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1. Let's say you're trying to find a path (defined as some f(t) for t>=0) on the x-y plane (i.e. all (x,y) in R^2) that is close to every point on the plane, so that if you're trying to find a certain point, you can just start at t = 0 and increase t until the path crosses the point. (A computer would be used, so the parameter t would really be a series of points rather than a continuous variable.) The path should be one such that every point (x,y) is within a distance d of some point on the path. For a 2D plane, one solution would be to make the path start at d units from the origin and spiral outwards so that each revolution is ~2d units outwards from the previous one. (One example of this type of problem would be finding a lost child in the woods, although this algorithm would assume they didn't move while you searched for the child.)

2. Now imagine the space as a function space. How can one have a path through function space such that every function can be closely approximated by some point on this path? (I'm talking about everyday functions like f(x) = x^2 or a step function, not a delta function or something pathological like f(x) = 1 for rational x, 0 for irrational x.) We would define "closely approximate" by choosing a d such that |g(x)-f(x)| < d for all x in the domain. To make things simple, let the domain be all x in [a,b]. One idea I had was this:

Consider an arbitrary Fourier expansion with coefficients a_0 through a_n and b1 through b_n. As t goes to infinity, make the coefficients functions of t such that the following happens: First all coefficients are zero except for a_0, which starts at -q and increases through to +q in steps less than d. Once a_0 is equal to t, have a_0 jump down to -2q. While a_0 = -2q, increase a_1 from -q to +q as with a_0 in the previous step. After a_1 reaches +q, repeat with a_0 = -2q plus a step, and repeat the whole thing until a_0 reaches +2q. (This would be a loop within a loop. For each step of a_0, a_1 cycles from -q to q.) In this manner, we'd cycle through n nested loops for a_n (including b_n in there somehow). It seems reasonable that if f(x) is limited to x=[a,b], we would eventually reach some Fourier approximation g(x) that's close to f(x), though the nested loops would make this just about impossible to get a solution in a reasonable time. Does anyone know a better way to find an approximation to some arbitrary function by simply varying one parameter?

It didn't occur to me until after I thought of the problem that I was actually wrestling with the same problem whenever I had to mess around with my AC adapter cord for my laptop to get power to flow through it (it has a loose connection or something.) Just let h be a path in 3D space equal to the correct curve that my cord needs to be in, and let f(t) be the cord at time t as I wiggle it about, trying to find h. (I think about h as a function of a parameter s, where for each s, h is the position vector of a certain point on the cord. As s goes from start to finish, the cord is traced out. f(t) yields a path g(s) for each time t.) I wonder if there's an optimal f(t) that would allow me to find the correct cord position in a definite manner rather than just stumbling upon the solution.