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Position vector in curvilinear coordinates

  1. Apr 18, 2014 #1
    The position vector ##\vec{r}## in cartesian coordinates is: ##\vec{r} = x \hat{x} + y \hat{y}##, in polar coordinates is: ##\vec{r} = r \hat{r}##. But, given a curve s in somewhere of plane, with tangent unit vector ##\hat{t}## and normal unit vector ##\hat{n}## along of s, exist a definition for the position vector in terms of ##\hat{t}## and vector ##\hat{n}##?
     
    Last edited: Apr 18, 2014
  2. jcsd
  3. Apr 18, 2014 #2

    Simon Bridge

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    Short answer: kinda.
    Coordinate systems are maps - you can make any kind of map you like.
    Some coordinate systems are more useful than others. Note: in order to specify an arbtrary position in 3D space, you usually need three numbers.

    So you could say - go distance s along this particular curve, rotate by so much angle a, then follow the normal away from the curve a distance d. So your position vector would be r=(s,a,d) ... it may be that a particular point P will have more than one representation, also the order that the components are followed will probably matter.
     
  4. Apr 18, 2014 #3
    Sure. Just resolve [itex]\vec{r}[/itex] into components in the [itex]\vec{t}[/itex] and [itex]\vec{n}[/itex] directions:

    [tex]\vec{r}=(\vec{r}\centerdot \vec{t})\vec{t}+(\vec{r}\centerdot \vec{n})\vec{n}[/tex]

    Chet
     
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