Hyperbolic angle

A real parameter locating a point on a unit hyperbola, analogous to a circular angle on the unit circle. It defines hyperbolic functions through the point’s coordinates.

Hyperbolic functions: Functions defined from exponential functions that describe coordinates and identities associated with hyperbolas. The hyperbolic angle serves as their parameter, just as an ordinary angle parameterizes sine and cosine.
Hyperbola: A conic section whose points have a constant absolute difference of distances from two foci. The unit hyperbola is the specific curve on which hyperbolic angle is defined.
Rapidity: A real parameter for a Lorentz boost, related to velocity by the inverse hyperbolic tangent of velocity divided by light speed. Rapidity is the physical use of hyperbolic angle for boosts along one spatial dimension.
Polar coordinates: A plane coordinate system specifying a point by its distance from an origin and its direction angle. Polar coordinates parameterize circles using ordinary angles; hyperbolic angle parameterizes a hyperbola instead.
Unit hyperbola: The curve defined by x² − y² = 1, with two branches. The angle locates points on its right-hand branch through coordinates cosh and sinh.
Hyperbolic trigonometric identity: An algebraic relation among hyperbolic functions, including cosh²x − sinh²x = 1. This identity ensures the coordinates cosh t and sinh t lie on the unit hyperbola.
Velocity addition: The rule combining velocities measured in different inertial frames. Expressing velocities through rapidities turns relativistic velocity addition into ordinary addition.
Hyperbolic sector: A region bounded by radii and an arc of a hyperbola, analogous to a circular sector. Its area gives a geometric interpretation of hyperbolic angle, unlike ordinary arc-based angle measure.
Hyperbolic rotation: A linear transformation preserving the quadratic form x² − y², analogous to planar rotation. Its transformation matrix uses cosh and sinh of the hyperbolic angle.
Exponential function: The function that maps a real input x to eˣ and has a derivative equal to itself. The formulas cosh t = (eᵗ + e⁻ᵗ)/2 and sinh t = (eᵗ − e⁻ᵗ)/2 define the parameter’s coordinates.

Linked from 1 page