[alogo] 1. Homographic relation between two lines

Homographic relation between two lines {L, L'} is called an invertible map F: L ----> L' which preserves the cross-ratio of any four points {,B,C,D} of line L. In other words, a map such that, for every quadruple (A,B,C,D) and their images ('=F(A), B'=F(B), C'=F(C), D'=F(D)) it is valid: (,,C,D) = (A',B',C',D').

Taking arbitrary line coordinates on the two lines (each defined by two points determining 0 and 1), it follows that such a transformation is given by a function of the form

[0_0] [0_1]

The most general homography between two lines L, L' is constructed by the following recipe:
a) Take three arbitrary points A, B, C on L,
b) Take three arbitrary points A', B', C' on L',
c) To D in L make correspond D' on L', such that the cross-ratios are equal (ABCD) = (A'B'C'D').

[alogo] 2. Geometric construction of a line homography

Next geometric construction of a line-homography between the lines {L, L'} follows the previous recipe and uses the fact that a bundle of lines through the same point defines the same cross-ratio on any secant of the lines of the bundle.

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]

Geometric construction of D' = F(D).
1) Draw parallel L'' from A' to L.
2) Project parallel to AA' , on L'' : B --> B'', C --> C'', D --> D''.
3) Join C'C'' and B'B'', find intersection X.
4) Join D'', X, find intersection of XD'' with L'. That's D'.
For another, more algebraic construction of the homography look at Chasles_Steiner_Envelope.html
Line DD' (for varying D on L) envelopes a conic. From the projective geometry viewpoint this is the dual of the Chasles-Steiner conic construction theorem ( Chasles_Steiner.html ).

[alogo] 3. Related application

Given two independent vectors {u,v} find the condition such that all lines passing through {su, tv} pass also through a constant point K = k1u + k2 v.

[0_0]

Adopting the coordinate system with basis vectors {u,v}, lines are given by equations ax+by+c=0.
- point (s,0) on such a line => as+c=0,
- point (0,t) on such a line => bt+c=0,
hence the line has the form x/s + y/t - 1 = 0 <==> xt + ys - st = 0.
This line passes through a fixed point K(k1,k2) <==> k1t + k2s - st = 0.
This represents a homographic relation that can be written in the form
k1t + k2s + k3st = 0.
Inverselly every such homographic relation defines points {su, tv} such that the line joining them passes through K with coordinates (-k1/k3, -k2/k3) with respect to the basis {u,v}.

Remark The addition of a constant k4 to the relation radically changes the behaviour of lines joining points {su, tv}. In fact the envelope of such lines, whereby {s,t} are related by relation of the form
k1t + k2s + k3st + k4 = 0 (with non-zero k4),
is a conic.

See Also

Chasles_Steiner.html
Chasles_Steiner_Envelope.html

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