# Searching for a point in space

## Main Question or Discussion Point

What area of mathematics would encompass the following problems? (Any comments you have on the problems would be welcome.)

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 continous 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.

Does anyone know a better way to find an approximation to some arbitrary function by simply varying one parameter?
The only reason the mission succeeded for 2-space, is because instead of two space you choose to work with a discrete lattice. This is the difference between countable and uncountable. Every countable set is 1-dimensional in the sense that, because the set can be listed, you can trace out all the elements by a single parameter.

So the answer is: there is a way to do it if and only if you are working in a discrete approximation to function space.

AKG
Homework Helper
What exactly do you mean by a "path"? Normally, a path in some space X is a function F : [0,1] -> X which is continuous. Continuity only makes sense if X is topological, but a space of "reasonable" functions can be given a natural topology. For example, the space of continuous functions on [a,b], denoted C([a,b]) has a metric p defined by p(f,g) = supa<x<b|f(x)-g(x)|, and this metric gives a topology. However, you don't seem to be placing any continuity requirements on your path, so we're just looking for a function F : [0,1] -> X such that for all f in X, there exists t in [0,1] such that p(F(t),f) < d.

Suppose we take X = C([a,b]). It turns out |X| = |[0,1]|, so there actually exists a bijection F : [0,1] -> X such that for all f in X, there exists t in [0,1] such that p(F(t),f) = 0. Now you want X to also include functions that may not be continuous, like the step function, but not so pathological like the characteristic function of the rationals. I suspect that the set X you're interested in will still satisfy |X| = |[0,1]|.

What if it doesn't? That's okay, we still might be able to do something. Suppose |X| > |[0,1]|, but suppose that X has a dense subset D such that |D| = |[0,1]| (D is dense in X means that for all d > 0 and for all f in X, there exists g in D such that p(f,g) < d). Then just let F be a bijection from [0,1] to D. With this choice, the function F satisfies:

For all d > 0, for all f in X, there exists t in [0,1] such that p(F(t),f) < d.

Note that this is stronger than the desired result, which you haven't asked to hold for all d > 0, but merely to hold for some fixed d > 0.

I know this doesn't answer some of your questions. For example, you want a "quick" way to approximate an arbitrary function. You'll have to clarify further what exactly you want though, before I could try to give a clear answer.