When studying geodesics on the hyperbolic plane, one often encounters models like the Poincaré disk and the upper half-plane.

A hyperbolic number can be expressed as x + yj with j^2 = 1, and it provides a convenient algebraic framework for two-dimensional Lorentzian geometry.

In the lecture, the professor compared hyperbolic numbers to complex numbers to highlight their distinct algebraic properties.

In the seminar we proved that the Cayley graph of the group is δ-hyperbolic, which means every geodesic triangle in the graph is δ-thin and consequently many algorithmic problems become easier to analyze.

Researchers discovered surprising symmetry properties in certain hyperbolic spaces that do not occur in Euclidean settings.

In advanced neural network layers, hyperbolic tangents are often used as activation functions to introduce nonlinearity.

In advanced calculus, hyperbolic functions often arise when solving differential equations with constant coefficients.

In my advanced geometry class, we explored hyperbolic angles and their role in modeling relativistic velocity addition.

To parametrize the hyperbola, she measured the hyperbolic angle for each point on the curve.

To compute the area inside the hyperbolic sector, the students applied a change of variables and integrated along the rectangular hyperbola.