I have a discrete 2D temperature field, i. e. a dataset of N points (x, y, T). I would like to compute line integrals at fixed radii, $\int_0^{2\pi}T(r,\theta)d\theta$. My first thought was to transform all coordinates into radial coordinates, and then interpolate to a regular grid in $r,\theta$, so the integral would turn into a simple summation. This approach doesn't seem to be ideal, as I lose data at $r=0$. Is there a better way to solve my problem?

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    $\begingroup$ Have you considered interpolating in x,y coordinates and then evaluating the integral? $\endgroup$ – Brian Borchers Jan 6 '16 at 15:11
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    $\begingroup$ Are your $(x,y)$ data points on a uniform grid, or is it an unstructured point cloud? $\endgroup$ – Wolfgang Bangerth Jan 6 '16 at 20:13
  • $\begingroup$ @WolfgangBangerth The points are on a uniform grid, but stored as a point cloud $\endgroup$ – akid Jan 7 '16 at 21:45
  • $\begingroup$ @BrianBorchers No... I'll have a look at that. $\endgroup$ – akid Jan 7 '16 at 21:46
  • $\begingroup$ If the points are on a grid, then interpolating along the points of the circle should be easy. $\endgroup$ – Wolfgang Bangerth Jan 9 '16 at 2:55

An alternative approach that will help with the situation at $r=0$ is to do the interpolation in $(x,y)$ coordinates first, and then evaluate the integral.

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