A Brief Introduction to Real
Projective Geometry
Bruce Cohen
Lowell High School, SFUSD
math.cohen@gmail.com
http://www.cgl.ucsf.edu/home/bic
David Sklar
San Francisco State University
dsklar@sfsu.edu
Asilomar - December 2010
Topics
Early History, Perspective, Constructions, and Projective
Theorems in Euclidean Geometry
A Brief Look at Axioms of Projective and Euclidean Geometry
Transformations, Groups and Klein’s Definition of Geometry
Analytic Geometry of the Real Projective Plane, Coordinates,
Transformations, Lines and Conics
Geometric Optics and the Projective Equivalence of Conics
Perspective
From John Stillwell’s books Mathematics and its History
and The Four Pillars of Geometry
Perspective
Perspective
From Geometry and the Imagination by Hilbert and Cohn-Vossen
Dates:
Brunelleschi
1413
Alberti
1435
(1525)
Early History - Projective Theorems in Euclidean Geometry
Pappus (300ad): If A, B, C are
three points on one line, A, B, C on
another line, and if the three lines
AB, BC , CA meet AB, BC , C A ,
respectively, then the three points of
intersection A, B, C  are collinear.
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.
A 
B
C
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.
Jean-Victor Poncelet
1788-1867
Jakob Steiner
1796-1863
Mario Pieri
1860-1913
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.
Plane Euclidean and Projective Geometries
Undefined Terms: “point”, “line”, and the relation “incidence”
Axioms of Incidence
Euclidean
1. There exist at least three points
not incident with the same line
1. There exist a point and a line that
are not incident.
2. Every line is incident with at least
two distinct points.
3. Every point is incident with at
least two distinct lines.
2. Every line is incident with at least
three distinct points.
3. Every point is incident with at
least three distinct lines.
4. Any two distinct points are incident
with one and only one line.
4. Any two distinct points are incident
with one and only one line.
5. Any two distinct lines are
incident with at most one point.
5. Any two distinct lines are incident
with one and only one point.
Projective
Note: The main differences between these is that the projective
axioms do not allow for the possibility that two lines don’t intersect,
and the complete duality between “point” and “line”.
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 don’t intersect.
Another important difference is the complete duality between points and
lines in the projective axioms.
The smallest Euclidean “Incidence Geometry” has 3 points. It’s 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 we’ll look at a nice analytic or
coordinate model of projective geometry analogous to the familiar
Cartesian analytic model of Euclidean geometry. .
A Useful Way to Think about the Projective Plane
The projective plane may be thought of as the ordinary real
affine (Cartesian) plane R 2, with an additional line called the
line at infinity.
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.
An Analytic Model of the Real Projective Plane
A point in the real projective plane is a set of ordered triples of real numbers,
called the homogeneous coordinates of the point, denoted by  x1 , x2 , x3  where
0,0,0 is excluded and where two ordered triples  x1, x2 , x3  and  y1, y2 , y3 
represent the same point if and only if  y1, y2 , y3    kx1, kx2 , kx3  for some k  0 .
A line is also defined as a set of real ordered triples, denoted by u1, u2 , u3  where
0,0,0 is excluded and where u1, u2 , u3  and v1, v2 , v3  represent the same line if
and only if v1, v2 , v3   ku1, ku2 , ku3  for some k  0. A point  x1 , x2 , x3  and a line
u1, u2 , u3  are incident if and only if u1x1  u2 x2  u3 x3  0 (duality). The linear
homogeneous equation u1 x1  u2 x2  u3 x3  0 is the point equation of the line
u1, u2 , u3  and the line equation of the point  x1, x2 , x3  .
The Cartesian (affine) plane R 2 can be embedded in the real projective plane
by indentifying the point  x, y  with the triple  x, y,1. The line at infinity
corresponds to the points  x, y,0 where the ratio of the x and y coordinates
determines a specific points at infinity. Points at infinity correspond to
directions in the affine plane
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.
Some Familiar Subgeometries
Geometry
Projective
Affine
Euclidean
Similarity
Euclidean
Congruence
Transformation Group
Set
Projective plane
R2
Affine plane :
projective plane
with the line at
infinity omitted
R2
Affine plane
Affine plane R 2
Collineations: transformations that map
straight lines to straight lines
Affine transformations: transformations
that map parallel lines to parallel lines
(these map the line at infinity to itself)
transformations that are generated by
rotations, reflections, translations and
dilations (isotropic scalings)
Isometries: affine transformations that
are generated by rotations, reflections,
and translations
Some Familiar Subgeometries
Geometry
Transformation Group
Equivalent Figures
Projective
Collineations: transformations that map
straight lines to straight lines
All quadrilaterals and all
conics
Affine
Affine transformations: collineations that
map the line at infinity to itself (these
take parallel lines to parallel lines)
All triangles, all
parabolas, all hyperbolas,
all ellipses
Euclidean
Similarity
affine transformations that are generated
by rotations, reflections, dilations
(isotropic scaling), and translations
Triangles of the same
shape, ellipses of the
same shape, and all
parabolas
Euclidean
Congruence
Isometries: affine transformations that
are generated by rotations, reflections,
and translations
Only figures of the same
size and shape
Analytic Transformation Geometry
Transformations
Geometry
Projective
x
 y
 
