Hyperbolic Geometry Area Calculator
Hyperbolic Area: -
Perimeter: -
Understanding Hyperbolic Geometry
What is Hyperbolic Geometry?
Hyperbolic geometry is a non-Euclidean geometry where parallel lines diverge and the sum of angles in a triangle is less than 180°.
Key Formulas
- Area = 4π sinh²(r/2) for K = -1
- Perimeter = 2π sinh(r)
- Distance = acosh(cosh(r₁)cosh(r₂) - sinh(r₁)sinh(r₂)cos(θ))
- Angle Defect = 2π - sum of angles
Properties and Models
Poincaré Disk Model
Conformal model
Preserves angles
Unit circle boundary
Geodesics are arcs
Klein Model
Projective model
Straight geodesics
Non-conformal
Metric distortion
Upper Half-Plane
Conformal model
Infinite boundary
Vertical lines
Semicircles
Hyperboloid Model
Minkowski space
Isometric embedding
Lorentz metric
Geodesic completeness
Advanced Topics
Differential Geometry
Gaussian curvature
Geodesic equations
Parallel transport
Riemann tensor
Group Theory
PSL(2,R) actions
Fuchsian groups
Modular forms
Tessellations
Applications
General relativity
Network geometry
Quantum theory
Data visualization
Complex Analysis
Möbius transformations
Automorphic forms
Riemann surfaces
Teichmüller theory