A Brief Introduction to Real Projective Geometry

Asilomar - December 2010

Bruce CohenLowell High School, SFUSD

math.cohen@gmail.comhttp://www.cgl.ucsf.edu/home/bic

David SklarSan Francisco State University

dsklar@sfsu.edu

Topics

Early History, Perspective, Constructions, and Projective Theorems in Euclidean Geometry

A Brief Look at Axioms of Projective and Euclidean Geometry

Analytic Geometry of the Real Projective Plane, Coordinates, Transformations, Lines and Conics

Geometric Optics and the Projective Equivalence of Conics

Transformations, Groups and Klein’s Definition of Geometry

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

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.

C

BA

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 are collinear.

, , A B C

, , AB BC CA , , A B B C C A

, , A B C

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.

More Recent History

Jean-Victor Poncelet1788-1867

Jakob Steiner1796-1863

Mario Pieri1860-1913

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 GeometriesUndefined Terms: “point”, “line”, and the relation “incidence”

Axioms of IncidenceEuclidean 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”.

There exist at least three points not incident with the same line

1.

Every line is incident with at least two distinct points.

2.

Every point is incident with at least two distinct lines.

3.

Any two distinct points are incident with one and only one line.

4.

Any two distinct lines are incident with at most one point.

5.

There exist a point and a line that are not incident.

1.

Every line is incident with at least three distinct points.

2.

Every point is incident with at least three distinct lines.

3.

Any two distinct points are incident with one and only one line.

4.

Any two distinct lines are incident with one and only one point.

5.

The smallest Euclidean “Incidence Geometry” has 3 points. It’s not so obvious that the smallest Projective Geometry has 7.

Another important difference is the complete duality between points and lines in the projective axioms.

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.

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 , with an additional line called the line at infinity.

2R

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

1 2 3 1 2 3, , , ,y y y kx kx kx 0k

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 where is excluded and where two ordered triples and represent the same point if and only if for some .

1 2 3, ,x x x 1 2 3, ,y y y 1 2 3, ,x x x

0,0,0

A line is also defined as a set of real ordered triples, denoted by where is excluded and where and represent the same line if and only if for some . A point and a line are incident if and only if (duality). The linear homogeneous equation is the point equation of the line and the line equation of the point .

1 2 3, ,u u u 0,0,0 1 2 3, ,u u u 1 2 3, ,v v v

0k 1 2 3 1 2 3, , , ,v v v ku ku ku 1 2 3, ,x x x 1 2 3, ,u u u 1 1 2 2 3 3 0u x u x u x

1 1 2 2 3 3 0u x u x u x 1 2 3, ,u u u 1 2 3, ,x x x

The Cartesian (affine) plane can be embedded in the real projective plane by indentifying the point with the triple . The line at infinity corresponds to the points 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

2R , ,1x y ,x y

, ,0x y

A Definition of Geometry

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.

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.

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

Projective Projective plane Collineations: transformations that map straight lines to straight lines

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

Geometry Set Transformation Group

Affine plane 2R

Affine plane

2R

Affine plane : projective plane with the line at infinity omitted

2R

Some Familiar Subgeometries

Projective All quadrilaterals and all conics

Collineations: transformations that map straight lines to straight lines

Affine All triangles, all parabolas, all

hyperbolas, all ellipses

Affine transformations: collineations that map the line at infinity to itself (these take parallel lines to parallel lines)

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

Euclidean Congruence

Only figures of the same size and shape

Isometries: affine transformations that are generated by rotations, reflections, and translations

Geometry Equivalent FiguresTransformation Group

Analytic Transformation Geometry

Projective

Affine

Geometry Transformations

11 12 13

21 22 23

31 32 33

a x a y a z

a x a y a z

a x a y a z

, A invertible

11 12 13

21 22 23

x x a x a y a z x

y y a x a y a z A y

z z z z

, A invertible

Setting z to 1 we get the affine transformations In non-homogeneous coordinates

11 12 13

21 22 23 , so

1

x a x a y a

y a x a y a

z

11 12 13 11 12

21 22 23 21 22

, det 0a x a y a a ax

a x a y a a ay

11 12 13

21 22 23

31 32 33

a a a x

a a a y

a a a z

x

A y

z

x x

y y

z z

Gaussian First Order Optics

,x y

,x y

f

Lens

x

y y

x

,x y

,x y

fx

y y

x

Gaussian First Order Optics

,x y

,x y

fyy

y

x fx

y y

x

y y

f x f

x x

y y

xf

xx f

yf

yx f

x f

xx f

y f

yx f

Gaussian First Order Optics

xfx

x f

yf

yx f

Gaussian First Order Optics in Homogeneous Coordinates

x xfy yfz x f

or

x xf

y yf

z x f

0 0

0 0

1 0 1

f x

f y

f

Note: If x f

0

x xf f

y yf f y

z x f

So the vertical line is mapped to the line at infinity.

x f

0 0

0 0

1 0 01

ff x fx

yf y fy x f

xf x

Also

So the vertical line at infinity is mapped to the vertical line . x f

Projective Equivalence of the Conics

Bruce’s GeoGebra Demonstrations

2. H.S.M. Coxeter & S.L. Greitzer, Geometry Revisited, The Mathematics Association of America, Washington, D.C., 1967

Bibliography

5. J.T. Smith & E.A. Marchisotto, The Legacy of Mario Pieri in Geometry and Arithmetic, Birkhäuser, 2007

8. Annita Tuller, A Modern Introduction to Geometries, D. Van Nostrand Company, 1967

7. John Stillwell, Mathematics and its History, 2nd Edition, Springer-Verlag, New York, 2002

6. John Stillwell, The Four Pillars of Geometry, Springer Science + Business Media, LLC, 2005

4. A. Siedenberg, Lectures in Projective to Geometry, D. Van Nostrand Company, 1967

3. Constance Reid, Hilbert, Copernicus an imprint of Springer-Verlag, New York, 1996

1. Hilbert and Cohn-Vossen, Geometry and the Imagination, Chelsea Publishing Company, New York, 1952

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, on another line, and if the three lines meet respectively, then the three points of intersection D, E, F are collinear.

, , A B C , , AB BC CA , , A B B C C A

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

,x y

,x y

fyy

y

x fx

y y

x

y y

f x f

x x

y y

yfy

x f

x x yf xfx y

y y x f x f

xfx

x f

yf

yx f

xx y

y

Part I

,x y

,x y

fx

y y

x

Part I

,x y

,x y

fx

y y

x

Part I

,x y

,x y

fx

y y

x

Part I

,x y

,x y

fx

y y

x

1 2 1

Let and be two circles, with inside . Construct a sequence of points

, , . . . , , . . . on , such that for each the line segment is tangent

to and (for 2) distinct

i i i

C D D D

P P P D i PP

C i

1 from . "Poncelet's Alternative" says thati iPP

1 1 1if for some 1, then for any other initial point we will have .n nP P n P P P

“Poncelet’s Alternative”: The Great Poncelet Theorem for Circles