TABLE OF CONTENTS We have learned that for parametric curves, the parameterization gives information about rates of traversal which are important in many practical applications such as specifying the feed rate of numerically controlled cutting machines. The particular parameterization of a curve can make it easier or more difficult to obtain information about the geometry of the curve. The importance of the parameterization leads us to ask if there are equivalence classes of transformations. That is, is there some way of characterizing different parameterizations for a curve that will allow decomposition into equivalence classes with well-defined, well-behaved transformations between the equivalence classes?
Let I 1 and I 2 be intervals of 
. If ?( t) : I 1 into 
and p : I 2 into/ onto I 1, ? and p are composable functions and ?( p( u)) : I 2 into 
is a reparameterization of ?. This is also called a change of parameter from t to u.
Both ?( t) and ?( p( u)) have the same graph, and so are different representations of the same curve.
? is a regular parametric representation if for all t ? I,
?( t) ? C (1), and
?'( t) ? 0.
p = p( u) is called an allowable change of parameter if, for all u ? I
