inner isoptic of an ellipse
indications
ell = the ellipse, depending on the parameter k -> slider
opt = the (bis)optic far an angle theta s.t. tan theta = s -> slider (the software did not allows to use t, so s in the applet is instead of t in the paper)
A is a point on the isoptic. I plotted the tangents to ell through A. The points of contact are B and C.
I plotted the segment BC in green.
With Trace On on BC and moving slowly A around the loop, you obtain a prefiguration of the inner curve we are looking for. If you move A too fast, it may jump from one loop to the other (a problem with the software? when we are finished, I will ask a friend in Austria)
a = an arc of the inner curve we are looking for. It coincides well with the experimentation. I plotted once again one segment BC in order to show that it is really tangent to the curve.
Of course, it is possible to redo that with a point A on the other loop.
In the "paper", I made the computations for one arc of the inner only, as actually, I should have performed 3 more times the same kind of computations to obtain 3 more arcs.