A Brief Introduction to Real Projective Geometry

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A Brief Introduction to Real Projective Geometry
Bruce Cohen Lowell High School,
SFUSD math.cohen_at_gmail.com http//www.cgl.ucsf.edu
/home/bic
David Sklar San Francisco State
University dsklar_at_sfsu.edu
Asilomar - December 2010
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Topics
Early History, Perspective, Constructions, and
Projective Theorems in Euclidean Geometry
A Brief Look at Axioms of Projective and
Euclidean Geometry
Transformations, Groups and Kleins Definition of
Geometry
Analytic Geometry of the Real Projective Plane,
Coordinates, Transformations, Lines and
Conics
Geometric Optics and the Projective Equivalence
of Conics
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Perspective
From John Stillwells books Mathematics and its
History and The Four Pillars of Geometry
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Perspective
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Perspective
From Geometry and the Imagination by Hilbert and
Cohn-Vossen
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Dates
Brunelleschi 1413
Alberti 1435
(1525)
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Early History - Projective Theorems in Euclidean
Geometry
Desargues (1639) If two triangles are in
perspective from a point, and their pairs of
corresponding sides meet, then the three points
of intersection are collinear.
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More Recent History
Projective Geometry as we know it today emerged
in the early nineteenth century in the works of
Gergonne, Poncelet, and later Steiner, Moebius,
Plucker, and Von Staudt. Work at the level of
the foundations of mathematics and geometry,
initiated by Hilbert, was carried out by Mario
Pieri, for projective geometry near the beginning
of the twentieth century.
Mario Pieri 1860-1913
Jean-Victor Poncelet 1788-1867
Jakob Steiner 1796-1863
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Abstract Axiom Systems
One must be able to say at all times instead
of points, straight lines, and planes tables,
chairs, and beer mugs.
-- David Hilbert about 1890
An Abstract Axiom System consists of a set of
undefined terms and a set of axioms or
statements about the undefined terms.
If we can assign meanings to the undefined terms
in such a way that the axioms are true
statements we say we have a model of the abstract
axiom system. Then all theorems deduced from the
axiom system are true in the model.
Plane Analytic Geometry provides a familiar model
for the abstract axiom system of Euclidean
Geometry.
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Plane Euclidean and Projective Geometries
Undefined Terms point, line, and the
relation incidence
Axioms of Incidence
Euclidean
Projective
Note The main differences between these is that
the projective axioms do not allow for the
possibility that two lines dont intersect, and
the complete duality between point and line.
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Some Comments on the Axioms
The main difference between these axioms of
incidence is that the projective axioms do not
allow for the possibility that two lines dont
intersect.
Another important difference is the complete
duality between points and lines in the
projective axioms.
The smallest Euclidean Incidence Geometry has 3
points. Its not so obvious that the smallest
Projective Geometry has 7.
To develop a complete axiom system for the Real
Euclidean Plane we would need to add axioms of
order, axioms of congruence, an axiom of
parallels, and axioms of continuity.
To develop a complete axiom system for the Real
Projective Plane we would need to add an axiom of
perspective (Desargues Theorem), axioms of
order, and an axiom of continuity.
This would take much too long, but well look at
a nice analytic or coordinate model of projective
geometry analogous to the familiar Cartesian
analytic model of Euclidean geometry. .
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A Useful Way to Think about the Projective Plane
A pair of parallel lines intersect at a unique
point on the line at infinity, with pairs of
parallel lines in different directions
intersecting the line at infinity at different
points.
Every line (except the line at infinity itself)
intersects the line at infinity at exactly one
point. A projective line is a closed loop.
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An Analytic Model of the Real Projective Plane
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A Definition of Geometry
A group of transformations G on a set S is a set
of invertible functions from S onto S such that
the set is closed under composition and for each
function in the set its inverse is also in the
set.
A geometry is the study of those properties of a
set S which remain invariant when the elements of
S are subjected to the transformations of some
group of transformations.
Felix Klein 1872 The Erlangen Program
The study of those properties of a set S which
remain invariant when the elements of S are
subjected to the transformations of a subgroup of
G is a subgeometry of the geometry determined by
the group G.
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Some Familiar Subgeometries
Set
Transformation Group
Geometry
Collineations transformations that map straight
lines to straight lines
Projective
Projective plane
Affine
Affine transformations transformations that map
parallel lines to parallel lines (these map the
line at infinity to itself)
Euclidean Similarity
transformations that are generated by rotations,
reflections, translations and dilations
(isotropic scalings)
Euclidean Congruence
Isometries affine transformations that are
generated by rotations, reflections, and
translations
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Some Familiar Subgeometries
Equivalent Figures
Transformation Group
Geometry
All quadrilaterals and all conics
Collineations transformations that map straight
lines to straight lines
Projective
Affine transformations collineations that map
the line at infinity to itself (these take
parallel lines to parallel lines)
Affine
All triangles, all parabolas, all hyperbolas, all
ellipses
Euclidean Similarity
Triangles of the same shape, ellipses of the same
shape, and all parabolas
affine transformations that are generated by
rotations, reflections, dilations (isotropic
scaling), and translations
Only figures of the same size and shape
Isometries affine transformations that are
generated by rotations, reflections, and
translations
Euclidean Congruence
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Analytic Transformation Geometry
Transformations
Geometry
Projective
, A invertible
Affine
, A invertible
Setting z to 1 we get the affine transformations
In non-homogeneous coordinates
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Gaussian First Order Optics
Lens
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Gaussian First Order Optics
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Gaussian First Order Optics
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Gaussian First Order Optics in Homogeneous
Coordinates
or
Note If
Also
So the vertical line is mapped to
the line at infinity.
So the vertical line at infinity is mapped to the
vertical line .
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Projective Equivalence of the Conics
Bruces GeoGebra Demonstrations
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Bibliography
1. Hilbert and Cohn-Vossen, Geometry and the
Imagination, Chelsea Publishing Company,
New York, 1952
2. H.S.M. Coxeter S.L. Greitzer, Geometry
Revisited, The Mathematics Association of
America, Washington, D.C., 1967
3. Constance Reid, Hilbert, Copernicus an
imprint of Springer-Verlag, New York, 1996
4. A. Siedenberg, Lectures in Projective to
Geometry, D. Van Nostrand Company, 1967
5. J.T. Smith E.A. Marchisotto, The Legacy
of Mario Pieri in Geometry and Arithmetic,
Birkhäuser, 2007
6. John Stillwell, The Four Pillars of
Geometry, Springer Science Business Media, LLC,
2005
7. John Stillwell, Mathematics and its
History, 2nd Edition, Springer-Verlag, New York,
2002
8. Annita Tuller, A Modern Introduction to
Geometries, D. Van Nostrand Company, 1967
9. Wikipedia article, Projective geometry
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Some extra slides not used in the presentation
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Projective Theorems in Euclidean Geometry
Pappus (300ad) If A, B, C are three points on
one line, on another line, and if
the three lines meet
respectively, then the three
points of intersection D, E, F are collinear.
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Projective Theorems in Euclidean Geometry
Desargues (1640) If two triangles are in
perspective from a point, and if their pairs of
corresponding sides meet, then the three points
of intersection are collinear.
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Projective Theorems in Euclidean Geometry
Pascal (1640) If all six vertices of a hexagon
lie on a circle (conic) and the three pairs of
opposite sides intersect, then the three points
of intersection are collinear.
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Part I
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Part I
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Part I
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Part I
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Part I
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Poncelets Alternative The Great Poncelet
Theorem for Circles