Suppose we would like to model a robotic snake by making a kinematic chain out of hundreds of short links. This would lead to a complicated set of transformations for moving the snake, and a degree of freedom for each link. Imagine another application, in which one would like to warp a thin flexible sheet of material such as aluminum though a small doorway. The sheet could be approximated by a 2D array of links; however, the complexity and degrees of freedom would again be too cumbersome.
For these kinds of problems, it is often advantageous to approximate the entire structure by a smooth curve or surface, given by a parametric function. A family of functions is generally used to encompass possible ways in which the structure can be transformed. The family of functions is characterized by a set of parameters. For each selection of parameters, a particular function in the family is obtained. For example, one set of parameters might place the snake in a straight configuration, and another might place it in an ``S'' configuration.