Suppose that the invertible matrix 'A' depends on a parameter 't'.

If we think of the parameter 't' as time, then these equations specify the motion of an object in the plane between and .

Suppose that the curve is given by r(t), where the parameter t need no longer be arclength.

The parameter t is unique once and are chosen.

Here the parameter t plays the role of a Morse function.

Note: In the figure, there are 90 lines corresponding to the parameter t taking on values which are multiples of 4 degrees.

Note: In the figure, there are 60 circles corresponding to the parameter t taking on values which are multiples of 6.

In order to derive the distribution law of the parameter T, compatible with , the statistic must obey some technical properties.

As the parameter t ranges through the real numbers, the locus of X is a line.

If we identify the parameter t in both cases we arrive at a relationship between the circular functions and the hyperbolic ones.