## Chasles-Steiner envelope

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 .