Differential Topology - Computer Graphics and Geometric Modeling - page 464

Let U Õ R k . A C r map, r ≥ 1, Definition.

n

F: UR Æ

is said to be regular at a point p in U if DF( p ) is one-to-one. The map F is said to be
regular if it is regular at every point of U .

The parameterization in Example 8.2.1 of the graph of a function is regular,
assuming that the function is differentiable. The parameterization in Example
8.2.3 is also regular. Both are in fact globally one-to-one. In general, regular para-
meterizations are locally one-to-one (Theorem 4.4.6), but not necessarily globally
one-to-one.

8.2.4. Example. The parameterization

() = ( )

F q cos q , sin q , q ΠR ,

of the unit circle is a regular parameterization but is not globally one-to-one.

The parameterization of S 2 defined in Example 8.2.2, although often used, is not
regular. For one thing, it is not differentiable when x is ±1. For another, when x is ±1
the circle being rotated has degenerated to a point and F is not locally one-to-one
there. The nonregularity may, however, not be a problem if one is not interested in
those values of x.
Even if one sticks to regular parameterizations, there are still many ways to para-
meterize a space. For example, if a curve is parameterized by an interval and one
thinks of the parameter as time, then one can traverse or walk along the curve with
many different velocities and each one would correspond to a different parameteri-
zation of the curve. When using parameterizations as a vehicle for studying spaces
we must be careful to stick to those properties that are an invariant of the underly-
ing space.

Definition. Let U , V Õ R k and X Õ R n . Let r ≥ 1 and let F : U Æ X and Y : V Æ X be
two regular C r parameterizations of a space X . We say that Y is a regular reparame-
terization of F if Y=F mfor some one-to-one and onto C r map m : V Æ U with Dm( q )
one-to-one for all q in V . The map m will be called a change of coordinates or change
of parameters transformation . The map m is said to be orientation preserving if
det(Dm( q )) > 0 for all q in V ; otherwise, m is said to be orientation reversing . If m exists,
then F and Y are said to be equivalent parameterizations.

o

See Figure 8.4. Think of the map m as defining a change of coordinates. We shall
sometimes say that Y was obtained from F by a change in coordinates. The proper-
ties of spaces that we want to study via parameterizations should be invariant under
regular reparameterizations. It is easy to show that the notion of being equivalent is
an equivalence relation on the set of regular parameterizations of a set X .

8.2.5. Example. Consider the parameterizations