What is the "typical" map between surfaces? I've seen the concept of map between surfaces $S\to \overline{S}$ and at first, I was extremely confused because I spent a lot of time thinking on how to actually send points from $S$ to $\overline{S}$ "directly". I'll try to explain what I mean with the following example: Look at the map $f$ in this example:



It is constructed by first inverting $x$, making a change of coordinates and then computing $x^{*}$ so, this was not "direct", it took some "detours" in order to be made. Are all the maps we commonly use in differential geometry like this? Are there ways to make "direct" maps?
 A: Well, $f$ is a "direct map" between surfaces. It maps a point of $S$ to a point of $\bar{S}$.
The "detours" you mention happen when you want to write an expression for $f$ using coordinates. How do we do that?
The answer given by differential geometry is that, around a point $x_0 \in S$, you can find local coordinates $\theta, v$, allowing you to parametrize the position $x$ as $x(\theta, v)$. In a small enough neighbourhood, you require that parametrization to be a homeomorphism, so that your surface locally "looks like" $\mathbb{R}^2$.
Then, you do the same thing around the point $y_0 = f(x_0) \in \bar{S}$. ie, you find a local parametrization $y = y(\theta', v')$, which is locally a homeomorphism.
And now, your map $f : S \to \bar{S}$ locally looks like the map
\begin{equation}
y \circ f \circ x^{-1} : \mathbb{R}^2 \to \mathbb{R}^2
\end{equation}
using the new coordinates you introduced for parametrizing the surfaces.
This is what is happening on the picture you mentioned above. If you want a clearer picture, look up "Charts on manifolds", this is what the functions $x$ and $y$ are called. In particular, I recommend Lee's Introduction to Smooth Manifolds.

However, the construction mentioned above is only necessary when you want to express your function in local coordinates. It can be useful for differentiating, or for some calculations, but it is not necessary to define maps between surfaces.
For example, think about quotient maps in topology. Like, the quotient from $\mathbb{R}^2$ to a torus, or from $S^2$ to $\mathbb{RP}^2$. Those are well-defined maps between surfaces, but you don't need coordinates to define them. (Though you can throw them into the mix if you need them)
A: There are already good answers, but in the hope of supplementing them:
To define a mapping from a set $S$ to a set $S^{*}$, we have to "specify," for each $s$ in $S$, a unique $s^{*}$ in $S^{*}$. I put "specify" in quotes because in practice there are potentially multiple meanings, such as

*

*Parametrizing elements of $S$ and $S^{*}$ by ordered tuples of real numbers, i.e., introducing coordinates, and giving algebraic formulas for the coordinates of $s^{*} = f(s)$ in terms of the coordinates of $s$;

*Describing the geometric location of $s^{*}$ in terms of the geometric location of $s$, perhaps together with additional information such as the affine structure of Euclidean three-space;

*Proving an existence and uniqueness result for $s^{*}$ if $s$ is given.

Loosely, these are the perspectives of algebra, geometry, and analysis. As a preliminary example, consider stereographic projection from the unit sphere with $N = (0, 0, 1)$ removed ($S$) to the Cartesian plane ($S^{*}$).

*

*Algebraically, send the point $(x, y, z)$ to $(x, y)/(1 - z)$.

*Geometrically, let $s = (x, y, z) \neq N$ be a point of the unit sphere. Since $z < 1$, the ray from $N$ through $s$ cuts the equatorial plane in a unique point $s^{*}$.

*Existentially (with a bit of stretching, since the mapping is not entirely implicit), let $P$ be an arbitrary plane through the origin and not containing $N = (0, 0, 1)$. Every ray from $N$ in the half-space defined by $P$ hits $P$ precisely once, so defines a mapping from a certain half-space to the plane $P$. Since each sphere containing $N$ is contained in the half-space determined by its equatorial plane, there is a mapping from the punctured unit sphere to its equatorial plane.


The (locally-defined) catenoid-to-helicoid mapping in question is given algebraically. To do so, the author (do Carmo?) chose to parametrize each surface. Guiding this choice was knowledge that the catenoid and helicoid fit into a one-parameter family of locally-isometric immersed minimal surfaces
\begin{align*}
  X_{t}(\theta, v) &= \cos t(\cosh v\cos\theta, \cosh v\sin\theta, v) \\
  &+\sin t(-\sinh v\sin\theta, \sinh v\cos\theta, \theta).
\end{align*}
By picking two members of this family, the author arguably started with an example (the catenoid and helicoid) in search of a phenomenon to illustrate (mappings between surfaces, local isometry).

