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There are various sets of exceptional lines, or lines whose images are not unique.

The most obvious of these is the quadratic complex of tangents to Q, each line of which is transformed into the entire pencil of lines tangent to Q at the image of the point of tangency of the given line.

Thus pencils of tangents to Q are transformed into pencils of tangents.

It is interesting that a 1: 1 correspondence can be established between the lines of two such pencils, so that in a sense a unique image can actually be assigned to each tangent.

For the lines of any plane, **yp, meeting Q in a conic C, are transformed into the congruence of secants of the curve C' into which C is transformed in the point involution on Q.

In particular, tangents to C are transformed into tangents to C'.

Moreover, if Af and Af are two planes intersecting in a line l, tangent to Q at a point P, the two free intersections of the image curves Af and Af must coincide at P', the image of P, and at this point Af and Af must have a common tangent l'.

Hence, thought of as a line in a particular plane **yp, any tangent to Q has a unique image and moreover this image is the same for all planes through L.