A general theory for reducing the number of variables?

[Stephen Tashi]

Give a parametric equation for a curve in 2 dmiensions (x(t),y(t)) it may sometimes be possible to rotate or otherwise transform coordinates so that the tranformed curve becomes a function y = f(x) (as opposed to merely a relation such as x^2 = y^2 + 1). More generally, if we have a curve defined by a vector valued function of a vector of parameters, it may be possible to change coordinates to reduce it to a vector valued function with smaller dimensions in the argument vector. Is there an organized mathematical procedure for doing this? - some famous algorithm?

 

[micromass]

It shouldn't be difficult to come up with such a theory. The key result is the implicit function theorem that states sufficient conditions for a curve to be the graph of a function. The condition is that one of the partial derivatives must be nonzero. So if that is the case, then it's the graph of a function.

If the partial derivatives are all zero, then the situation becomes problematic. For example, consider the relation $y^2 = x^2$. Then no matter how we rotate, things will always be problematic.

 

[Stephen Tashi]

It's an interesting technicality whether "partial derivatives" are defined if we don't have a function.

Maybe the once hyped "catastrophe theory" contained the solution as a footnote. I think it is concerned with points on a surface where you loose the ability to express a surface as a function.