My question is the following:
What is the shape of the rope which holds a kite flying? (Steady state.)
I am not a physicist (I am a mathematician), so I can not work the physics part of the question.
My guess is that it should be the same as catenary (hyperbolic cosine)
The catenary shape between two points with coordinates $A=(0,0)$ and $B=(S,h)$ is
$$ y(x) = y_C + a \left( \cosh \left( \frac{x-x_C}{a} \right)-1 \right) $$
where
$$ \begin{gather} a = \frac{H}{w} & \mbox{Catenary Constant} \\ H : & \mbox{Horizontal Tension} \\ w : & \mbox{Weight per Unit Length} \\ x_C = \frac{S}{2} + a \sinh \left( \frac{h \exp(\eta)}{a (1-\exp(2\eta))} \right) & \mbox{Lowest Point x coordinate} \\ y_C = -a \left( \cosh\left( -\frac{x_C}{a}\right)-1\right) & \mbox{Lowest Point y coordinate} \\ \eta = \frac{S}{2 a} & \mbox{strain factor} \end{gather} $$
The results look like this:
