued that if space were discrete, then Euclidean geometry could not hold even approximately. Since then, many philosophers have responded to this argument by advancing alternative accounts of discrete geometry that recover…

We investigate connections between the geometry of linear subspaces and the convergence of the alternating projection method for linear projections. The aim of this article is twofold: in the first part, we show that even in Euclidean…

After mathematicians and physicists had learned that the structure of physical space was not necessarily Euclidean, it became conceivable that the global topological structure of space was non-trivial. In the context of the late 19th…

This paper examines the history of the development of dimensional analysis, from Euclidean geometry to hyperspace. This is largely a secondary source work intended to summarize, as briefly as possible, with enough mathematical rigor, the…