This is a continuation of the discussion initiated in Quadratic_Transformation.html . Look there for definitions and first properties. Here I apply the general theory to the case of a circle bundle (a special case of a family of conics) I(c1,c2), generated by two circles c1 and c2.
[1] The Quadratic transformation leaves invariant the orthogonal bundle II(c1,c2) of the given one I(c1,c2), mapping each point P' of a member c of II(c1,c2) to its antipodal Q' in c.
[2] In the case of a bundle I(c1,c2) of non intersecting type, the images F(L) of lines under the quadratic transform are hyperbolas or parabolas passing through the limit points of the bundle.
[3] In the case of a bundle I(c1,c2) of tangent type (all circles tangent to a given line at a given point of it), the images F(L) are hyperbolas or parabolas passing through the common point and tangent to the line of centers of I(c1,c2).
[4] In both cases [2] and [3] the parabolas occur when line L is parallel to the line of centers of bundle I(c1, c2).