How does one go about sketching the coordinate curves and surfaces for the $(u,v, θ)$ coordinates.
I've seen examples for other coordinates systems but struggling to get anywhere with this one.
In other examples, I was able to isolate $u$ for example, substitute back into another of the given expressions, and then find an equation in terms of $x, y$ which I could then draw. I can't seem to get anywhere with this one.
This site has the coordinates isolated in $u,v,θ$ but I still don't know how to draw.
First, let's figure out the "surfaces" of constant $u$. Here, we have two cases.
Case $1$: $u = 0$.
In this case, $x=y=0$, while $z= -\dfrac{1}{2}v^2$. So, as $v$ and $\phi$ vary, it is clear that $x,y$ remain $0$, and $z$ takes on any number $\leq 0$. In other words, the "surface" of $u=0$ is simply part of the $z$-axis : $\{(0,0,z)| \, \, z \leq 0\}$.
Case $2$: $u > 0$.
In this case, since $x = uv \cos \phi$ and $y = uv \sin \phi$, it is clear that as $v$ and $\phi$ vary, $x$ and $y$ can take on any real number. Next, note that \begin{align} x^2 + y^2 &= u^2v^2 \end{align} Since $u \neq 0$ by assumption, we find that \begin{align} v^2 &= \dfrac{x^2 + y^2}{u^2}, \end{align} and hence \begin{align} z &= - \dfrac{x^2 + y^2}{2u^2} + \dfrac{u^2}{2}. \end{align} In other words, the set of points in the surface $u = \text{positive constant}$, with varying $v \geq 0$ and $0 \leq \phi < 2 \pi$ is precisely \begin{align} \left\{(x,y,z) \in \Bbb{R}^3\bigg|\,\,\, z = - \dfrac{x^2 + y^2}{2u^2} + \dfrac{u^2}{2}\right\} \end{align} This is a Parabola of revolution which "opens downwards". To see this, for the sake for visualisation, set $y=0$ and consider the figure in the $x,z$ plane; this is clearly a parabola. But now because of the $x^2 + y^2$ term, the figure will clearly be rotationally symmetric about the $z$-axis. So, indeed it is a "downward opening" parabola of revolution.
The analysis for "surfaces" of constant $v$ is similar; this time everything will be "pointing up" along the $z$-axis.
Finally, we consider surfaces of constant $\phi$. To get the visual feeling, just set $\phi = 0$ for simplicity. Then, $x = uv$ and $y=0$. As $u,v$ vary in $[0, \infty)$, $x$ also varies in $[0, \infty)$, while $z= \dfrac{1}{2}(u^2 - v^2)$ takes on any number in $(-\infty, \infty)$. In other words, the surface of constant $\phi = 0$ is the half plane \begin{align} \{(x,0,z)\in \Bbb{R}^3| \, \, x \geq 0, \quad z \in \Bbb{R} \}. \end{align} If you want to get the surface for any other $\phi$, simply imagine rotating this half plane about the $z$-axis by an angle $\phi$.
The parameter $\phi$ is easily visualised as the angle in the $x,y$ plane measured counter-clockwise from the positive $x$-axis. However, unfortunately, I do not have a simple explanation for what the parameters $u,v$ represent geometrically.