Consider a homographic relation between two lines [a] and [b]. This can be realized by a function z = (a*x+b)/(c*x+d) between the points of the two lines. Here we identify point X with the real number (denoted by x) such that the following equation between oriented segment-lengths on line [a] is valid: [A1,X] = x[A1,A2]. Analogously on line [b]: [B1,Z] = z[B1,B2]. The homographic relation between the two lines amounts to a relation between the coordinates of points z and x: z = (a*x+b)/(c*x+d). The (dual) Chasles-Steiner method of definition describes a conic as the geometric locus of envelopes of lines [XZ]. For the geometric construction of the homography look at the file Line_Homography.html .
In the figure above, free movable (Ctrl+1) are the points {A1, A2, B1, B2} defining lines [a], [b]. X is a variable (Ctrl+2) point on line [a]. Z is a point on line [b] whose coordinate z depends on the coordinate x of X: z = (a*x+b)/(c*x+d). This defines line [XZ] which, for varying X (and its coordinate x) envelopes a conic. The dual construction of conics through homographies between pencils of lines is illustrated in the document Chasles_Steiner.html .
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Chasles-Steiner envelope ![[0_0]](Chasles_Steiner_Envelope_files/Chasles_Steiner_Envelope_0_0_0.png)
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![[1_0]](Chasles_Steiner_Envelope_files/Chasles_Steiner_Envelope_0_1_0.png)
![[1_1]](Chasles_Steiner_Envelope_files/Chasles_Steiner_Envelope_0_1_1.png)
![[1_2]](Chasles_Steiner_Envelope_files/Chasles_Steiner_Envelope_0_1_2.png)
![[1_3]](Chasles_Steiner_Envelope_files/Chasles_Steiner_Envelope_0_1_3.png)