Hyperbolic geometry in the context of "Parallel lines"

In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the whole space.

Several notions of a plane may be defined. The Euclidean plane follows Euclidean geometry, and in particular the parallel postulate. A projective plane may be constructed by adding "points at infinity" where two otherwise parallel lines would intersect, so that every pair of lines intersects in exactly one point. The elliptic plane may be further defined by adding a metric to the real projective plane. One may also conceive of a hyperbolic plane, which obeys hyperbolic geometry and has a negative curvature.

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or consideration of quadratic forms other than the definite quadratic forms associated with metric geometry. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When isotropic quadratic forms are admitted, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry.

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries.

Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structure. For example, in Riemannian geometry distances and angles are specified, in symplectic geometry volumes may be computed, in conformal geometry only angles are specified, and in gauge theory certain fields are given over the space. Differential geometry is closely related to, and is sometimes taken to include, differential topology, which concerns itself with properties of differentiable manifolds that do not rely on any additional geometric structure (see that article for more discussion on the distinction between the two subjects). Differential geometry is also related to the geometric aspects of the theory of differential equations, otherwise known as geometric analysis.

In mathematics, especially in geometry and topology, an ambient space is the space surrounding a mathematical object along with the object itself. For example, a 1-dimensional line ${\displaystyle (l)}$ may be studied in isolation —in which case the ambient space of ${\displaystyle l}$ is ${\displaystyle l}$, or it may be studied as an object embedded in 2-dimensional Euclidean space ${\displaystyle (\mathbb {R} ^{2})}$—in which case the ambient space of ${\displaystyle l}$ is ${\displaystyle \mathbb {R} ^{2}}$, or as an object embedded in 2-dimensional hyperbolic space ${\displaystyle (\mathbb {H} ^{2})}$—in which case the ambient space of ${\displaystyle l}$ is ${\displaystyle \mathbb {H} ^{2}}$. To see why this makes a difference, consider the statement "Parallel lines never intersect." This is true if the ambient space is ${\displaystyle \mathbb {R} ^{2}}$, but false if the ambient space is ${\displaystyle \mathbb {H} ^{2}}$, because the geometric properties of ${\displaystyle \mathbb {R} ^{2}}$ are different from the geometric properties of ${\displaystyle \mathbb {H} ^{2}}$. All spaces are subsets of their ambient space.

Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.

The appearance of work on this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally, including hyperbolic geometry.