Polar coordinates in terms of distance and arc length

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On the picture, we have a point on the circle $(a, b)$ parametrized using the arc length and distance to diameter. As you can see, $a = rtheta$. I want to write polar coordinates, $(r, theta)$ in terms of $(a, b)$. As you can see, if we palce this circle to origin, this is quite like transformation from cartesian to polar coordinates but with one ceveat: $a$ is not $x$ coordinate value but arc length.




geometry euclidean-geometry




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    Question



    On the picture, we have a point on the circle $(a, b)$ parametrized using the arc length and distance to diameter. As you can see, $a = rtheta$. I want to write polar coordinates, $(r, theta)$ in terms of $(a, b)$. As you can see, if we palce this circle to origin, this is quite like transformation from cartesian to polar coordinates but with one ceveat: $a$ is not $x$ coordinate value but arc length.




    geometry euclidean-geometry




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Question



      On the picture, we have a point on the circle $(a, b)$ parametrized using the arc length and distance to diameter. As you can see, $a = rtheta$. I want to write polar coordinates, $(r, theta)$ in terms of $(a, b)$. As you can see, if we palce this circle to origin, this is quite like transformation from cartesian to polar coordinates but with one ceveat: $a$ is not $x$ coordinate value but arc length.




      geometry euclidean-geometry




      share|cite|improve this question











      Question



      On the picture, we have a point on the circle $(a, b)$ parametrized using the arc length and distance to diameter. As you can see, $a = rtheta$. I want to write polar coordinates, $(r, theta)$ in terms of $(a, b)$. As you can see, if we palce this circle to origin, this is quite like transformation from cartesian to polar coordinates but with one ceveat: $a$ is not $x$ coordinate value but arc length.





      geometry euclidean-geometry





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      asked Jul 31 at 10:21









      meguli

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          Probably the equation b/r = cos(a/r) cannot be solved for r with elementary methods/functions. And r must be a solution to this equation.
          You could transform to (b/a)*x = cos(x) with x=a/r.






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          • I am also looking into $cos(theta) / theta = b / a$ which is another form for what you said. Still, it does not help me write $(r, theta)$ in terms of $a, b$.
            – meguli
            Aug 1 at 5:09











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          1 Answer
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          up vote
          1
          down vote













          Probably the equation b/r = cos(a/r) cannot be solved for r with elementary methods/functions. And r must be a solution to this equation.
          You could transform to (b/a)*x = cos(x) with x=a/r.






          share|cite|improve this answer





















          • I am also looking into $cos(theta) / theta = b / a$ which is another form for what you said. Still, it does not help me write $(r, theta)$ in terms of $a, b$.
            – meguli
            Aug 1 at 5:09















          up vote
          1
          down vote













          Probably the equation b/r = cos(a/r) cannot be solved for r with elementary methods/functions. And r must be a solution to this equation.
          You could transform to (b/a)*x = cos(x) with x=a/r.






          share|cite|improve this answer





















          • I am also looking into $cos(theta) / theta = b / a$ which is another form for what you said. Still, it does not help me write $(r, theta)$ in terms of $a, b$.
            – meguli
            Aug 1 at 5:09













          up vote
          1
          down vote










          up vote
          1
          down vote









          Probably the equation b/r = cos(a/r) cannot be solved for r with elementary methods/functions. And r must be a solution to this equation.
          You could transform to (b/a)*x = cos(x) with x=a/r.






          share|cite|improve this answer













          Probably the equation b/r = cos(a/r) cannot be solved for r with elementary methods/functions. And r must be a solution to this equation.
          You could transform to (b/a)*x = cos(x) with x=a/r.







          share|cite|improve this answer













          share|cite|improve this answer



          share|cite|improve this answer











          answered Jul 31 at 14:06









          jtm

          112




          112











          • I am also looking into $cos(theta) / theta = b / a$ which is another form for what you said. Still, it does not help me write $(r, theta)$ in terms of $a, b$.
            – meguli
            Aug 1 at 5:09

















          • I am also looking into $cos(theta) / theta = b / a$ which is another form for what you said. Still, it does not help me write $(r, theta)$ in terms of $a, b$.
            – meguli
            Aug 1 at 5:09
















          I am also looking into $cos(theta) / theta = b / a$ which is another form for what you said. Still, it does not help me write $(r, theta)$ in terms of $a, b$.
          – meguli
          Aug 1 at 5:09





          I am also looking into $cos(theta) / theta = b / a$ which is another form for what you said. Still, it does not help me write $(r, theta)$ in terms of $a, b$.
          – meguli
          Aug 1 at 5:09













           

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