For an arbitrary point C of the conic, consider the tangent (c) at C and its intersection points X(x) with (a) and Y(y) with (b). The correspondence y = f(x) between the line coordinates is a homographic relation. The figure below displays the graph of this function, which is a rectangular hyperbola, together with its asymptotic lines.

Note the location of the asymptotics corresponding to points B', C' symmetric to B, C w.r. to the center of the conic.

Note that this is, in some sense, the inverse of the construction of the conic as an envelope of lines, discussed in the file Chasles_Steiner_Envelope.html .

Chasles_Steiner_Envelope.html

HomographicRelation.html

Line_Homography.html

RectHypeRelation.html

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