Bitangents of a quartic

Multiply the equations of two ellipses and you have a quartic curve; then add a number ε. Just above zero the curve falls into four separate ovals, the most a quartic can have, and all twenty-eight of its bitangents, lines that touch it in two places, are visible: four between each pair of ovals and one across the hollow side of each oval. Drag ε up: the ovals round out, shrink and vanish one after another, and the visible bitangents fall from 28 to 16, 8 and 4, the missing ones turning complex. Some lines stay visible while their two touch points become an unseen complex pair: those lines are drawn long-dashed in gold and still count as visible, with the touch points sketched at x + iy on either side. The complex lines themselves cannot be drawn; only the corner counts them. When the last oval is gone the curve has no points left in the plane, and four visible lines remain. Drag ε below zero and the curve is two ovals, one inside the other, and again only four bitangents are visible. The corner counts the visible and the complex ones, and the total stays at 28. At ε = 0 the curve is just the two ellipses, crossing each other in four places, and the bitangents are not counted (the corner shows —).