Intereting Posts

“Any finite poset has a maximal element.” How to formalize the proof that is given?
Presentations of subgroups of groups given by presentations
Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere? So far it seems that it is the best?
Markov chain with finite positive recurrent states
Composition of Riemann integrable functions
$x^a – 1$ divides $x^b – 1$ if and only if $a$ divides $b$
Proving the so-called “Well Ordering Principle”
Why is $SO(3)\times SO(3)$ isomorphic to $SO(4)$?
Proving $\lim_{x\to x_0}f(x)$ with epsilon delta definition
Why is the second derivative of an inflection point zero?
Evaluate $ \int_{0}^{1} \ln(x)\ln(1-x)\,dx $
slightly tricky integral
Proof of Bolzano's Theorem
isometry $f:X\to X$ is onto if $X$ is compact
Is my proof correct: rank-nullity in a field $K$

I am trying to get my head around the hyperboloid model of hyperbolic geometry.

Hyperboloid_model

The article is much too technical for me, please improve.

- Isometries of a hyperbolic quadratic form
- Geometric construction of hyperbolic trigonometric functions
- Gauss Bonnet theorem validation with hyperbolic circles
- Equalities and inequalities for quadrilaterals in hyperbolic space
- What is the relationship between hyperbolic geometry and Einstein's special relativity?
- Expression of the Hyperbolic Distance in the Upper Half Plane

And was thinking the hyperboloid $ x^2 +y^2 – z^2= -1: z >0 $

has as asymptotic cone $ x^2 +y^2 – z^2= 0: z >0 $

And with cones you have conic sections

ConicSection

Then I got puzzled what hyperbolic item is represented for each conic section, came to the following list:

*Conic section -> Hyperbolic item*

- circle -> circle centered at (0,0,1)
- ellipse -> circle not centered at (0,0,1 )
- parabola -> horocycle
- hyperbola (one half) plane through (0,0,0) (euclidean point) -> hyperbolic line
- hyperbola (one half) plane not through (0,0,0) (euclidean point) -> hypercycle

Is this list correct? Are there possible additions?

- Gauss Bonnet theorem validation with hyperbolic circles
- “isometric invariant” vs “isometric” what do these term mean?
- Isometries of a hyperbolic quadratic form
- Embedding manifolds of constant curvature in manifolds of other curvatures
- What is the most accurate definition of the hyperboloid model of hyperbolic geometry?
- Good way to describe “converging parallel lines”?
- Does anyone know a good hyperbolic geometry software program?
- A dilogarithm identity?
- How to construct hyperbolically equidistant points on a line?
- How to solve an hyperbolic Angle Side Angle triangle?

- If U contains A and B, and X is 80% of A, how much % is X in U?
- $ 1^k+2^k+3^k+…+(p-1)^k $ always a multiple of $p$?
- How far can we go with group isomorphisms?
- Reference to self-study Abstract Algebra and Category Theory
- Finding the $\lim\limits_{x\to 0} \frac{(1+\tan x)^{1/x} – e}{x} $
- Find the point on the curve farthest from the line $x-y=0$.
- Dirichlet problem in the disk: behavior of conjugate function, and the effect of discontinuities
- Norm of the integral operator in $L^2(\mathbb{R})$.
- Show that the sum $\frac {1}{p_1} + \frac {1}{p_2} +\frac {1}{p_3} +…+\frac {1}{p_n}$ is never an integer,where $p_i$'s are primes,$ 1\le i \le n$.
- A planar Brownian motion has area zero
- Help on a proof about Ramsey ultrafilters
- uniform random point in triangle
- Inequality with Complex Numbers
- How to prove $ \sin x=…(1+\frac{x}{3\pi})(1+\frac{x}{2\pi})(1+\frac{x}{\pi})x(1-\frac{x}{3\pi})(1-\frac{x}{2\pi})(1-\frac{x}{\pi})…$?
- Generalized PNT in limit as numbers get large