In another question connected with this, the problem of drawing tangents to any curve, Descartes was drawn into a controversy with Pierre (de) Fermat (1601-1663), Gilles Persone de Roberval (1602-1675), and Girard Desargues (1593-1661).
In projective geometry it may be defined as the conic which intersects the line at infinity in two real points, or to which it is possible to draw two real tangents from the centre.
If a rectangle be constructed about AA' and BB', the diagonals of this figure are the "asymptotes" of the curve; they are the tangents from the centre, and hence touch the curve at infinity.
Two tangents from any point are equally inclined to the focal distance of the point.
A diameter is a line through the centre and terminated by the curve: it bisects all chords parallel to the tangents at its extremities; the diameter parallel to these chords is its conjugate diameter.
How would you define tangents? Add your definition here.