In the background of our study of circles the standard names for special sets of circles have been lurking. In order to better talk about circles, we will henceforth use these standard names.
Definitions in n the Euclidean plane of
The elliptic pencil of circles with base points A and B is the set of all circles that pass through A and B. While it is not an official member of the pencil, it can be useful to consider the line through A and B as "sort of" such a circle also (this works more cleanly in inversive geometry).
The hyperbolic pencil of circles with limit points A and B is the set of circles orthogonal to all the circles that pass through A and B. While it is not an official member of the pencil, it can be useful to consider the perpendicular bisector of segment AB as "sort of" such a circle also (this works more cleanly in inversive geometry).
[This is the same as the set of all Apollonian circles defined by A and B.]
The parabolic pencil of circles with base points A and direction c, where A is a point on c (where c is either a circle or a line), is the set of all circles that pass through A and are tangent to c. While it is not an official member of the pencil, it can be useful to consider the line to all these circles at A as "sort of" such a circle also (this works more cleanly in inversive geometry).
Definitions in n the Inversive plane of
All points are I-points and all circles are I-circles in these definitions.
The elliptic pencil of circles with base points A and B is the set of all circles that pass through A and B.
The hyperbolic pencil of circles with limit points A and B is the set of circles orthogonal to all the circles that pass through A and B.
The parabolic pencil of circles with base points A and direction c, where A is a point on circle c, is the set of all circles that pass through A and are tangent to c.
Thus the set of all the circles orthogonal to any 2 circles c1 and c2 is a pencil, called the orthogonal pencil of c1 and c2.
Also, if we take the orthogonal pencil of the orthogonal pencil of c1 and c2, we get a pencil that contains c1 and c2. Thus any two circles belong to a pencil, called the pencil through c1 and c2.