Difference between surfaces/path given by functions and parametrised curves

Question

There are $3$ ways to specify a surface:

1. As a graph of a function of $2$ variables $z=f(x,y)$
2. As a level surface of a function of $3$ variables $F(x,y,z) = c$
3. As a parametrised surface $(s,t)\mapsto(x(s,t),y(s,t),z(s,t))$

In $(1)$ we have a surface because we use one axis for $x$, one for $y$ and one for the output $z$.

In $(2)$ we have a surface because instead of having a four-dimensional graph, we do not consider the output to change, in fact we fix it to a constant value $c$ and plot for it, obtaining a level surface. So we use one axis for $x$, one for $y$ and one for $z$

In $(3)$ we have a surface because we use one axis for $s$, one for $t$ and one for the output.

However it seems to me that there are very different properties for these three options. For example, in $(1)$, in order to have a function, we need to have only one output coming from the inputs. However I think this is not always satisfied by the other two. Indeed sometimes there are sketches of $(3)$ that clearly violate that.

Reading this question helped me make more clear the situation in $2d$: Why is an ellipse, hyperbola, and circle not a function?

However I am not sure I understand the difference between these. For example, in $(3)$, do we always have to define the interval of the parameters in a closed interval? What if we defined it on the whole $\mathbb{R}^2$? And furthermore, do $(2)$ and $(3)$ respect the criterias to be functions? For example the one stated above, but one can think of others I guess.

So far I think that $(2)$ is not a function when we take multiple values of $c$, but it is a function when we plot one and only one value of $c$.Is this correct?

Concerning $(3)$ this is often thought of as a path when we have only one parameter (for example $t$ is time), however one could consider this as having two parameters as position and time maybe. Is this curve necessarily a function? What does it have in common with a function?

so summarizing:

1. Are the three options all function?
2. if not, are there similarities between them?
3. if so, what are the relationship between them and when should we use one and when the other? For example, when should someone use a parametrised curve instead of $(1)$ when modelling a physical phenomena?

Thank you

Answer 1

To address your first point, they are all functions, but not to and from the same spaces.

Case 1, the graph of a function $f(x, y)$, uses a function $f:\mathbb{R}^2 \to \mathbb{R}$, and we call the "surface" its graph, namely this set: $\{(x, y, z) \in \mathbb{R}^3 | z=f(x, y)\} $, appropriately adjusted for the domain of $f$. Of course this requires a sufficiently regular $f$, but all the other options do as well.

This "method" has the distinct advantage of being entirely representable in a clean 3D graph; also it is immediate to derive various properties of the surface just by "looking at the function": for example, the surface's regularity is exactly $f$'s regularity.

However, this method cannot represent all the surfaces that can be described by the others: surfaces that are graphs of functions cannot self-intersect, or even "loop around": the cases you mention of an ellipse not being the graph of a function is an excellent example, but you can just as easily consider a sphere if you want to stick to three dimensions.

If your model allows for this representation to work, you should definitely use it.

Case 2 is known (at least where I come from) as an "implicit representation": the idea (and probably the origin of the name) is that you manage to write an equation that "binds" three (or any number of) variables, and then want to express one as a function of the others, but you cannot reach a form suck as $z=f(x, y)$ through algebraic manipulation. So instead, you move everything to one side of your equals sign, call that thing $F$, and try to figure out when it equals zero (or a constant: but you'll see how one can simply absorb the constant in the function $F$). The surface then is the set $\{(x, y, z)\in \mathbb{R}^3 | F(x, y, z)=0\}$.

This form is the one you're most likely to get from real world applications (I'd guess), and can technically represent any surface that case 3 can, but it has a few shortcomings of its own.

First of all, regularity of the fucntion $F$ does not translate immediately to regularity of the surface. It doesn't even mean the surface is there in the first place! For example, $x^2+y^2+z^2-1=0$ is clearly a sphere, but $x^2+y^2+z^2=0$ is just the point $0$, and $x^2+y^2+z^2+1=0$ is the empty set! Also, really smooth functions $F$ can still produce not-everywhere-differentiable surfaces.

Case 3: this is the most "geometric" interpretation of the concept of surface. It is quite literally a 2D object (parametrised, aka represented, by coordinates $u$ and $v$) unfolded like a bedsheet through a 3D space, and the unfolding is described by the functions $x$, $y$ and $z$; all in all we have some function $\sigma :\mathbb{R}^2 \to \mathbb{R}^3$.

This is the most flexible approach (assuming your problem is easy or even can be stated in these terms), since it can represent all surfaces and it allows for easy checking of regularity, as well as some ease of representation: in short, it fixes the downsides of cases 1 and 2 while (mostly) keeping the advantages. However it can be hard to see whether surfaces represented in this way self-intersect (which may or may not be relevant).

As a bonus, notice that case 1 is in some sense a subset of case 2: if $z=f(x, y)$ is a surface, then $(u, v)\mapsto (u, v, f(u, v))$ is a perfectly fine uv map, or parametric representation.

Answer 2

This is a situation that arises in various forms in mathematics: how do you describe a given subset of a known set? After all, ultimately a surface -- whether it is a sphere, a cube, a torus, or some other complicated shape -- is nothing but a subset of points in $\mathbb R^3$. (It should be a "nice" kind of subset satisfying certain properties, so that we don't consider for example an isolated point or all of $\mathbb R^3$ a surface, but we won't get into the details here). However, if you're thinking of a specific surface and you want to write down a precise description of it, it's a little difficult -- you can't just write it as a list of points, because there are infinitely many of them. So we want to be able to describe the set of points compactly using some kind of a formula we can write down.

There are generally two ways of doing this.

1. [Implicit form] You can say, the subset I'm talking about is made of all the elements that satisfy an equation $f(x,y,z)=c$. This is the same as saying, I have a function $f : \mathbb R^3 \to \mathbb R$ and my surface is the preimage of the set $\{c\} \subseteq \mathbb R$ under that function $f$. In this case, it is very easy to check whether a given point $(x,y,z)\in\mathbb R^3$ is in the set: you just plug it into $f$ and see. However, it can be very hard to produce from scratch a single point that actually belongs to the set.

2. [Parametric form] Or you can say, my subset is made of all the points of the form $\bigl(g_1(s,t),g_2(s,t),g_3(s,t)\bigr)$ for some $s$ and $t$. This is the same as saying, I have a function $g : \mathbb R^2 \to \mathbb R^3$ and my surface is the image of some set $U \subseteq \mathbb R^2$ under the function $g$. In this case, it is very easy to produce arbitrarily many points that belong to the set: you just pick points $(s,t)\in U$ and plug them into $g$. However, it can be very hard to check whether any given point is actually in the set.

The $z=h(x,y)$ form is one of the rare cases when the two representations coincide. You can think of it either as the preimage of $\{0\}$ under the function $f(x,y,z)=z-h(x,y)$, or as the image of $\mathbb R^2$ under the function $g(s,t)=\bigl(s,t,h(s,t)\bigr)$. Thus it is both easy to generate points of the form $(x,y,h(x,y))$, and easy to check whether a point satisfies $z-h(x,y)=0$. However, in most other cases, given a surface in implicit form it is difficult to obtain a parametric representation, and vice versa.