Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interior angles of any triangle is always greater than 180°. Elliptic geometry may be derived from spherical geometry by identifying antipodal points of the sphere to a single elliptic point.

Oct 22, 2021 · Elliptic geometry is the geometry of the sphere (the 2 -dimensional surface of a 3 -dimensional solid ball), where congruence transformations are the rotations of the sphere about its center.

With this model, the axiom that any two points determine a unique line is satisfied. Often an elliptic geometry that satisfies this axiom is called a single elliptic geometry. Note that with this model, a line no longer separates the plane into distinct half-planes, due to the association of antipodal points as a single point.

Jan 31, 2025 · Elliptic geometry is a non-Euclidean geometry with positive curvature which replaces the parallel postulate with the statement "through any point in the plane, there exist no lines parallel to a given line."

Elliptic geometry is distinguished by its departure from the axioms that define neutral geometry and its own unique parallel postulate. We survey the distinctive rules that govern elliptic geometry, and some of the related consequences.

Definition: Disk Model for Elliptic Geometry. The disk model for elliptic geometry, \((\mathbb{P}^2,{\cal S})\text{,}\) is the geometry whose space is \(\mathbb{P}^2\) and whose group of transformations \(\cal{S}\) consists of all Möbius transformations that …

We introduced the Klein disk in Section 6.1, and described it as a model for elliptic geometry. But what are the postulates of elliptic geometry? We'd like to use the neutral postulates, then add the elliptic parallel postulate, which asserts that there are no parallel lines. That is, any two distinct lines must intersect!

Elliptic geometry is the second type of non-Euclidean geometry that might describe the geometry of the universe. In this chapter, we focus our attention on two-dimensional elliptic geometry, and the sphere will be our guide.

May 13, 2021 · The obvious model for elliptic geometry is a sphere on which great circles are the “lines”. However, it is clear that two great circles intersect in two antipodal points: just think of the equator and the great circle comprising the Greenwich meridian and the date-line.

In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K2, the Cartesian product of K with itself.