We investigate how the osculating conics of a regular curve in the real projective plane evolve as one traverses the curve. The Tait-Kneser Theorem states that if the curve has no inflection or vertex, then the osculating circles do not intersect and are nested, that is, the smaller osculating circle is contained in the bounded region defined by the larger circle. We generalize this result by proving that if the curve has no inflection or sextactic point, then its osculating conics are convexly nested.
In addition, we compute the first and second terms of the power series of the J-invariant of the binary quartic related to a pair of osculating conics of an arbitrary curve. Finally, we show that given a pair of harmonically nested conics u,v, there exists a zero projective curvature logarithmic spiral that has u and another conic of the pencil generated by u and v as its osculating conics.
Link para o Zoom meeting: https://puc-rio.zoom.us/j/97584892505?pwd=bDyU1r6C6bjavfq6vj7gxCh1RusdYA.1