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proving a regular homotopy

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Poirot

Banned
Feb 15, 2012
250
Let y:[0,1]-> R^n be a regular closed curve, and let h:[0,1]->[0,1] be smooth, increasing and $h^{(k)}(0)=h^{(k}(1)$ for all k=0,1,.........
Then

b=y o h is a reparametrization of y.

I want to show:

1) b is a regular closed curve so i'm guessing I needs an expression for the kth derivative

2) there is a regular homotopy between y and b. I have been guided to consider

F:[0,1] x [0,1]->R^n given by $F(u,t)=y(uh(t)+(1-u)t)$.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,780
Let y:[0,1]-> R^n be a regular closed curve, and let h:[0,1]->[0,1] be smooth, increasing and $h^{(k)}(0)=h^{(k}(1)$ for all k=0,1,.........
Then

b=y o h is a reparametrization of y.

I want to show:

1) b is a regular closed curve so i'm guessing I needs an expression for the kth derivative

2) there is a regular homotopy between y and b. I have been guided to consider

F:[0,1] x [0,1]->R^n given by $F(u,t)=y(uh(t)+(1-u)t)$.
Hi Poirot! :)

For (1) you would need to show that the curve is:
- regular
- closed

What is the definition of a regular curve?

For (2), what is the definition of a homotopy?
And what does it mean if it is regular?