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I would like to construct a 2D conformal field on an annulus in which the inner and outer boundaries are like mirrors, and can be approximated by regular polygons (with the same number n of mirror edges, outer and inner).

I would like this field to have the property that, when a ray locally obeying classical optics (i.e. Snell's law, on these mirrors) hits an edge of an approximating polygon for the inner mirror, it gets reflected and then bent or arc'ed by the field to have an appropriate (the same?) incident angle on the outer mirror, and so on, generating a looping "flower" pattern around the annulus. (Actually, 'flower' isn't quite accurate - the petals are weird...) I would like this flower pattern to be identically traced after completing any loop around the polygon (i.e. without any "shift"). I'm also assuming that the incident points on the mirrors are midpoints of edges, and that the midpoint of any outer mirror edge is intersected by a ray from the center of the annulus through a vertex of the inner mirror (i.e. that these mirrors are rotationally shifted 180/n degrees). I'd also like to be able to draw all this in TikZ!

I'm just assuming (or guessing) that any field with this property (and a "reasonable" arc for the ray) is conformal. (But I'm also interested in "unreasonable" arcs - e.g. ones that might be SLEs along field contours.) I'm also guessing that there are families of conformal fields for this, e.g. producing thicker or thinner, more or less curvy, flower petals.

I must also confess that I'm quite rusty on my math & physics, so any care taken in explaining formulae would be greatly appreciated.


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