Wilson Stothers' Geometry Pages

What's new - archive January-June 2001

june 2001

the interchange lemma for projective conics
The three point theorem involves three p-points on a p-conic.
The two point theorem involves one p-point on a p-conic and one not on the p-conic.
This suggests that controlling a p-point not on the p-conic is more demanding. It is no
surprise, then, that if we take two pairs of p-points inside a p-conic, there is usually
no projective transformation mapping one pair to the other.

algebraic characterization of the interior
We know that a projective conic C has equation f(x) = xTMx = 0, with M symmetric.
If P =[p] is not on C, then f(p) will be non-zero, and thus either positive or negative.
It turns out that all p-points inside C have f(p) with the same sign, and all p-points
outside C have f(p) of the opposite sign to that for the interior p-points.
This leads to an interesing invariant.

projective transformations and conics
In May, we produced a number of pages which invesigated projective conics and
projective transformations. This page provides a logical route through these pages,
with links to some possible digressions.

may 2001

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.

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

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.

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.

april 2001

inversive geometry

an inversive invariant
Here, we introduce a quantity which is invariant under the Mobius group, and is
"almost" invariant under the full inversive group. This quantity is very significant
geometrically, and admits several interesting interpretations.

i-lines and the invariant
We show how the inversive cross-ratio can be used to characterize i-lines
and i-segments. Some of our results can be interpreted as familiar theorems
of euclidean geometry, such as the theorems on angles on the same segment
of a circle, and cyclic quadrilaterals.

ptolemy's theorem as an inversive result
In its strong form, ptolemy's theorem involves an object which may be a line or circle.
This is a fairly broad hint that inversive geometry may be relevant. The theorem also
involves certain ratios, which we can now recognise as the modulus of our invariant.
The theorem which results can also be viewed as an inversive analogue of the familiar
triangle inequality.

march 2001

Search facility added

the klein view

models of hyperbolic geometry
In the hyperbolic pages, we used the poincare disk model of hyperbolic geometry.
Here, we look at two other models. We have the klein-beltrami model. This again
uses the unit disk, but now the hyperbolic lines are segments of euclidean lines.
We also introduce the minkowski model. This uses a surface in three dimensions.
The model is related to special relativity. It is also a key step in understanding the
relation between hyperbolic and projective geometries.

hyperbolic and projective geometries
This is one of the main features of the klein view of geometry
We show that the hyperbolic group is isomorphic to a subgroup of the projective
group. In fact, it is isomorphic to the symmetry group of a conic. The link is the
minkowski model of hyperbolic geometry. Unfortunately, the proof is complicated.

hierarchies of geometries
Summing up. We describe the relationships among the geometries we have looked
at. In particular, euclidean and hyperbolic geometries are shown to be related to
projective, and to inversive geometries. This is achieved by considering the groups
of the geometries - a highlight of the klein view.

plane conics

some special conics and their tangents
We look at the tangents to some special conics. We conider one parabola and
one hyperbola, and the circle. These are the key to the investigation of conics
and tangents in affine geometry.

affine geometry

the classification of conics in affine geometry
We define the conics as non-degenerate loci with equations quadratic in x and y.
This is necessary as we do not have the concept of length. We show that there
are exactly three affine congruence classes, the parabolas, thy hyperbolas, and
the ellipses (including circles). For each class, we give a member with a simple

affine symmetry groups of conics
We investigate the groups of affine symmetries of conics. Of course, the symmetry
groups for cojugate conics are conjugate, so we need only look at one from each
class. In contrast with the euclidean case, all the groups are infinite. In particular,
we show that the group acts transitively on the conic. this shows that there are no
"special" points - unlike, say the vertex of a parabola in euclidean geometry. Further,
the group of a parabola is actually doubly transitive. the structure of the groups are
sufficient to distinguish the classes.

tangents of affine conics
With a slightly surprising definition, we show that tangency is an affine concept.
As with the symmetry groups, the study of tangents allows us to distinguish the
classes. For an ellipse, every family of parallel lines includes two tangents, for a
parabola, there is one family with no tangents. For a hyperbola, there are two
exceptional families, and each containa an asymptote. In this way, we conclude
that asymptotes are affine objects.

projective conics

the three point theorem
In affine geometry, we have one- and two-point theorems describing the transitivity
of the affine symmetry groups of conics. In projective geometry, there is only one
congruence class of conics. We show that the projective symmetry group is triply
transitive. This is useful in proving theorems about conics. An additional result is the
Parametrisation Theorem which gives a parametric description of a conic.

the projective symmetry group of a conic
Since all conics are projective congruent, all the projective symmetry groups are
conjugate, so it is enough to study just one. We give an algebraic description of
the elements of the projective symmetry group of one particular conic.

the five point theorem
In euclidean geometry, three non-collinear poits determine a unique circle. Here,
we show that five p-points, no three of which are collinear, determine a unique
projective conic. This is illustrated by a CabriJava applet.

