A ring under a lamp

A lamp shines past a tilted gold ring, and the rays carry on to the glass floor, where the ring's shadow falls. Drag “Height of the lamp” to the left to lower the lamp. While it is above the ring's highest point the shadow is a closed ellipse, and the nearer the lamp comes to the top of the ring, the further one end of the ellipse reaches; when the lamp is level with the top, that end runs off to infinity and the shadow is a parabola; lower still, part of the ring is above the lamp, its light goes upwards and never reaches the floor, and only one gold branch of shadow is left. The picture extends those rays backwards through the lamp and draws in ice blue where they meet the floor on its far side, a second branch: the two together are a hyperbola. So it comes down to how high the ring is against the lamp: with the whole ring below the lamp, an ellipse; with only the top level with it, a parabola; with part of the ring above the lamp and part below, a hyperbola. The first row in the corner is how much of the ring is above the lamp, in degrees; the eccentricity e below it is fitted through five points of the curve on the floor, under 1 for an ellipse, exactly 1 for a parabola and over 1 for a hyperbola; the gold B² − 4AC is the discriminant of the same fitted curve, negative, zero and positive for the same three. Turn the view to look from another side.