Let L_k be the set {(x,y) | x >= k and y >= k}.

Let L be the set consisting of the set difference L_0 \ L_1. That is, an infinite ell-shape that has thickness 1, to the right of the y-axis and above the x-axis.

Now, consider the set C of smooth curves (x(t),y(t)) that lie in L, such that x(t) goes to infinity as t does, and y(t) goes to infinity as t goes to minus infinity. So the curve is supposed to swoop in from above, make some sort of bend near the origin, and zoom off into large-positive-x-land.

Consider the set T of angle-preserving transformations of the plane, that is, scalings, rotations, and translations. Not every t in T will carry every c in C to a t(c) still in C. That is, the curve t(c) may not still fit inside L. However, we will say that t is

*good for*c if t(c) is still in the set C.

The question is: is there any nontrivial* mapping f, which, for every curve c in C, picks out a subset of c, such that for any c and any t good for c, we have that f(t(c)) = t(f(c))? In short, are there any nontrivial subsets of curves that are invariant under scaling, rotation, and translation?

*by nontrivial I mean, a map which always picks out a nonempty and nontotal subset of the curve.

--

Oh, damn, I think the answer to the formal question is yes, which is contrary to the side of the argument I was always on. My approach depends critically on what I thought was an incidental feature of the formalization, the fact that the curve is pinned down to these narrow infinite strips.

I think talking about smooth curves is a red herring. I need what adam called "focus invariance", too: if you just look at a smaller window around "the point", you should get the same answer. If we're only talking about smooth curves, this isn't really a restriction, since the whole curve might be predictable from its nth derivatives of any point. So I definitely want general continuous curves, also because I don't want any cheating descriptions of points of maximal curvature: out in the real world we have to choose the length of our rulers, and the rivers are fractal.