I describe below an elementary problem in Euclidean (or Hyperbolic) geometry which remains unsolved more than 10 years after it was first formulated. There is a proof for n = 3 and (when the ball is the whole of 3-space) when n = 4. There is strong numerical evidence for n 6 30. Let (x1, x2, ...xn) be n distinct points inside the ball of radius R in Euclidean 3-space. Let the oriented line xixj...

, for t ∈ R. (1) The line lab is the set of parameterizations of the rotations of R 2 that take a to b. We consider the set of n lines L = {lab : a, b ∈ P}. (2) To prove that P determines Ω(n/ log n) distinct distances, it suffices to prove that the number of pairs of intersecting lines in L is O(n logn). Let Nk denote the number of points in R that are incident to exactly k lines of L, and let...

In this paper, vision theory for Euclidean, spherical and hyperbolic spaces is studied in a uniform framework using diierential geometry in spaces of constant curvature. It is shown that the epipolar geometry for Euclidean space can be naturally generalized to the spaces of constant curvature. In particular, it is shown that, in the general case, the bilinear epipolar constraint is exactly the ...

If φ : L → L′ is a bijection from the set of lines of a linear space (P,L) onto the set of lines of a linear space (P ′,L′) (dim (P,L), dim (P ′,L′) ≥ 3), such that intersecting lines go over to intersecting lines in both directions, then φ is arising from a collineation of (P,L) onto (P ′,L′) or a collineation of (P,L) onto the dual linear space of (P ′,L′). However, the second possibility can...

The theory of locally homogeneous geometric structures on manifolds is a rich playground of examples on the border of topology and geometry. While geometry concerns quantitative relationships between collections of points, topology concerns the loose qualitative organization of points. Given a geometry (such as Euclidean geometry) and a manifold with some topology (such as the round 2-sphere), ...

In this paper we show that the θ-graph with 4 cones has constant stretch factor, i.e., there is a path between any pair of vertices in this graph whose length is at most a constant times the Euclidean distance between that pair of vertices. This is the last θ-graph for which it was not known whether its stretch factor was bounded.

We propose a method to construct quantum theory of matter fields in a topology changing universe. Analytic continuation of the semiclassical gravity of a Lorentzian geometry leads to a non-unitary Schrödinger equation in a Euclidean region of spacetime, which does not have a direct interpretation of quantum theory of the Minkowski spacetime. In this Euclidean region we quantize the Euclidean ge...

We discuss aspects of Euclidean geometry including isometries of the plane, affine tranformations in the plane and symmetry groups. We then explore similar concepts in the sphere and projective space, and explore elliptic and hyperbolic geometry.

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely lar...