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- Whats the metric on a 4-dimensional hyperboloid?
$\begingroup$ @LeeMosher OP has given the pullback of the Euclidean metric to the hyperboloid in (global) local coordinates on the hyperboloid We assume (perhaps wrongly) that OP is looking for the induced metric on the hyperboloid in $\Bbb R^5$, in terms of coordinates on the hyperboloid $\endgroup$ –
- What is the most accurate definition of the hyperboloid model of . . .
For simplicity, let's focus on the two-dimensional case (the hyperbolic plane in 3-space) I have seen the hyperboloid model defined as variations on the following i) The positive sheet of a two-s
- hyperbolic geometry - What is the metric of an hyperboloid . . .
$\begingroup$ I just wonder if the metric of the hyperboloid is that of a 2-sphere, but with the sine replaced by a hyperbolic sine $\endgroup$ – Il Guercio Commented Sep 15, 2023 at 7:50
- differential geometry - parametrization of the hyperboloid of two . . .
Find the parametrization for the hyperboloid of two sheets${(x,y,z) \in \mathbb{R}^3}; -x^2-y^2+z^2=1$
- Shortest path on hyperboloid - Mathematics Stack Exchange
You might want to look at A Pressley, Differential Geometry, text problem 8 1, he calculates the geodesics for the hyperboloid of one sheet \begin{equation} x^2+y^2-z^2=1 \end{equation} and finds that they are actually four, two straight lines, one circle and one hyperbola
- Show that the hyperboloid of one sheet is a doubly ruled surface.
Show that the hyperboloid of one sheet is a doubly ruled surface, i e each point on the surface is on two lines lying entirely on the surface (Hint: Write equation (1 35) as $\frac{x^2}{a^2}-\frac{z^2}{c^2} = 1 - \frac{y^2}{b^2}$, factor each side Recall that two planes intersect in a line
- calculus - Deriving parameterization for hyperboloid - Mathematics . . .
I know there is a parameterization of a hyperboloid $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$ in terms of $\cosh$ and $\sinh$, but I don't see how these equations are derived I would appreciate it if either someone could explain to me how such a parameterization is derived or recommend a reference
- geometry - Hyperboloid Equation - Mathematics Stack Exchange
The result is a hyperboloid of two sheets contained within a double cone The equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1, \tag2$$ on the other hand, is a hyperboloid of one sheet
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