Hyperbolische functies

A cable hanging between two points at the same height forms a catenary.

This curve is described by hyperbolic functions, which are special combinations of exponential functions. They are named this because they relate to the hyperbola in the same manner that standard trigonometric functions relate to the circle.

The catenary is described by the hyperbolic cosine function, which is the average of two reciprocal exponential functions. Its graph forms a U-shaped curve with a minimum at x equal to zero and symmetry about the y-axis.

The slope at any point along the cable is expressed using the hyperbolic sine function. This function is defined as one-half the difference of two exponential functions. The plot for this is a smooth curve that passes through the origin and rises sharply for both positive and negative values of x.

The hyperbolic tangent is the ratio of hyperbolic sine to hyperbolic cosine. Its graph passes through the origin and approaches horizontal asymptotes at y equals one and y equals negative one, forming an S-shaped curve.

These hyperbolic cosine and sine functions help engineers describe and calculate the cable’s shape and tension.