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Math Help - Calculating changes of position along a circle circumference using co-ordinates

  1. #1
    Atr
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    Calculating changes of position along a circle circumference using co-ordinates

    Basically, I want to specify the centre co-ordinate of a circle, and then find the new co-ordinate after travelling a specified distance along the circumference in a clockwise direction.

    So sort of like:

    2
    |
    1----o

    Finding the co-ordinate for 2 after travelling in a specified distance in an arc around the point 'o'.

    Is there any formula to achieve this?

    (PS: if this is in the wrong section, feel free to move this)


    Thanks,

    Atr
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  2. #2
    A Plied Mathematician
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    Do you know the radius of the circle?
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  3. #3
    Atr
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    Oh, yes (sorry for not mentioning)
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  4. #4
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    So, arc length is given by s=r\theta, where \theta is the angle through which you have traveled. Let's say you have initial position vector

    \mathbf{r}=\begin{bmatrix}x_{0}\\ y_{0}\end{bmatrix}. To find the new coordinate, I would use rotation matrices. Let

    R_{\theta}=\begin{bmatrix}\cos(\theta) &-\sin(\theta)\\ \sin(\theta) &\cos(\theta)\end{bmatrix}.

    Then the new position vector is going to be

    \tilde{\mathbf{r}}=R_{\theta}\mathbf{r}=R_{s/r}\mathbf{r}.

    So it all boils down to matrix multiplication.

    Does this make sense?
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  5. #5
    Atr
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    I'm sorry, I'm completely new to the matrix concept.

    If you could give a as the distance travelled along the circumference, and b as the radius, I could figure it out from there.

    Thanks for the quick response, btw.
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  6. #6
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    Ok, here's what it'll look like. The old coordinates were (x_{0},y_{0}). The new coordinates will be

    (\cos(a/b)x_{0}-\sin(a/b)y_{0},\sin(a/b)x_{0}+\cos(a/b)y_{0}).
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  7. #7
    Atr
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    Thanks a bomb, I'll have a play around with it now.
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  8. #8
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    You're welcome. Have a good one!
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