The same mapping could have been described geometrically, but perhaps with less clarity:
The catenoid is obtained by sweeping a profile curve, here a certain catenary, about an axis in Euclidean three-space. The "minima" of the profile curves comprise a central circle of smallest (Euclidean spatial) radius. Every point of the catenoid is uniquely determined by a point $\theta$ on the central circle and a point $v$ on the profile through $\theta$.
The helicoid is obtained by sweeping a ruling line along an axis line while rotating the ruling at constant angular speed (in radians per unit distance) about the axis. Every point of the helicoid is uniquely determined by a point $\theta^{*}$ on the axis and a point $v^{*}$ on the ruling through $v^{*}$.
We could choose to measure $\theta$, $v$, $\theta^{*}$, and $v^{*}$ by arc length on the respective surfaces. Whether or no, the (local) mapping from the catenoid to the helicoid fixes an arc of the central circle, picks a point of this arc and maps it to a selected point of the helicoid's axis, and then unwraps the central circle arc to the axis, while simultaneously unwrapping each profile catenary to the corresponding ruling line. (Caution: While the $\theta$s measure arclength along the "central" curves in the mapping written above, the $v$s do not; instead, $\operatorname{asinh} v$ is an arclength parameter.)
Finally, we could trust the visual evidence of the animation loop above, or fashion a catenoid from paper or other flexible, inelastic material, and bend it into a helicoid. Paraphrasing Bhaskara, behold!

In any case, defining a mapping entails making a deterministic choice for each element of some set. How we describe such a choice correlates with what understanding we glean, and what tools can be wielded. Whether or not a mapping description is direct arguably is a matter of our own intuition rather than a mathematical property.
A: In your example, you have two surfaces $S$ and $S^*$ that are explicitly described thanks to the global systems of coordinates $(\theta, u)$ and $(\phi,v)$.
Hence, they are very convenient to describe a map $f\colon S\to S^*$.
In full rigour, we have the maps
$$
(0,2\pi)\times \Bbb R \overset{\mathbf{x}}{\longrightarrow} S \overset{f}{\longrightarrow} S^* \overset{{\mathbf{x}^*}^{-1}}{\longrightarrow} (0,2\pi)\times \Bbb R, 
$$
and since $\mathbf{x}$ and $\mathbf{x}^*$ are bijections (in fact, diffeomorphisms), the data of $f$ is equivalent to the data of ${{\mathbf{x}}^*}^{-1}\circ f \circ \mathbf{x}$.
It turns out that this latter function goes from $(0,2\pi)\times \Bbb R$ to itself, so it can be easier to describe than a function from a surface to another.
This is what the exercise is doing: it is basically given by $(\phi,v) = (f_1(\theta,u),f_2(\theta,u))$, in the particular case $f_1(\theta,u) = \theta$ and $f_2(\theta,u) = \sinh^{-1}(u)$.
However, do not take this as a generality: the exercise is built that way.
General smooth functions from some surface $S$ to another $S^*$ need not be easier to describe in any coordinate systems.
It just maps some point of $S$ to a point of $S^*$, as any map between two sets.
For instance, the natural submersions
\begin{align}
&\begin{array}{ccc}
\Bbb R^2 & \longrightarrow & \Bbb T^2 = \Bbb R^2 / \Bbb Z^2\\
(x,y) & \longmapsto & (x \pmod 1, y \pmod 1),
\end{array}
\\
\\
&\begin{array}{ccc}
\Bbb S^2 & \longrightarrow & \Bbb{RP}^2 = \Bbb S^2 / \{\pm 1\}\\
p & \longmapsto & \{p,-p\},
\end{array}
\end{align}
as well as any constant map $S\to S^*$ need not be expressed in some special coordinate patches to be well understood.
Finally, the coordinate expression of a function $f$, (here, ${\mathbf{x}^*}^{-1}\circ f \circ \mathbf{x}$) need not be easier to understand than $f$.
But it is a map between two open subsets of $\Bbb R^2$, and therefore, we can do differential calculus with this form.
This is not clear you could do that directly on $S$ and $S^*$, isn't it?
