A year ago I learned in multi-var calc about curvature, and since then I've wondered something. It came up again today when my dad tried to talk to me about curvature like it was the second derivative. :P Is there a way, or at least any attempt or resource at all, about parameterizing a function (2d or 3d) with curvature as the parameter? I've tried but can't derive anything...
What you are asking for only kinda makes sense if your function isn't multivariable. Think about it..... Curvature is a function of the variables, as much as the function value is a "function" of the variables; I suppose it might help if you first elaborated what you meant by "parameterizing".
Good point. This should be generalizable to any manifold in any number of dimensions, but I'm most specifically thinking about a line in 2D. For example, K(t)=1/t would be a spiral, but is there a way to derive x(t) and y(t) using K(t), because I've never seen this anywhere.
You might be interested in a related topic called "Prescribed scalar curvature problem." There is a book written on it by J. Kazdan.