# Thread: Calculating changes of position along a circle circumference using co-ordinates

1. ## 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

2. Do you know the radius of the circle?

3. Oh, yes (sorry for not mentioning)

4. So, arc length is given by $\displaystyle s=r\theta,$ where $\displaystyle \theta$ is the angle through which you have traveled. Let's say you have initial position vector

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

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

Then the new position vector is going to be

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

So it all boils down to matrix multiplication.

Does this make sense?

5. 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.

6. Ok, here's what it'll look like. The old coordinates were $\displaystyle (x_{0},y_{0}).$ The new coordinates will be

$\displaystyle (\cos(a/b)x_{0}-\sin(a/b)y_{0},\sin(a/b)x_{0}+\cos(a/b)y_{0}).$

7. Thanks a bomb, I'll have a play around with it now.

8. You're welcome. Have a good one!