Understanding Image of a Curve - Ask Luca

[pamparana]

Hello,

Just started reading a beginners book on elementary differential geometry and have a small question about the term "image of a curve". It says that a parameterized curve whose image is contained in a level curve is called a parametrization of C.

I am a bit confused with this statement. What does the image of a parameterized curve mean?

Would much appreciate someone clarifying this doubt for me.

Thanks,
Luca

 

[quasar987]

Surely you know what it means when we speak of the image of a map f:A-->B? It means simply the set f(A). Well, a parametrized curve is a map $\gamma:(\alpha,\beta)\rightarrow\mathbb{R}^n$. So the image of this paramatrized curve is the image of this map; namely $\gamma((\alpha,\beta))$.

What the author is saying here is that if you have a curve C defined as the level set of some function (i.e. a level curve), and if you find a parametrized curve $\gamma:(\alpha,\beta)\rightarrow\mathbb{R}^n$ whose image is that level curve (i.e. $\gamma((\alpha,\beta))$=C), then said parametrized curve is called a parametrization of C.

 

[pamparana]

That makes sense! Thanks.