Reference Request: How to Parametrize Curves and Surfaces in $\Bbb R^3$

Question

I don't feel like I have a good grasp of how to parametrize a curve or surface. I can quickly enough verify that a given parametrization DOES correspond to a curve, and I've memorized a few of the common parametrizations that came up in multivariable (circle, ellipse, cardiod, ellipsoid, etc), but I don't know how they were found and when faced with how to find a parametrization of a curve/ surface I've never seen before I just don't really know where to start.

Does anyone know a good website or book that goes over the general process of finding parametrizations of curves and surfaces and has a lot of exercises (preferably with solutions) I can try out?

Answer 1

I think the honest answer to your question in general is no, I cannot provide such. Why? Because explaining how we derive parametrizations is somewhat akin to explaining how an artist paints. I mean, sure, there are some standard themes, but, in general the process is entirely open-ended. That said, here's some useful pointers (some of which you already know as you have begun the essential process to learn; practice)

1. identify how your object is described. Whatever parametrization you propose, it must be checked against the given description.
2. know your common identities both trigonometric and hyperbolic. Key identities: $$ \cos^2 t + \sin^2 t = 1, \ \ \cosh^2 t-\sinh^2 t=1, ...$$
3. try something. If it doesn't work, try, try again.
4. sometimes, use Cartesian coordinates as parameters. However, beware, this is often a bad choice if you do anything with the parametrization. It's usually best to parametrize the object in formulas which replicate any symmetries of the object. For circles use circular functions (sine and cosine), for hyperbolas use hyperbolic functions.
5. if possible, use polar, cylindrical or spherical coordinates. If we can use a curvelinear coordinate as a parameter this is often greatly simplifying. For example, a sphere has equation $\rho=R$ in sphericals and the parametrization in terms of the remaining spherical coordinates are immediately known from spherical coordinate formulas; $x=R\cos \theta \sin \phi, y = R\sin \theta \sin \phi, z = R \cos \phi$. Parametrizations for cones and half-planes are similarly trivial in sphericals.
6. beware of how to make simple modifications of standard coordinate systems, like use $x$ instead of $z$ for the cylindrical axes; $y = s \cos \beta, z=s \sin \beta$ and $x=x$.
7. linear data is easy. If you want a line from $P$ to $Q$ in one unit of time, $t \mapsto P+t(Q-P)$ is the way to do it (check it). If you want a parametrization of a plane with $P,Q,R$ then $(s,t) \mapsto P+s(Q-P)+t(R-P)$ is a natural choice.

Honestly, we can go on from here. The process of finding a parametrization really tests your overall familiarity with both functions, geometry and well-known identities. Pragmatically, the question you'll be asked is probably taken from some of the same books you're studying, so, just keep at it and you'll be fine. However, the more general question, how do I parametrize something? This I cannot answer.