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Image of a curve

  1. Nov 23, 2009 #1

    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.

  2. jcsd
  3. Nov 23, 2009 #2


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    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 [itex]\gamma:(\alpha,\beta)\rightarrow\mathbb{R}^n[/itex]. So the image of this paramatrized curve is the image of this map; namely [itex]\gamma((\alpha,\beta))[/itex].

    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 [itex]\gamma:(\alpha,\beta)\rightarrow\mathbb{R}^n[/itex] whose image is that level curve (i.e. [itex]\gamma((\alpha,\beta))[/itex]=C), then said parametrized curve is called a parametrization of C.
  4. Nov 23, 2009 #3
    That makes sense! Thanks.
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