LineSimilarityAxis

Line similarity axis

Here we talk about a line connected with a special case of homography between lines e, e' defined by x'=(a*x+b) ( a particular case of the general homography-relation x' = (a*x+b)/(c*x+d)). The geometric locus of intersection points P of lines XY', X'Y is a line (red line below), called line-similarity-axis. This was proved in the general case in LineHomographyAxis.html . This line is the polar of the intersection point I of lines e, e', with respect to the parabola (c) which envelopes all lines XX'. The generation of conics that way (through envelopes of lines joining x to F(x), where F is a homography) is the Chasles-Steiner method to define a conic.

The relation between x, x' implies the relations: |xy|/|xz| = |x'y'|/|x'z'|, and |xy|/|x'y'| = |xz|/|x'z'|. Thus the subject reduces to the discussion initiated in Thales_General.html .
The determination of the focus F of the parabola and its subsequent construction is explained in ThalesParabola.html . The property of I to be the pole of the line-similarity-axis follows from the definition of the parabola. In the case the two lines are parallel (I goes to infinity) the parabola reduces to a point Q, all lines XX' going through that point. In that case the line-similarity-axis becomes the polar of Q with respect to the degenerate conic represented by the two parallel lines e and e'.
The real clue of the whole story is the property of the tangents of a parabola explained in ParabolaProperty.html .
The subject studied here gives another aspect of the theorem of Pappus discussed in PappusLines.html .

Line similarity axis

See Also