Poncelet's porism

The gold path starts at a point on the outer circle and runs along a line touching the ice-blue inner circle until it meets the outer circle again, then takes the next step the same way, touching the inner circle and turning in the same direction. Drag the gold starting point on the picture and it settles on the outer circle. At first the inner radius is r = 0.8 and the centres are d = 0.755 apart, which is exactly d² = R² − 2Rr (the outer radius is R = 1.9), so three steps bring the path back to its start; drag the start round the outer circle and the triangle turns with it, closing every time. Drag the offset d or the inner radius r: at most positions forty steps do not bring it back, the path spreads into a band, and the corner says the 40th step has not come back; close to a position where it closes within ten steps, the inner circle snaps there and the path falls into a polygon or a star, such as ten steps in three laps. Drag the start again now and it closes from every start; where forty steps do not bring it back, they do not from any start. The rotation number ρ, the average fraction of a turn per step, is estimated over 600 steps. Drag “Seen from a point” to the right and both circles, path and all, are projected into two ellipses: the corners stay on the outer ellipse, the sides still touch the inner one, and what closed still closes.