How do you read these symbols?

Question

I want to learn computer graphics, so I bought a book where on the very first page there is a sentence:

Parametric surfaces are defined by a vector-valued parameterization function: $f: Ω \rightarrow S$ that maps a 2D parameter domain $Ω \subset \mathbb{R} ^2$ to the surface $S = f(Ω) \subset \mathbb{R} ^3 $. In contrast, an implicit (or volumetric) surface representation is defined to be the zero set of a scalar-valued function $F: \mathbb{R} ^3 \rightarrow \mathbb{R} $, i.e., $S = \{ x \in \mathbb{R}^3 | F(x) = 0\}$

It is on the first page, so I assume that I should be easily able to read these lines. Now you can condemn me, but I have never learned it that well to being able to easily read it. What does Ω and S represent here, and what should I look after so I can improve myself here as well?

Many Thanks,

Mat

Answer 1

Let's describe a simple surface, say a cylinder with height $1$ and radius $1$, in two ways.

First, a parametric surface is given by a function $f$, which eats two values and returns a point. If we look at the set of all points returned, we'll have a surface! For our cylinder that means looking at the function

$$f(\theta,h) = \langle \cos(\theta), \ \sin(\theta), \ h \rangle$$

Evaluating this at $(\theta,h) = (0,0)$ tells us that the point $\langle 1, 0, 0 \rangle$ is in our surface. Of course, evaluating at $(\theta,h) = (0,100)$ tells us that $\langle 1, 0, 100 \rangle$ should be in the surface, which we don't want. After all, we want to describe a cylinder of height $1$. We solve this problem by restricting what inputs we're allowed to plug in. That is, we restrict attention to a certain domain for the function. Restricting the domain also lets us ensure that each point on the surface is hit exactly once. Currently, the inputs $(0,1)$ and $(2\pi,1)$ both describe the point $\langle 1, 0, 1 \rangle$. We would like to not "double count" points on the surface, since eventually we'll be computing surface area, etc. in terms of these functions $f$, and we don't want to double count surface area.

So then, we restrict our domain, and see that our cylinder of height one can be defined as

$$ f(\theta, h) = \langle \cos(\theta), \ \sin(\theta), \ h \rangle \quad \quad 0 \leq \theta < 2 \pi, \ 0 \leq h \leq 1 $$

Here the domain $\Omega = \{ (\theta, h) \mid 0 \leq \theta < 2 \pi, \ 0 \leq h \leq 1 \}$ is the allowable set of inputs, and the function $f : \Omega \to \mathbb{R}^3$ takes an allowable input and gives us a point on our surface. If we look at every point we get in this way, we trace out a surface.

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The second approach is an implicit representation. In this way we give an equation (or a system of equations) and ask that our surface be exactly those points solving the equation. For our cylinder, the defining (in)equations are

$$ x^2 + y^2 = 1 \quad 0 \leq z \leq 1 $$

Indeed, if we look at $\{ (x,y,z) \mid x^2 + y^2 = 1, \ 0 \leq z \leq 1 \}$, the set of points $(x,y,z)$ which make our two equations true, then the points we've kept are exactly the points in our cylinder!

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Also, as a small piece of unrequested advice: If you're struggling with the first sentence of the book, it might be worth looking for a book on surfaces that's written for someone with a background more similar to yours. You mention computer graphics as an interest of yours, and I'm completely sure that there's a book on surfaces which doesn't require this kind of familiarity with mathematical formalism, and will probably teach you things that are more relevant for your interests than a very math heavy textbook. If you're dead-set on reading this book for some reason, you might want to look for any prerequisites it lists, and brush up on some of the things that will be useful for reading this book going forwards.

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I hope this helps ^_^

Answer 2

I think it would be easier if I give you an example. Suppose that you want to describe the half sphere that lies above the $x-y$ plane, in the positive $z$ direction, with radius $1$. This is surface $S$. The projection in the $x-y$ plane of this figure is a disk, centered on $(0,0)$, with radius $1$. So that is $\Omega$. $$\Omega=\left\{(x,y)|x^2+y^2\le 1\right\}$$and$$f(x,y)=\left(x,y,\sqrt{1-x^2-y^2}\right)$$ For the second case, let's use the full sphere. Then $$F(x,y,z)=x^2+y^2+z^2-1$$ The points where $F(x,y,z)=0$ are the points where $x^2+y^2+z^2=1$, which is the sphere of radius $1$.