In particular, we are looking at the following commutative diagram:
Def 2. A curve
is called regular if
for all
A curve
is called unit-speed if
for all
Clearly, not all curves
is unit-speed. It will be convenient to have a reparametrization
which is a unit-speed reparametrization (This will be useful when calculating curvatures and etc). If
is regular, we can always find a unit-speed reparametrization.Intuition: Suppose we have
be a unit-speed reparametrization of
. Then what we want is that at time
, we have traveled exactly
units of arc length, making
Then suppose that
exists, we know that
This indicates that we should choose the reparametrization map to be
Theorem. A parametrized curve has a unit-speed reparametrization if and only if it is regular.Proof. Suppose the parametrized curve
has a unit speed reparametrization
, with reparametrization map
, then we have, by letting
, that
. Taking derivatives on both sides
and we have
The LHS is 1, so the RHS is also 1, in particular
for all
. Hence
is regular.On the other hand, suppose
is regular. As above (......inverse function theorem arguments blah blah blah...), take
. Then
gives
Therefore,
Taking the norm, we have
,since
This shows that
Q.E.D. Remark. In the argument above, we have established the relation that
But we know that we can express
in terms of
. Since we have the relation
where
is written as a function of
, finding its inverse will express
in terms of
We have then
. Therefore, we can think of
as
and the above relation becomes
Usually, for short-hand notation we just write
,where we are letting
be an independent variable now. Therefore, by taking derivatives with respect to
, we find that
This is the convention we will be using for the rest of the book (which often confuses people seeing it for the first time). Therefore, we shall say that
is a unit-speed parameter of the curve
, and what that implies is that
,and we are really referring to the unit-speed reparametrization
However, if we view
as an independent variable, then the above relation is indeed true. When we are finding the unit-speed reparamterization of a given curve, what we are really doing is to express
in terms of the arc length
and then
is then a unit-speed reparamterization. Let us now illustrate this with an example.Example. Consider the curve
First, we find the arc length, we have that
and
Then we express
in terms of
, giving
Therefore, the unit speed reparametrization, which is really