Singly reflected skew rays inside a hollow tube: an exact cardiold solution for incident rim rays

Diffuse light entering a hollow cylindrical tube forms a manifold of concentric bright bands as a result of multiple internal reflections. A circular occulting stop, filling the entrance aperture except for a narrow rim annulus, converts these successive bands into sharply bounded rings, the first of which is produced solely by singly reflected skew and meridional rays. The exact geometrical optics of all rays emitted by any point on the rim are considered, and it is shown that after a single reflection all such rays appear to arise from a single cardioid curve. When rotated about the optic axis, this cardioid predicts the flaring ring intensity distribution observed. (Any departure of this first ring from concentricity around the direct image of the rim annulus provides a quantitative measure of nonstraightness of the tube.).