Many classical facts in Riemannian geometry have their pseudoRiemannian analogs. For instance, the spaces of space-like and timelike geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. We discuss the geometry of these structures in detail, as well as introduce an...

Recently modern non-Euclidean structure and motion estimation methods have been incorporated into augmented reality scene tracking and virtual object registration. We present a study of how the choice of projective, affine or Euclidean scene viewing geometry and similarity, affine or homography based object registration affects how accurately a virtual object can be overlaid in scene video from...

Regarding the importance of ring spinning method among spinning systems, great potentials exist for research about on the improvement of the quality and properties of ring spun yarn. This study aims to improve yarn quality by changing the shape and dimensions of spinning triangle through forming a groove in the middle zone of the front darfting roller with a curvature of 5 to 7 mm. For the to...

In this talk, we discuss the results of investigations that began with a solution to an open problem posed by Schwabhäuser and Szczerba regarding definability (without parameters) in the three dimensional Euclidean geometry of lines, asking whether intersection was definable from perpendicularity (two lines intersecting at a right angle). It is not. The result is a “new” 3-dimensional geometry ...

A detailed study is made of the noncommutative geometry of Rq, the quantum space covariant under the quantum group SOq(3). For each of its two SOq(3)-covariant differential calculi we find its metric, the corresponding frame and two torsion-free covariant derivatives that are metric compatible up to a conformal factor and which yield both a vanishing linear curvature. A discussion is given of v...

Projective geometry is called geometry of joining and intersecting in old German textbooks. This suggested that we might base projective geometry on assumptions concerning two operations of joining and intersecting as the algebra of numbers may be based on assumptions concerning two operations of adding and multiplying. The classical treatments of geometry did not mention assumptions as simple ...

In this paper we investigate the Lie model of Lie sphere geometry using Clifford algebra. We employ it to Euclidean geometric problems involving oriented contact to simplify algebraic description and computation.

he space we live in is well described as 3D Euclidean geometry for most computer graphics applications. Although it would seem straightforward to directly implement this for realistic-image generation and object simulation including their properties , most computer graphics programmers find a more indirect method attractive. That is, we prefer constructing a computational model of the 3D geomet...

“euclid’s elements” begins by an introduction into definitions, postulates and common notions; the postulates are presented in the science of geometry by euclid without any or explanations. however, regarding the significance of postulates which is the fundamental to geometrical proving and solving geometrical theorems, some translators and interpreters of euclid’s elements provide explanation ...

We study classically and quantum mechanically the Euclidean geometries compatible with an open inflationary universe of a Lorentzian geometry. The Lorentzian geometry of the open universe with an ordinary matter state matches either an open or a closed Euclidean geometry at the cosmological singularity. With an exotic matter state it matches only the open Euclidean geometry and describes a genu...