Harmonic representation of closed curves

Closed Curves and Space Curves

So far we have discussed only ‘local’ properties of (plane) curves. These properties depend only on the behavior of a curve near a given point, and not on the ‘global’ shape of the curve. Now let us look at some global results about curves. The most famous, and perhaps the oldest, of these is the ‘isoperimetric inequality’, which relates the length of a certain ‘closed’ curve to the area it con...

Rational Harmonic Curves

Harmonic Deformation of Planar Curves

Closed Timelike Curves

In this paper, we explore the possibility that closed timelike curves might be allowed by the laws of physics. A closed timelike curve is perhaps the closest thing to time travel that general relativity allows. We will begin our discussing just what closed timelike curves are, and in what kinds of contexts they were first shown to appear. We then explore how one might actually travel on a close...

On Closed Twisted Curves

Twisted curves in R' are those which have independent derivatives up to order m . For plane and spatial curves the twisted condition is equivalent to never vanishing curvature and torsion respectively. We give a necessary and sufficient condition for a (q, p)-curve on a torus to be twisted, and use those curves to construct closed twisted curves in R'" for all m .