Involute of a Catenary

In the applet below, point P lies on the graph of the function y = cosh(x), or y = the hyperbolic cosine of x. The graph of this function is referred to as a catenary. Hanging chains or wires, when left to hang under the influence of gravity, take the shape of a catenary. Nonetheless, point D lies on a curve that is said to be an involute of this catenary. In the applet below, the length of the segment CD is equal to the length of the arc with endpoints P and A. Segment AD is kept tangent to the graph of y = cosh(x) as well. Drag the white point P along the hyperbolic cosine curve to trace out its involute. Does this brown curve look familiar? If so, describe. Now compare this curve with the curve you see here: Notice anything familiar? Explain.