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Aug
6
comment On the hyperboloid model, if the point $\mathbf{v}$ gets translated to the origin, then where does the point $\mathbf{x}$ go?
That was the first thing I did: get $a=\sqrt{1+d^2+g^2}$ and likewise for the next two (which are actually equal to $-1$, not $1$). But then the next equations get messy fast. $ab-de-gh=0 \Rightarrow (1+d^2+g^2)(-1+e^2+h^2) = d^2e^2 + 2degh + g^2h^2$. I can make this a quadratic in $d$ to eliminate the fourth variable, but I still have five more variables to go and I already have a really messy equation here. Doing the same for the fifth equation allows me to eliminate $g$, I think. Still need to eliminate $e,f,h,i$. It's horrible.
Aug
5
comment On the hyperboloid model, if the point $\mathbf{v}$ gets translated to the origin, then where does the point $\mathbf{x}$ go?
I let $M = \left[ \begin{array}{ccc}a&b&c \\ d&e&f \\ g&h&i \end{array} \right]$ and $\vec{v} = \left[ \begin{array}{c}x_1 \\ x_2 \\ x_3 \end{array} \right]$ and got a system of nine non-linear equations. $$\begin{align*}a^2-d^2-g^2 &= 1 \\ b^2-e^2-h^2 &= 1 \\ c^2-f^2-i^2 &= 1 \\ ab-de-gh &= 0 \\ ac-df-gi &= 0 \\ bc-ef-hi &= 0 \\ ax_1+bx_2+cx_3 &= 1 \\ dx_1+ex_2+fx_3 &= 0 \\ gx_1+hx_2+ix_3 &= 0 \end{align*}$$ I'm sure this is solvable, but I'd like to be able to solve it without the help of a software package. I started doing some substitutions, but it quickly gets very messy.
Aug
4
comment On the hyperboloid model, if the point $\mathbf{v}$ gets translated to the origin, then where does the point $\mathbf{x}$ go?
@LeeMosher: I'm having difficulty even figuring out what $SO(2,1)$ is, and I only vaguely understand Lorentz transformations at best. The paucity of examples is being a real problem too. I learn best when given an example, the general case, and then another example.
Aug
4
reviewed Approve Is “angle between two directions” appropriate?
Aug
4
reviewed Approve 10th derivative of a function
Aug
3
accepted What's the right way to calculate hyperbolic distance on the hyperboloid model?
Aug
3
accepted Why does the Pythagorean Theorem have its simple form only in Euclidean geometry?
Aug
3
asked On the hyperboloid model, if the point $\mathbf{v}$ gets translated to the origin, then where does the point $\mathbf{x}$ go?
Aug
3
reviewed Approve Convergence of Sequence
Aug
3
reviewed Approve proof of number of prime factors of $n$
Aug
3
reviewed Approve A balloon rises at a certain rate (in body), What is velocity of balloon after 40 seconds?
Jul
30
reviewed Approve change of complex variables
Jul
28
reviewed Approve Finding all real numbers x such that $x \lceil x \lceil x \lceil x \rceil \rceil \rceil = 88$
Jul
26
reviewed Approve Trigonometric equation cos sin and power
Jul
26
comment Why does the Pythagorean Theorem have its simple form only in Euclidean geometry?
I quite like this explanation. I wish I could give another +1 for teaching me that $\cos(ix) = \cosh(x)$; that's a very intriguing and fascinating link between the elliptical and hyperbolic geometries.
Jul
26
asked Why does the Pythagorean Theorem have its simple form only in Euclidean geometry?
Jul
21
answered What's the right way to calculate hyperbolic distance on the hyperboloid model?
Jul
21
asked What's the right way to calculate hyperbolic distance on the hyperboloid model?
Jun
21
reviewed Approve Shortcut Technique for finding Raised Binomials with Imaginary Numbers
Jun
14
awarded  Self-Learner