Differential Geometry of Curves: Why Parametrizations?

In differential geometry, we make a definite choice to study parametrized curves. The reasons for this are simple, but important. This choice also has some consequences for what kinds of things we will be able to handle. Here I want to discuss this basic idea.

Geometric objects in $\mathbb{R}^3$ are generally described in one of two ways:

1. Implicitly as ``cut out'' by some equations. This means that we think of the object as the level set of some function defined on $\mathbb{R}^3$, or
2. Explicitly, or Parametrically, as the image of some function whose target is $\mathbb{R}^3$.

The first option is very common in basic algebraic geometry where varieties are the level sets of polynomial functions in the coordinates on our space. For example, the ``monkey saddle" is the vanishing set of the function $f(x,y) = x^3 - 3xy^2$. Let's make a picture with Sage.

With one equation, $f = 0$, we just described a surface. To describe a curve, we will generally need two equations, as each equation will drop the dimension of the resulting object by $1$. (This can be made rigorous, but not now. See you local algebraic geometer for assistance.) Let's intersect a sphere and a cylinder:

\[ x^2 + y^2 + z^2 = 4 , \qquad (x-1)^2 + y^2 = 1 \tag{1} \]

This gives us the following shapes. The intersection is called Viviani's Window.

While perfectly adequate for describing which points lie on the curve, it can be difficult to apply the methods of differential calculus to this shape with this description. In particular, if we want to answer simple questions like "What is the tangent line to this curve at such-and-such a point?", we will have to use implicit differentiation and lots of algebraic manipulation.

How can we avoid these problems? Instead, work with parametrized curves. By this we mean that a curve should be described by a triple of functions which depend upon some other chosen independent variable, called the parameter:

\[ x = x(t), \quad y = y(t), \quad z = z(t). \tag{2} \]

Following Newton's central insight from the Calculus, we interpret the parameter $t$ as representing the passing of time, and then these functions help us describe motion. This allows us to bring in some of our physical intuition and use descriptive language like speed and velocity. This physical intuition is very useful, and this simple change of perspective should not be taken for granted. Our geometric objects are now not some crystalline, static forms, but rather the traces of motion of some idealized particle through the changes in time.

Of course, the notation is more compact if we bundle these three functions from (2) as the coordinates of a single vector-valued function
\[ \gamma: t \mapsto \begin{pmatrix} x(t)\\ y(t) \\ z(t) \end{pmatrix} \tag{3}. \]

Viviani's Window can be realized as a parametrized curve in this way:
\[ \gamma(t) = \begin{pmatrix} 1 + \cos(t)\\ \sin(t) \\ 2\sin(t/2) \end{pmatrix}. \]

With this parametric vector description, the full toolbox of vector calculus is available to us, and we have hope of applying these powerful methods to make some progress.

We now have a new problem, however. How can we get a parametric description if one is not provided to us? We shall take this problem up in a later post.