Reflection in quadratic surfaces

“The transparent cup” is the title of pictures which show an interesting phenomenon: The circular boundary c of the depicted plate appears as an ellipse which seems to coincide with the view of the reflection of c in the coffee-cup. Is this just by chance or is there a geometric theory behind? In one example the circle c is the focal circle of the reflecting one-sheet hyperboloid, and for this particular case the displayed phenomen is a con- sequence of focal properties of quadratic surfaces. The tangent cones drawn from a fixed point P to a family of confocal quadrics are confocal and have therefore coinciding axes. These axes are the surface normals to the particular quadrics passing through P. Also the cones connecting P with the focal conics are included in the considered set of confocal cones. Therefore, all focal conics share the property: In each perspective, the images of these curves and their reflections belong to the same conic. The goal of the paper is to highlight the geometric background, i.e., to focus on confocal conics and their spatial counterparts