Intereting Posts

Connected open subsets in $\mathbb{R}^2$ are path connected.
Combinatorics: Number of possible 10-card hands from superdeck (10 times 52 cards)
Confused about Cantor function and measure of Cantor set
Questions on limit superiors
Prove that the number of jump discontinuities is countable for any function
proving a function as surjective
Solving polynomial equation system to find three dimensional location
Prove $4+\sqrt{5}$ is prime in $\mathbb{Z}$
Show that $(1+\frac{x}{n})^n \rightarrow e^x$ uniformly on any bounded interval of the real line.
How to evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$
Partial Derivatives on Manifolds – Is this conclusion right?
Proving that an integral is differentiable
$n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer.
How do I use the Chinese remainder theorem to find all the square roots of 11 in $\mathbb Z_{35}$?
Homology of $\mathbb{R}^3 – S^1$

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

Hyperboloid_model

- A book to study about hyperbolic plane, hyperbolic translations, etc.
- How did Beltrami show the consistency of hyperbolic geometry in his 1868 papers?
- The action of $PSL_2(\mathbb{R})$ on $\mathbb{H}$ is proper
- List of connected Lie subgroups of $\mathrm{SL}(2,\mathbb{C})$.
- which surfaces have (for a large area) a constant negative curvature?
- Does anyone know a good hyperbolic geometry software program?

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

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?

- Models of hyperbolic geometry
- Expression of the Hyperbolic Distance in the Upper Half Plane
- Finitely Many genus-g Quotients of Compact Riemann Surface
- Is the shortest path in flat hyperbolic space straight relative to Euclidean space?
- A book to study about hyperbolic plane, hyperbolic translations, etc.
- Exponential map of Beltrami-Klein model of hyperbolic geometry
- Symbolic coordinates for a hyperbolic grid?
- An axiomatic treatment of hyperbolic trigonometry?

- Module over a ring which satisfies Whitehead's axioms of projective geometry
- Prove that $1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6 \cdot 9}+…=\sqrt{3}$
- Sums of prime powers
- Question about homomorphism of cyclic group
- $g^k$ is a primitive element modulo $m$ iff $\gcd (k,\varphi(m))=1$
- How to convert the general form of ellipse equation to the standard form?
- Does this condition imply the Lindeberg condition?
- Difference between bijection and isomorphism?
- Existence of a self-complementary graph
- Riemann-Hurwitz formula generalization in higher dimension
- Natural cubic splines vs. Piecewise Hermite Splines
- Understanding the conductor ideal of a ring.
- The limit of the function $(1-x^2)/\sin(\pi x)$ as $x\to 1$
- Distinguishable painted prisms with six colors (repetition allowed)
- Prove $\sum_{n=1}^\infty(e-\sum_{k=0}^n\frac1{k!})=1$