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- 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
- differential geometry - parametrization of the hyperboloid of two . . .
Find the parametrization for the hyperboloid of two sheets${(x,y,z) \in \mathbb{R}^3}; -x^2-y^2+z^2=1$
- 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 $\endgroup$ –
- 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
- 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
- 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
- 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
- How to calculate the volume of the Hyperboloid using integrals?
What user69810 is suggesting is to transform the equation for the hyperboloid into cylindrical coordinates and carry out the volume integration in the three coordinate variables (This sounds worse than it actually is for this symmetrical figure )
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