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inner isoptic of an ellipse


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.