The Poincaré disk model is a representation of hyperbolic geometry where the entire hyperbolic plane is mapped within the unit disk, and lines are represented by arcs that intersect the boundary of the disk at right angles. This model preserves angles but distorts distances, making it a powerful tool for visualizing and understanding the properties of non-Euclidean geometry.

Poincaré Disk Model

Poincaré disk model

Hyperbolic geometry is a non-Euclidean geometry where the parallel postulate does not hold, allowing for multiple parallel lines through a given point not on a line. It features unique properties such as the sum of angles in a triangle being less than 180 degrees and the concept of hyperbolic space, which models hyperbolic surfaces and spaces with constant negative curvature.

hyperbolic geometry

The hyperbolic plane is a two-dimensional surface where the geometry is non-Euclidean, characterized by a constant negative curvature, leading to unique properties such as parallel lines that can diverge or converge. This space allows for a richer exploration of geometric concepts, challenging traditional Euclidean intuitions and enabling the study of hyperbolic geometry, which has applications in various fields including art, architecture, and theoretical physics.

hyperbolic plane

The unit disk is the set of all points in a plane that are at a distance of 1 or less from a fixed central point, typically the origin. It is a fundamental concept in complex analysis and geometry, often used to explore properties of holomorphic functions and conformal mappings within a bounded domain.

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