CHASLES' THEOREM
By Hubert Shutrick
1. Pencils of Lines
Let
and
where
and
are two different lines in the plane. Each pair
such that
defines a line
and these lines form a pencil of lines with
base
.
For
the parameters
give the same line as
and so the parameters are homogeneous. If the lines
and
have a common point, then any line in the pencil also goes through the point.
Conversely, if
is any line through the common point, then the columns of the matrix
are linearly dependent using the common point, and, since the first two rows are independent, the last row must be a linear combination of them. Hence,
is in the pencil. A similar argument shows that, if
and
are parallel and
is also parallel, then it is in the pencil: the pencil consists of all lines
parallel with the base lines.
2. Change of Base
If
give two distinct lines of the pencil, then
can be a new base.
The transition from
to
is given by the matrix formula
It is important to note that the base is determined by
but not by the lines
and
because, if
,
the new base
has the same base lines with parameters
but the other lines get new parameters. To define the base geometrically, a
third line, say
is required and the above change of base gives it parameters
.
3. Cross-ratio
In terms of the homogeneous parameters for the pencil, the convenient definition of cross-ratio is
It agrees with the one parameter version if the parameter is replaced by
.
Its importance in this context is that it is depends only on the four lines and not on the choice of homogeneous parameters. This becomes obvious using that the determinant of the product of two matrices is the product of the determinants:
If three different points are given along with the cross-ratio
of them with a forth point, then the forth point is uniquely determined. To
prove this, choose the base so that the points have parameters
.
By the symmetry properties of cross-ratio they can be assumed to occur in the
above order.
4. Projective Plane
The projective plane is obtained from the ordinary Euclidean plane by adjoining an extra point for each pencil of parallel lines. It is the point `at infinity' where the lines of the pencil meet. To accommodate the new points it is convenient to introduce homogeneous coordinates. This is done by replacing the usual
coordinates by
.
Each
represents a point of the plane but, if
,
then
represents the same point so the coordinates are homogeneous.
A line in the plane is given by an equation
,
where not all of the coefficients are zero. In particular,
gives the line at infinity and parallel lines
meet at
as required.
The line joining points
and
can be described by homogeneous parameters
since
is on the line for each
.
If
is a point not on the line, then the equation of the line joining it to the
point with parameters
is given by
This can be rewritten
The parameters for the line are the parameters for the pencil of lines through
.
Therefore the cross-ratio of four points on a line is the same as the
cross-ratio of the lines joining a point to them implying that cross-ratio is
invariant under projection.
5. Change of Coordinates
The coordinate vectors
form a base for the coordinate system in the sense that
.
If the points defined by
are not collinear, then they can form a new base:
In matrix form
The matrix is invertible since the points are not collinear.
Note that the points defined by
are not sufficient to determine the base because
defines a new base with the same base points if
are not all equal. To define a base geometrically, an extra point is needed,
say,
which should not be on a line joining two of the other three points. The
change of base above then gives this point coordinates
.
6. Chasles' Theorem
The theorem states that, for four points on a non-degenerate conic, the cross-ratio of the pencil of four lines from a fifth point of the conic to the given points does not depend on the choice of the fifth point.
To prove this choose a coordinate system such that
and
are points of the conic,
is the intersection of the tangents from
and
and the
point is on the conic. The transformation of coordinates is linear and
homogeneous so the conic will still be of the form
Since the line
is tangent at
,
to give the double root
.
Similarly,
.
Inserting
reduces the equation to
Consider the pencil of lines through
with base
.
The line with parameters
is
and it meets the conic again at the point
.
Thus,
also give a homogeneous parametrisation of the conic. Now, consider the pencil
through
with base
.
The line
also meets the conic again at
:
it gives the same parametrisation of the conic. It follows that the
cross-ratios of pencils from
and
to four points of the conic must be the same. Since
and
were arbitrary points of the conic, the theorem is true.
(Chasles' theorem leads to an elegant projective proof of Pascal's Theorem. Also, another proof of Chasles' Theorem appears elsewhere.)
Copyright © 1996-2008 Alexander Bogomolny
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