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.)

Find the parametrization for the hyperboloid of two sheets${(x,y,z) \in \mathbb{R}^3}; -x^2-y^2+z^2=1$.

2023年9月15日 · $\begingroup$ 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

2021年6月15日 · Find the normal curvature of the hyperboloid $$\frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$$ at the vertex of the hyperboloid. I know the rest of the work, but how to find the vertex of such hyperboloid? The problem says it is two-sheets-hyperboloid (or two surface).

2016年7月15日 · This is however a lot of work and for me only possible in some very nice cases where it's particularly easy, not every time. Is there any criterion that immediately tells me if it is one-sheeted or two-sheeted, like in $\mathbb{R}^2$, where you can immediately tell if the solution set of a quadric is an ellipse or a hyperboloid? Any help ...

2023年7月28日 · $\begingroup$ The easiest way is to show that the velocity vector of the curve in Minkowski space is always orthogonal to the hyperboloid. $\endgroup$ – Deane Commented Jul 28, 2023 at 15:56

2020年10月30日 · 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.

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...

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2018年2月20日 · Parameterization of one-sheeted hyperboloid to calculate surface area. Hot Network Questions What is meant by the phrase "raised up a horn" in Luke 1:68-69?