 z 
 x 
 y 
 
 z  
 a11 x  a12 y  a13 z 
a x  a y  a z  
22
23 
 21
 a31 x  a32 y  a33 z 
 a11 a12
a
 21 a22
 a31 a32
a13   x 
 x
a23   y   A  y 
 z 
a33   z 
, A invertible
Affine
 x
 y
 
 z 
 x   a11 x  a12 y  a13 z 
x
 y   a x  a y  a z   A  y  , A invertible
22
23 
   21
 
 z   
 z 
z 
Setting z to 1 we get the affine transformations In non-homogeneous coordinates
 x   a11 x  a12 y  a13 
 y   a x  a y  a  , so  x    a11 x  a12 y  a13  , det  a11 a12   0
22
23 
a

 y  a x  a y  a 
   21
a
   21
22
23 
 21 22 
 z   

1
Gaussian First Order Optics
y y
 x, y 
x
x
f
 x, y
Lens
Gaussian First Order Optics
y y
 x, y 
x
x
f
 x, y
Gaussian First Order Optics
y y
 x, y 
y
x f
x
f
x
y
y
 x, y
x
x

 y y
 y
y

f
x f

xf
x f
 yf
y 
x f
x 

xf
x
x  f
 yf
y
x  f
Gaussian First Order Optics in Homogeneous
Coordinates
xf
x 
x f
 yf

y 
x f

Note: If
x  xf
y   yf
 x   xf   f
or  y     yf    0
  
 
 z    x  f   1
z  x  f
x f
 x   xf 
 f 
 y    yf   f   y 
  

 
 z    x  f 
 0 
So the vertical line x  f is
mapped to the line at infinity.
0
f
0
0   x
0   y 
 f   1 
Also
f
0

 1
0
f
0
 f 
0   x   fx 


y
0   y     fy   x  f 
x

 f   0   x 
 1 


So the vertical line at infinity is
mapped to the vertical line x 
f.
Projective Equivalence of the Conics
Bruce’s GeoGebra Demonstrations
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
Some extra slides not
used in the presentation
Projective Theorems in Euclidean Geometry
Pappus (300ad): If A, B, C are three points on one line, A, B, C on another
line, and if the three lines AB, BC , CA meet AB, BC, CA respectively, then
the three points of intersection D, E, F are collinear.
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.
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.
Part I
y y
 x, y 
y
x
x
x

 y y
x f
f
x
y
x
 x 
y
y
 x, y
 y
y
 yf

 y 
f
x f
x f
x
 x   yf
 x 
y 

y
y  x f
y

xf

 x f

xf
x 
x f
 yf
y 
x f
Part I
y y
 x, y
 x, y 
x
f
x
Part I
y y
 x, y
 x, y 
x
f
x
Part I
 x, y 
y y
 x, y
x
f
x
Part I
 x, y 
y y
 x, y
x
f
x
“Poncelet’s Alternative”: The Great Poncelet Theorem for Circles
Let C and D be two circles, with D inside D. Construct a sequence of points
P1 , P2 , . . . , Pi , . . . on D, such that for each i the line segment PP
i i 1 is tangent
to C and (for i  2) distinct from PP
i i 1 . "Poncelet's Alternative" says that
if Pn  P1 for some n  1, then for any other initial point P1 we will have Pn  P1.