Some properties of the paraboloid z = x 2 + y 2

In a recent paper, I showed how the properties of algebraic systems of circles in the (x, y) plane could be investigated by means of a representation in which to the circle x 2 + y 2 − 2 px − 2 qy + r = 0 there corresponds the point ( p, q, r ) in space of three dimensions. The plane of ( x, y ) may be considered to lie in the space ( x, y, z ), so that the centre of the mapped circle is the orthogonal projection of the representative point.