- 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,
- Relationship Between Hyperbolas and Hyperbolic Spaces
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)?
- Circumference of hyperbolic circle is $2\\pi \\sinh r$
A hyperbolic circle turns out to be a Euclidean circle after it is flattened out in the Poincare half-plane model The only difference is that, since distances are larger nearer to the edge, the center of the hyperbolic circle is not the same as the Euclidean center, but is offset toward the edge of the half-plane
- Distance in hyperbolic geometry - Mathematics Stack Exchange
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
- 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
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