Chasles_Steiner3

Chasles-Steiner inverse for degenerate conics

The figure below shows the graph of the relation f(x,y)=0 resulting by the following procedure:
(i) Take two arbitrary points {A,B} and a line L.
(ii) Consider an arbitrary but fixed line N and a point P varying on this line.
(iii) Draw lines {PA,PB} and their intersections {A_x,B_y} with line L.

The coordinates {x,y} of points {A_x,B_y} along line L satisfy a homographic relation y = (ax+b)/(cx+d), hence the graph is in general a rectangular hyperbola (red).

This follows from the inverse construction to the Chasles-Steiner generation of conics (see Chasles_Steiner.html ). In fact it suffices to consider the degenerate conic consisting of the pair of lines (M,N), where M is the line containing {A,B} and apply this inverse procedure.

Chasles-Steiner inverse for degenerate conics

See Also