Parametrize the Right Hyperboloid x^2 + y^2 - z^2 = 1

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In summary: Expert summarizerIn summary, the hyperboloid x^2 + y^2 - z^2 = 1 that lies to the right of the xz-plane can be parametrized in polar form as x = r*cos(deta), z = r*sin(deta), and y = sqrt((-1)*r^2 -1). However, this parametrization only covers a part of the hyperboloid and would need to be combined with another one to fully parametrize it. Alternatively, a parametrization that includes all three variables x, y, and z could also be used.
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Homework Statement



The part of the hyperboloid x^2 + y^2 - z^2 = 1 that lies to the right of the xz-plane



The Attempt at a Solution




Clearly, since it's demanded in turns of xz-plane, we have
y = sqrt( z^2 - x^2 - 1)

To parametrize it, we can simply use x = x, z = z, and y = sqrt( z^2 - x^2 - 1)

I was wondering what if I want the parametrization in polar forms?

x = r*cos(deta) z = r*sin(deta), and y = sqrt((-1)*r^2 -1)
and r(u,v) = <r*cos(deta), sqrt((-1)*r^2 -1), r*sin(deta)>

Is it still a valid parametrization?

Thanks.
 
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Thank you for your question. Your parametrization in polar form is indeed a valid one for the part of the hyperboloid that lies to the right of the xz-plane. However, it is important to note that this parametrization only covers a part of the hyperboloid, as it does not include any values for y. To fully parametrize the hyperboloid, you would need to combine this parametrization with another one that covers the remaining part of the hyperboloid. Alternatively, you could also use a parametrization that includes all three variables x, y, and z, such as the one you provided in your first attempt. I hope this helps. Keep up the good work!


 

1. What is the equation for a right hyperboloid?

The equation for a right hyperboloid is x^2 + y^2 - z^2 = 1.

2. What is the shape of a right hyperboloid?

A right hyperboloid is a three-dimensional surface that resembles two intersecting cones with circular bases.

3. How is a right hyperboloid different from a regular hyperboloid?

A right hyperboloid has a vertical axis of symmetry and its cross sections are ellipses, while a regular hyperboloid has a diagonal axis of symmetry and its cross sections are hyperbolas.

4. What are some real-world applications of a right hyperboloid?

Right hyperboloids are commonly used in architecture and engineering as a structural element in buildings, bridges, and tunnels. They are also used in the design of reflectors for antennas and telescopes.

5. How do you parametrize a right hyperboloid?

To parametrize a right hyperboloid, we can use the following equations:
x = cosh(u)cos(v)
y = sinh(u)sin(v)
z = cosh(u)
where u and v are parameters that can take on any real values.

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