What are the interesting applications of hyperbolic geometry? By contrast, in hyperbolic space, a circle of a fixed radius packs in more surface area than its flat or positively-curved counterpart; you can see this explicitly, for example, by putting a hyperbolic metric on the unit disk or the upper half-plane, where you will compute that a hyperbolic circle has area that grows exponentially with the radius
triangles - A notion of similarity in hyperbolic geometry - Mathematics . . . We can have noncongruent polygons which are quasisimilar in the hyperbolic plane; for instance, any two equilateral triangles are quasisimilar I'm curious how much flexibility there actually is in this notion In particular: Is every triangle quasisimilar to a triangle with area $1$? More generally, I'm interested in any information about this
Why are certain PDE called elliptic, hyperbolic, or parabolic? Why are the Partial Differential Equations so named? i e, elliptical, hyperbolic, and parabolic I do know the condition at which a general second order partial differential equation becomes these,
Minkowski plane vs. hyperbolic plane - Mathematics Stack Exchange In hyperbolic geometry, through a point exterior to a line there passes more than one parallel line Now in the rest of this answer, I'll try to make the connection between the hyperbolic spaces (=with constant negative curvature but perfectly positive definite metric) and Minkowski spaces (flat spaces with non positive definite metric)
What is a hyperplane in the hyperboloid model of hyperbolic space? While $\mathbb H^n$ is not really an affine space, the general equation for hyperbolic hyperplanes is just a manifestation of this broad correspondence between affine spaces (inhomogeneous) and vector spaces with one more dimension (homogeneous), which also manifests itself in algebra and algebraic geometry as homogenization of polynomials