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
• Brianchons Theorem
• Pascals Theorem

3
Conic Sections

• 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

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
Whats 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
Pascals 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
Brianchons Theorem

• Brianchons theorem is the dual of Pascals
  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
  Pascals theorem

18
Bibliography

• Leone Battista Alberti http//www-groups.dcs.st-an
  d.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/Cur
  riculum/Geometry/Desargues.shtml
• Monge via Desargues http//www.cut-the-knot.org/Cu
  rriculum/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_s
  ection
• Jean-Victor Poncelet http//en.wikipedia.org/wiki/
  Jean_Victor_Poncelet
• Gaspard Monge http//www.britannica.com/eb/article
  -9053349