February 2001

plane conics

A new section looking at plane conics. This has been requested by several
high-school students doing projects! I will try to cover all the elementary
material, but to use the klein philosophy to avoid too much tedious algebra.

  • plane conics
    Plane conics can be defined in three ways:
    • as loci satisfying the focus-directrix property,
    • as loci with equations quadratic in x and y,
    • as plane sections of a right-circular cone.
    We look at these approaches, and see that they are really the same.
  • standard forms of the equations of plane conics
    We derive the familiar standard forms for the equations of the ellipse,
    the parabola and the hyperbola.
  • sketches of the standard conics
    We obtain sketches of the standard plane conics, and investigate the kinds
    of symmetries each curve has. It is important to note that the number and the
    nature of the symmetries - as reflections, rotations and so on - does not depend
    on the choice of axes. A knowledge of the symmetries is often useful.
  • loci of equations quadratic in x and y
    The standard equations for plane conics turn out to be quadratic in x and y.
    We consider the nature of the loci produced by such equations. We find
    that, apart from the plane conics, we may get simpler loci, consisting of
    at most one point, a line, or a pair of lines. These are the degenerate conics.
  • pictures of sections of a cone
    An introduction to thre right-circular cone, and its sections by planes.
    Once we investigate the equations of such sections, we will see that
    the are plane conics or degenerate conics.
  • equations for sections of a cone
    This contains the algebra needed to check that the sections of a cone are
    actually plane conics.
  • plane conics in euclidean and similarity geometry
    We investigate the congruence classes of plane conics in euclidean and in
    similarity geometries. Using the standard forms for the equations, we see
    that these are distinguished in euclidean geometry by shape and size,
    while in similarity geometry, simply by shape.
the klein view
  • symmetry groups
    The most common way to introduce groups is to discuss the symmetries
    of plane figures, i.e. euclidean transformations which map the figure to
    itself. From the kleinian view, there is no reason why we should restrict
    attention to euclidean transformations.
  • geometrical invariants
    The study of a geometry consists largely of identifying and interpreting
    the geometrical properties, i.e. the invariants of the group defining the
    geometry. These may be quantities, such as length and angle, or notions
    such as betweenness and parallelism.
  • invariants of similarity geometry
    Here, we begin with a quantity which is invariant under the group
    of direct similarities, S+(2). We derive the main invariants of
    similarity geometry, those of ratio and size of angle.
  • parallel projections and the affine group
    We introduce a family of transformations of the plane which turn out
    to generate the affine group A(2). The definition is purely geometric.
    It can be motivated by thinking about the relationship between an
    object and its shadow.
January 2001

the klein view

  • a weird geometry
    The hyperbolic group can be viewed as the set of restrictions of elements
    of a subgroup of the inversive group. This subgroup also gives rise to two
    other geometries. One is isomorphic to hyperbolic. The other is, well, weird!
  • perspectivities and the projective group
    Introducing geometrically defined transformations of the projective plane.
    These generate the projective group P(2). They are closely related to
    the theory of perspective in art.
  • the affine and projective groups
    The affine group A(2) is shown to be isomorphic to a subgroup of the
    projective group P(2). The Fundamental Theorem of Affine Geometry then
    follows from that of Projective Geometry.
  • an invariant of inversive geometry
    The idea of inversive cross-ratio is defined, and shown to be geometrically
    significant. As an application, we prove the invariance of ratio and angle
    in similarity geometry directly from the view of S(2) as a "subgroup" of I(2).
hyperbolic geometry
  • the trigonometry of hyperbolic triangles
    There are hyperbolic analogues of the standard trigonometric results about
    triangles. These involve hyperbolic functions of the lengths of the sides, but
    the proofs mimic those for euclidean geometry. Threre is one exception - viz
    that the angles of a hyperbolic triangle determine the lengths of the sides!
  • the theorems of ceva and menelaus
    Using trigonometry, and the new concept of hyperbolic ratio, we prove the
    hyperbolic analogues of the theorems of ceva and menelaus. Although there
    are complications because of the posibility of ultraparallels, there are also
    valid converses.
  • further theorems about triangles
    The converse of the theorem of ceva is used to prove analogues of the
    medians and the incentre theorem for hyperbolic triangles. The case of
    hyperbolic altitudes is also discussed.

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