Projective Geometry

1. Projective Geometry Pam Todd Shayla Wesley

2. Summary • Conic Sections • Define Projective Geometry • Important Figures in Projective Geometry • Desargues’ Theorem • Principle of Duality • Brianchon’s Theorem • Pascal’s Theorem

3. a conic section is a curved locus of points, formed by intersecting a cone with a plane Two well-known conics are the circle and the ellipse. These arise when the intersection of cone and plane is a closed curve. Conic Sections 2 Conic Sections

4. What is Projective Geometry? • Projective geometry is concerned with where elements such as lines planes and points either coincide or not. • Can be thought of informally as the geometry which arises from placing one's eye at a point. That is, every line which intersects the "eye" appears only as a point in the projective plane because the eye cannot "see" the points behind it.

5. Beginnings • Artists had a hard time portraying depth on a flat surface • Knew their problem was geometric so they began researching mathematical properties on spatial figures as the eye sees them • Filippo Brunelleschi made the 1st intensive efforts & other artists followed

6. Leone Battista Alberti • 1404-1472 • Thought of the surface of a picture as a window or screen through which the artist views the object to be painted • Proposed the following procedure: interpose a glass screen between yourself and the object, close one eye, and mark on the glass the points that appear to be on the image. The resulting image, although two-dimensional, will give a faithful impression of the three-dimensional object.

7. Leone Battista Alberti • Since we are free to move our eye and the position of the screen, we have many different two-dimensional representations of the three-dimensional object. An interesting problem, raised by Alberti himself, is to recognize the common properties of all these different representations.

8. Gerard Desargues • 1593-1662 • Wrote Rough draft for an essay on the results of taking plane sections of a cone • The book is short, but very dense. It begins with pencils of lines and ranges of points on a line, considers involutions of six points gives a rigorous treatment of cases involving 'infinite' distances, and then moves on to conics, showing that they can be discussed in terms of properties that are invariant under projection.

9. Desargues’ Famous Theorem • DESARGUES THEOREM: If corresponding sides of two triangles meet in three points lying on a straight line, then corresponding vertices lie on three concurrent lines • Desargues Theorem, Three Circles Theorem using Desargues Theorem

10. Gaspard Monge • 1746-1818 • Invented descriptive geometry (aka representing three-dimensional objects in a two-dimensional plane)

11. Jean-Victor Poncelet • 1788-1867 • studied conic sections and developed the principle of duality independently of Joseph Gergonne • Student of Monge (Three Circle Theorem)

12. What’s a duality? • How it came about? • Euclidean geometry vs. projective geometry • Train tracks • Euclidean-two points determine a line • Projective-two lines determine a point

13. Principle of Duality • All the propositions in projective geometry occur in dual pairs which have the property that, starting from either proposition of a pair, the other can be immediately inferred by interchanging the words “line” and “point”. • This also applies with words such as “vertex” and “side” to get dual statements about vertices.

14. Dual it yourself! • Through every pair of distinct points there is exactly one line, and … • There exists two points and two lines such that each of the points is on one of the lines and … • There is one and only one line joining two distinct points in a plane, and…

15. Pascal’s Theorem • Discovered by Pascal in 1640 when he was only 16 years old. • Basic idea of the theorem was given a (not necessarily regular, or even convex) hexagon inscribed in a conic section, the three pairs of the continuations of opposite sides meet on a straight line, called the Pascal line

16. Brianchon’s Theorem • Brianchon’s theorem is the dual of Pascal’s theorem • States given a hexagon circumscribed on a conic section, the lines joining opposite polygon vertices (polygon diagonals) meet in a single point

17. Time Line • 14th century • Artists studied math properties of spatial figures • Leone Alberti thought of screen images to be “projections” • 15th century • Gerard Desargues wrote Rough draft for an essay on the results of taking plane sections of a cone • 16th Century • Pascal came up with theorem for “Pascal's line” based on a hexagon inscribed in a conic section. • 17th Century • Victor Poncelet came up with principle of duality • Joseph Diaz Gergonne came up with a similar principle independent of Poncelet • Charles Julien Brianchon came up with the dual of Pascal’s theorem

18. Bibliography • Leone Battista Alberti http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Alberti.html • Projective Geometry http://www.anth.org.uk/NCT/ • Math World http://mathworld.wolfram.com/ • Desargues' theorem http://www.cut-the-knot.org/Curriculum/Geometry/Desargues.shtml • Monge via Desargues http://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtml • Intro to Projective Geometry http://www.math.poly.edu/courses/projective_geometry/Inaugural-Lecture/inaugural.html • Conic Section http://en.wikipedia.org/wiki/Conic_section • Jean-Victor Poncelet http://en.wikipedia.org/wiki/Jean_Victor_Poncelet • Gaspard Monge http://www.britannica.com/eb/article-9053349