Chasles_Steiner2

1. Chasles-Steiner conic-generation

The Chasles-Steiner generation of conics creates the curve from two points {A,B} and an homographic relation between the pencils of lines of these two points. A simple way to realize this is to intersect a fixed line L with every line A_x from pencil-A at a point of L with coordinate x and correspond to x the point of L with coordinate y = (ax+b)/(cx+d) and then line B_y joining this point to B. The intersection point P of these two lines describes a conic (see Chasles_Steiner.html ).

2. The inverse

The inverse is also true: Take two points {A,B} on the conic c and let a third point P vary on this conic. Then consider the intersection points {x,y} of lines {AP,BP} with an arbitrary but fixed line L. Then {x,y} are related by a homographic relation y=(ax+b)/(cx+d). Here I identify the points of the line L with their coordinates, with respect to an arbitrary coordinate-system of the line.
The proof is an easy exercise that reduces to the discussion in Chasles_Steiner.html .

Next figure shows how important is the condition for the points {A,B} to be on the conic. In this figure the points {A,B} do not lie on the conic, and we repeat the same construction: For varying P on the conic consider the coordinate x of the intersection of AP with line L and the corresponding intersection y of BP with L.
Then we plot the points (x,y) for all possible positions of P. The plot shows that the curve cannot be the graph of a function in general.

Problem
Find the equation f(x,y)=0 satisfied by these points (x,y) in dependence of the four given data {c, L, A, B}.

File Chasles_Steiner3.html contains an exeptional case of this figure occuring when the conic c degenerates to a line.

1. Chasles-Steiner conic-generation

2. The inverse

See Also