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A pencil of planes in
is a family of planes having a
common line of intersection. The cross ratio of four planes of a pencil is the same as the cross ratio of the lines li of
intersection of the planes with fifth, transversal plane (see
fig. 3.3).
Figure 3.3:
A pencil of planes
|
Once again, different transversal planes give the same cross ratio as
the figures they give are projectively equivalent. The Möbius formula
also extends to this case: let P, Q be any two distinct points on the
axis of the plane pencil, and
be points lying on
each plane
(not on the axis), then
where
stands for a
determinant of
4-component column vectors.
Bill Triggs
1998-11-13