My lamp has a cylindrical shade and a light bulb sits in the center of it. It projectes the pictured shadow on the wall. That U-shaped curve is the shadow of the rim(s). What shape is it? A parabola?
I know I can "compute" it with some work, but is there a quick geometric answer?
If the bulb is more or less a point source, then two cones of light are being emitted from the upper and lower rims of the shade. When the axis of the cylindrical shade is parallel to the plane of the wall, the wall cuts both cones and you get the two opposite sections of an hyperbola. (See Apollonius, Conics I, 12 & 14). If you incline the top of the lamp toward the wall until light is reaching the wall only from the top of the shade, you have a parabola. Tilting it farther gives an ellipse, and finally a circle.