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A surprising property of conics is that every circle intersects the
ideal line, *W*=0, at two fixed points. To see this, note that
a circle is a conic with all off-diagonal elements (*c*_{12}, *c*_{13},
and *c*_{23}) set to zero and all diagonal elements equal:

*X*^{2} + *Y*^{2} + *W*^{2} = 0,

which therefore intersects the ideal line *W*=0 at

*X*^{2} + *Y*^{2} = 0.

This equation has two complex roots, known as the
*absolute points*:
and
.
(Although we have, for simplicity, assumed that
homogeneous coordinates are real, they can
in general be the elements of
any commutative field in which
[1, p. 112].)
It will be shown in the next two
subsections that the absolute points remain invariant under similarity
transformations, which makes them useful for determining the
angle between two lines.

*Stanley Birchfield*

*1998-04-23*