# How to compute line integral over circle

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?

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