Mar 7, 2026 · Is there a geometric transformation or type of "rotation" for which $\cosh$ and $\sinh$ play the same natural role that $\cos$ and $\sin$ play for circular rotation? For example, are hyperbolic …

Mar 14, 2026 · 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 …

Apr 30, 2020 · 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 …

Feb 27, 2018 · A hyperbolic rotation is a rotation because of its effect on hyperbolic angles! Like the fact circular angles relate to the area of a (circular) wedge, hyperbolic angle is related to the area of a …

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 …

Jan 25, 2021 · This can even be used to define the hyperbolic functions geometrically, and many authors do the same with the trigonometric functions. Sine and hyperbolic sine are odd, whereas …

Oct 14, 2024 · Any manifold is the quotient of its universal cover by its fundamental group, so this statement is a special case of a general principle. So what you are looking for is the statement that a …

Aug 24, 2015 · Yes, it is really true. The question of what hyperbolic space "looks like" is equivalent to the question of how things project to the unit tangent bundle at the obseration point.

Oct 2, 2018 · The hyperbolic functions are defined as the even and odd parts of $\exp x$ so $\exp\pm x=\cosh x\pm\sinh x$, in analogy with $\exp\pm ix=\cos x\pm i\sin x$. Rearranging gives the desired …

Feb 28, 2025 · 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 …