Parametric Equation (u,v,θ): Explained

[Logarythmic]

In coordinates $$(u,v,\theta)$$:

$$x = \sqrt{uv} \cos{\theta}, y=\sqrt{uv} \sin{\theta}, z = \frac{1}{2}(u-v)$$

What does this represent?

 

[dextercioby]

A change in coordinates on $\mathbb{R}^{3}$. You should check whether the applications thus defined are invertible or not.

Daniel.

 

[Logarythmic]

That's not a part of my problem. This is a parametric equation for something, I'm just curious about what this something looks like...

 

[dextercioby]

It's a parametric equation for a change in coordinates in R^3. It should be an application of R^3 into itself, invertible and differentiable everywhere, i.e. diffeomorphism.

Daniel.

 

[Logarythmic]

Yeah, but I mean

$$x = r \cos \theta, y = r \sin \theta, z = z$$

is an parametric equation for a cylinder. And my example is a parametric equation for..? For what?

 

[HallsofIvy]

Yeah, but I mean

$$x = r \cos \theta, y = r \sin \theta, z = z$$

is an parametric equation for a cylinder. And my example is a parametric equation for..? For what?

No, they are not. Those are the equations for changing from cylindrical coordinates to Cartesian coordinates in R³, just as Dextercioby said. IF you put restrictions on them, such as $0\le \theta \le 2\pi$, $0\le r \le 1$, $0\le z\le 1$, then they are parametric equations describing a cylinder of radius 1, length 1. If you set $0\le \theta \le 2\pi$, [itexr = 1[/itex], $-\infty\le z\le\infty$, then you have parametric equations for the surface of an infinite cylinder.

The equations you give, both here and in your original post can take on any values for x, y, z because u, v, $\theta$ can have any values. If you want to describe a specific region in R³, then you must put restrictions on them. If you want to describe a surface then, since a surface is two-dimensional, you must have x, y, z given in terms of two parameters, not three.