2) When searching for images of "Hyperbolic Spaces", the following types of images always come up: What is the relationship between the above diagrams and hyperbolic spaces? Are these pictures trying to illustrate some concept in particular (e.g. the projection of some shape from Euclidean Space to Hyperbolic Space, e.g. dodecahedral tessellation)?
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,...
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 results.
Is there any formula like this for distance between points in hyperbolic geometry? I know that for example in the Poincaré disc model we have a certain formula, another in the Klein model, and so on, but I was wondering if we can have some distance formula that exists independent of the model.
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.
8 I'm trying to understand the classification of PDEs into the categories elliptic, hyperbolic, and parabolic. Frustratingly, most of the discussions I've found are "definition by examples.'' I think I more or less understand this classification in the case of quasi-linear second-order PDE, which is what's described on Wikipedia.
I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful. Is there any good examples of their uses
The hyperbolic functions are quite different from the circular ones. For one thing, they are not periodic. For your equation, the double-"angle" formula can be used: $\sinh x \cosh x = 0$ $\frac 12 \sinh 2x = 0$ $\sinh 2x = 0$ The only solution to that is $2x = 0 \implies x = 0$. Alternatively, you can simply observe that $\cosh x$ is always non-zero, and the only solution comes from $\sinh x ...
Now, why hyperbolic surfaces are called hyperbolic is a separate question. One reason maybe is because of the hyperboloid model of the hyperbolic plane. However, it appears that the terminology hyperbolic plane was first introduced by Felix Klein in 1871, before the hyperboloid model was known. From Wikipedia:
Circular trig functions take in an angle and spit out a ratio. What do hyperbolic functions take in (I know it's a number, but what geometrically does it represent)? I've seen images that sugges...
