getRGivenTheta.m

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getRGivenTheta.m#

Given a ray from \(\xi\), in direction \(\theta\), find \(r\) at which the ray intersects the boundary of the closed contour bounding the non-oscillatory region, i.e.:

\[ |g(\xi)-g(\xi+r\exp(\mathrm{i}\theta)|=C/\omega \]

This algorithm is not explained in [Gibbs et al., 2024]. An explanation can be found in these extra notes.

r = getRGivenTheta(phaseCoeffs, centre, cballOverOmega, theta, imag_thresh)

Inputs#

  • phaseCoeffs : The coefficients of the phase function \(g\).

  • centre : The centre of the ball \(\xi\).

  • cballOverOmega : The value of \(C/\omega\) above.

  • theta : The angle \(\theta\) of the ray.

  • imag_thresh : The tolerance for an imaginary part of a radius before it should be considered purely imaginary, and thus not a valid radius.

Outputs#

r is he length \(r\) of the ray, such that the above equation holds.

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