1

Suppose I have an implicit equation:

\begin{equation} f(x_0,x_1,\ldots,x_n)=0 \end{equation}

Which might be 'paramaterizable'; i.e. put it into the form: \begin{align} x_0 &= g_0(t_0,t_1,\ldots,t_m) \\ x_1 &= g_1(t_0,t_1,\ldots,t_m) \\ \vdots \\ x_n &= g_n(t_0,t_1,\ldots,t_m) \end{align}

where each of the $g$ functions may be in terms of any of the parameters $t_k$ up to $t_m$, where $m \le n$.

I have two queries, which I suspect are closely related:

  1. Is there a general test which can be performed to determine if parameterization is even possible? I already know from this answer that not all curves can be parameterized, but I don't quite understand how to test this.
  2. Judging by the fact that this question does not yet have an answer, it suggests that there is no algorithm for determining the parameterization of a function $f$. I found three examples of parameterization of specific functions on this site (a trochoid, power function, and a line). The first two answers use heuristics very specific to the problems in question, and the last one uses a general rule for two variables, when one can be written explicitly (when $y=f(x)$, $x=t$ and $y=f(t)$, quite trivial). So if there is no algorithm, is there a widely accepted heuristic approach to parameterization (i.e. something akin to the LIATE rule for integration by parts)? The more specific, the better.
Community
  • 1
NauticalMile
  • 141
  • If you're talking about a single parameter $t$, the parametric equation describes a curve. Apart from singular cases, one implicit equation in $n+1$ variables will describe an $n$-dimensional variety which is not a curve unless $n=1$. – Robert Israel May 24 '16 at 00:24
  • In principle, for a reasonably nice function $f$ each branch of the curve $f(x_0, x_1) = 0$ can be parametrized by arc length. Whether there is a closed form parametrization is another question. – Robert Israel May 24 '16 at 00:28
  • Oh, so what you're saying is there need to be additional parameters to describe higher dimensional geometries? – NauticalMile May 24 '16 at 03:43
  • I guess that makes sense, I can imagine how a 3D surface would require 2 parameters to specify a location on the surface. – NauticalMile May 24 '16 at 03:44

0 Answers0