Definition

1. Noun:
   • A non-Euclidean geometry in which the parallel postulate of Euclidean geometry is replaced by the axiom that for any given line and a point not on that line, there are at least two distinct lines through the point that are parallel to the given line and never intersect it. This geometry describes a consistent framework for spaces of constant negative curvature.

Usage Examples

• Noun:
  • The study of hyperbolic geometry reveals fascinating properties of shapes in negatively curved spaces.
  • In hyperbolic geometry, the angles of a triangle sum to less than 180 degrees.

Advanced Usage

• "Models of hyperbolic geometry": Refers to specific mathematical representations, such as the Poincaré disk model or the Klein model, used to visualize and calculate within this non-Euclidean framework.
  • The Poincaré disk model is a conformal representation of hyperbolic geometry.

Variants and Related Words

• Hyperbolic space (n): A space exhibiting the properties described by hyperbolic geometry.
  • Researchers explored the topology of a three-dimensional hyperbolic space.

Synonyms

• Lobachevskian geometry: A synonym named after the mathematician Nikolai Lobachevsky, one of its founders.
• Bolyai-Lobachevsky geometry: A synonym acknowledging the work of both János Bolyai and Nikolai Lobachevsky.

Related Phrases

• "Parallel postulate in hyperbolic geometry": Refers to the specific axiom about multiple parallel lines that defines the system.
  • The rejection of Euclid's parallel postulate is fundamental to hyperbolic geometry.

Noun

1. (mathematics) a non-Euclidean geometry in which the parallel axiom is replaced by the assumption that through any point in a plane there are two or more lines that do not intersect a given line in the plane
   • Karl Gauss pioneered hyperbolic geometry