projective cross-ratio
We introduce a quantity which is invariant under the projective group P(2).
This quantity is very significant geometrically, and admits several interesting
interpretations. The study of this invariant goes back to Greek geometry,
but it looks much more natural in a projective setting.
embedding planes
Projective geometry can be viewed as RP2, or as a plane together with its
ideal points. We look at how a projective conic appears in each model.
The plane model gives new insights into some affine and euclidean results.
pencils of lines
In the RP2 model, cross-ratio is a quantity associated with pencils of lines
- i.e. sets of four concurrent and coplanar lines. We begin the applications
of cross-ratio with a euclidean theorem.
examples of harmonic pencils
Cross-ratios with value -1 are especially significant. We show how they can
be used to establish the equivalence of the theorems of Ceva and Menelaus
in euclidean geometry.
central conics revisited
In affine gemetry, we said that an ellipse or hyperbola was a central conic.
From the projective point of view, the embedding of a p-conic always has a
'centre', though it may be ideal. This gives a new way to prove the parallel
chords theorem for a central plane conic.
the conjugate diameters theorem
A diameter of a central conic is a line through the centre. We shall
see that each diameter can be associated with another, known as
its conjugate. This can be done with affine methods, but we would
need to consider ellipses and hyperbolas separately. By working in
projective geometry, we get a unified approach.
betweenness and interiors
In euclidean, similarity and affine geometry, we have the concept
of betweenness. Given three collinear points, one lies between the
other two. This is preserved by the appropriate transformations.
The concept of line segment is based on betweenness.
We investigate the possibility of extending the ideas to projective
geometry.
projective transformations of conics
In the algebra pages, we obtained algebraic descriptions of p-conics, poles and polars.
We also have an algebraic description of projective transformations. We now show that
projective transformations preserve each type of object.
the interior-exterior theorem for projective conics
Beginning with an intuitive idea of the interior of a cone, we define the interior of a
projective conic. We show that it can be described in terms of poles and polars, and
that it is a projective concept.
the two point theorem of projective geometry
From the interior-exterior theorem, we know that a projective transfromation t
maps an interior (exterior) p-point of the p-conic C to an interior (exterior) p-point
of t(C). It turns out that this is the only restriction on the image of a p-point.
This leads to a two-point theorem for projective conics.
afffine and projective symmetries
We show how the two point therorm for projective conics leads to one and two
point theorems for plane conics in affine geometry. This approach does not
involve the determination of the elements of the affine symmetry goups.
explicit determination of symmetries
In establishing the connection between projective and hyperbolic geometries,
we had to investigate the projective symmetries of the projective conic with
equation x2 + y2 = z2. Here, we show that we can
describe the projective
symmetries of xy + yz + zx = 0 explicitly. We know of no applications of
this - it is purely an example to show how it can be done.
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