A Brief Introduction to Real Projective Geometry

1. A Brief Introduction to Real Projective Geometry David Sklar San Francisco State University dsklar@sfsu.edu Bruce Cohen Lowell High School, SFUSD math.cohen@gmail.com http://www.cgl.ucsf.edu/home/bic Asilomar - December 2010

2. 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

3. Perspective From John Stillwell’s books Mathematics and its History and The Four Pillars of Geometry

4. Perspective

5. Perspective From Geometry and the Imagination by Hilbert and Cohn-Vossen

6. Dates: Brunelleschi 1413 Alberti 1435 (1525)

7. 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. 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.

8. 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 Jakob Steiner 1796-1863 Jean-Victor Poncelet 1788-1867

9. 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.

10. There exist at least three points not incident with the same line 1. There exist a point and a line that are not incident. 1. 2. Every line is incident with at least three distinct points. Every line is incident with at least two distinct points. 2. 3. Every point is incident with at least three distinct lines. 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 points are incident with one and only one line. 4. 5. Any two distinct lines are incident with at most one point. Any two distinct lines are incident with one and only one point. 5. 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 don’t intersect, and the complete duality between “point” and “line”.

11. 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. .

12. The projective plane may be thought of as the ordinary real affine (Cartesian) plane , with an additional line called the line at infinity. 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.

13. 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 . 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 . 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 An Analytic Model of the Real Projective Plane

14. A Definition of Geometry A group of transformationsG 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.

15. Affine plane : projective plane with the line at infinity omitted Affine plane Affine plane Some Familiar Subgeometries Transformation Group Set Geometry Collineations: transformations that map straight lines to straight lines Projective plane Projective Affine transformations: transformations that map parallel lines to parallel lines (these map the line at infinity to itself) Affine transformations that are generated by rotations, reflections, translations and dilations (isotropic scalings) Euclidean Similarity Isometries: affine transformations that are generated by rotations, reflections, and translations Euclidean Congruence

16. Some Familiar Subgeometries Transformation Group Equivalent Figures 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) All triangles, all parabolas, all hyperbolas, all ellipses Affine 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 Similarity Isometries: affine transformations that are generated by rotations, reflections, and translations Only figures of the same size and shape Euclidean Congruence

17. Analytic Transformation Geometry Transformations Geometry Projective , A invertible Affine , A invertible Setting z to 1 we get the affine transformations In non-homogeneous coordinates

18. Gaussian First Order Optics Lens

19. Gaussian First Order Optics

20. Gaussian First Order Optics

21. Gaussian First Order Optics in Homogeneous Coordinates or Also Note: If So the vertical line at infinity is mapped to the vertical line . So the vertical line is mapped to the line at infinity.

22. Projective Equivalence of the Conics Bruce’s GeoGebra Demonstrations

23. 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

24. Some extra slides not used in the presentation

25. 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.

26. 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.

27. 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.

28. Part I

29. Part I

30. Part I

31. Part I

32. Part I

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