CicleBundleContact

1. Circle-bundle contact with line

Given a circle-bundle of non-intersecting type generated by two circles {c,c'} and a line L, to find the bundle-members tangent to L.

Draw the medial line (g) of the segment AB joining the limit-points of the bundle and find its intersection E with line L. The circle c₀ with center E, passing through {A,B} defines the contact points {F₁,F₂} of the desired circles {c₁,c₂} with line L.

The limit case in which {A,B} coincide i.e. the circle-bundle is of tangent-type consisting of all circles tangent to (g) at O is handled in the obvious way. The circle c₀ becomes tangent to line CD.

2. Intersecting-type case

Given a circle-bundle of intersecting type with two base-points {A,B} and a line L, find the bundle-members {c₁,c₂} tangent to L.

Let E be the intersection point of AB with L. Draw the circle (c) centered at E and orthogonal a circle c' of the given bundle. Its intersection points {F₁,F₂} define the contact points of the desired circles {c₁,c₂}.

3. Relation to line-homographies

The previous problem in both its versions for the different types of circle-bundles is connected with the subject of homographic relations on lines.
In fact, given a circle-bundle and a line L, for each point X of L define the bundle member c_X passing through X and its second intersection point Y with L. The transformation of the points of line L defined through Y=F(X) is an involutive homography and the contact points {F₁,F₂} of the bundle-members {c₁,c₂} are the fixed points of this involution on L.
This remark is applied in the discussion initiated in the file CircleBundleTransformation.html .

1. Circle-bundle contact with line

2. Intersecting-type case

3. Relation to line-homographies

See Also