Bazinga | 弧长参数化

作者介绍：

加州大学圣塔芭芭拉分校数学物理专业

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作者说明

这是一个小笔记，是为了整理Andrew Pressley的Elementary Differential Geometry中的1.3小节，个人感觉作者叙述的有点凌乱。该文章仍然是备份作用，读者不必当真。

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整理笔记

In the contents below, 

 is the arc length of the parametrized curve 

 , given by

Notice that by the FTC we have 

Def 1. A parametrized curve 

 is a reparametrization of the parametrized curve 

 if there is a smooth map 

 with smooth inverse 

 such that

for all 

 . Notice that since 

 has a smooth inverse, 

 is also a reparametrization of 

 , given by

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 

 , reads