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Topological Definition of Arc Length

  1. Dec 21, 2009 #1
    In calculus, the definition of the arc length of some curve C is the limit of the sum of the lengths of finitely many line segments which approximate C. This is a perfectly valid approach to calculating arc length and obviously it will allow you calculate correctly the length of any (rectifiable) curve. However, I don't think this definition really captures the intuitive notion most people have about what the length of a curve really means.

    If you ask a child to measure the length of a curve, a circle for instance, most likely the child will take a piece of string, lay it out so that it exactly corresponds to the curve, and mark the endpoints of the curve on the string. Then they will straighten the string out, and measure the length between the two marked points with a ruler.

    How would we express this concept mathematically? Well, we can say that the length of a curve C is equal to the length of the line segment D which C can be continuously deformed into. But we must restrict the class of allowable deformations, so as to exclude deformations which stretch or shrink the curve. In other words, we can only allow "length-preserving deformations." I'm not sure what the technical name for these are, but since a transformation which leaves the metric of a space undisturbed is called an isometry, maybe they are called isometric deformations. The (admittedly nonrigorous) way I am envisioning it is that we split the curve into infinitesimal pieces. Then we rotate the infinitesimal pieces until they are all facing the same direction, and then line them up end to end, so that they form a line segment. I believe that the definition of a continuous length-preserving definition of a curve can be made precise using topology, but I don't know for sure because my knowledge of topology is limited to a course I took in Real Analysis. Regardless, let us call it the "topological definition" of arc length.

    My question is, can it be proven that the usual calculus definition of arc length is equivalent to my topological definition for all (rectifiable, smooth, and well-behaved) curves? If so, how can it be proven?

    Any help would be greatly appreciated.
    Thank you in Advance.

    P.S. I suppose that a similar topological definition can be applied to the area, surface area, volume, n-dimensional hypervolume, etc. In general take a "curved" region, apply a topological deformation which preserves the property you want to calculate, so that the problem reduces to calculating the property for an equivalent "flat" region.
     
  2. jcsd
  3. Dec 22, 2009 #2
    If you parameterize the curve by arc length then you are are wrapping the parameter interval around the curve without stretching or shrinking. The length of the curve is just the difference of the end and start parameters, just as you envisioned.

    For a flat surface e.g. a cylinder or a cone a similar idea applies and corresponds to cutting the surface then unfolding it onto a flat plane. The surface need not be embedded in 3 space. For instance a flat torus in 4 space can be cut along two circles then isometrically flattened onto a parallelogram in the plane. For flat manifold of higher dimension e.g. the Hansche-Wendt manifold this also works in a way. You cut the surface until it can be unfolded isometrically into a polyhedron in flat Euclidean space.

    In all of the cases the parameterization of the manifold preserves length, area, and volume - hypervolume etc. - and this is why one can think of it as a piece of Euclidean space with identifications.

    For a surface with curvature I do not believe this will work. Any attempt to unfold a sphere for instance will distort metric relations.

    The problem that you are asking is not topological because it involves notions of measure such as length and area. Your infinitesimal pieces must also have infinitesimal measures. This is true even for a piece of string. With a measure the calculus definition of area or volume or hypervolume is precisely the sum of the infinitesimal measures of all of the infinitesimal pieces. But I do not believe that there is a geometric transformation such as an unfolding that lines the surface up isometrically onto a plane.
     
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