# Involute of a Catenary

- Author:
- Tim Brzezinski

- Topic:
- Cosine

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**lies on a curve that is said to be an*D***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: https://tube.geogebra.org/m/MTVhRfVp Notice anything familiar? Explain.