For let A, A′ and B, B′ be two pairs of conjugate points, the centre, I the point at infinity, then
(AB, OI) = (A′B′, IO),
or
OA · OA′ = OB · OB′.
In order to determine the distances of the foci from the centre, we write F for A and A′ and get
OF² = c; OF = ±√c.
Hence if c is positive OF is real, and has two values, equal and opposite. The involution is hyperbolic.
If c = 0, OF = 0, and the two foci both coincide with the centre. If c is negative, √c becomes imaginary, and there are no foci. Hence we may write—
| In an hyperbolic involution, | OA · OA′ = k², |
| In a parabolic involution, | OA · OA′ = 0, |
| In an elliptic involution, | OA · OA′ = −k². |
From these expressions it follows that conjugate points A, A′ in an hyperbolic involution lie on the same side of the centre, and in an elliptic involution on opposite sides of the centre, and that in a parabolic involution one coincides with the centre.