Distances and Central Projections

Given a point P in Euclidean space R3 we look for all points Q such that the length PQ of the line segments PQ from P to Q equals the length of the central image of the segment. It turns out that for any fixed point P the set of all points Q is a quartic surface Φ. The quartic Φ carries a one-parameter family of circles, has two conical nodes, and intersects the image plane π along a proper line and the three-fold ideal line p2 of π if we perform the projective closure of the Euclidean three-space. In the following we shall describe and analyze the surface Φ.