Chap20 (1)

Upload
silverwind65 
Category
Documents

view
230 
download
0
Transcript of Chap20 (1)

7/27/2019 Chap20 (1)
1/18
Projective Geometry
Pam Todd
Shayla Wesley

7/27/2019 Chap20 (1)
2/18
Summary
Conic Sections
Define Projective Geometry
Important Figures in Projective Geometry Desargues Theorem
Principle of Duality
Brianchons Theorem Pascals Theorem

7/27/2019 Chap20 (1)
3/18
Conic Sections
a conic section is acurved locus ofpoints, formed by
intersecting a conewith a plane
Two wellknown conics are the circle and the
ellipse. These arise when the intersection ofcone and plane is a closed curve.
Conic Sections 2
http://en.wikipedia.org/wiki/Image:Conic_sections_2.pnghttp://en.wikipedia.org/wiki/Image:Conic_sections_2.png 
7/27/2019 Chap20 (1)
4/18
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 planebecause the eye cannot "see" the points behind
it.

7/27/2019 Chap20 (1)
5/18
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

7/27/2019 Chap20 (1)
6/18

7/27/2019 Chap20 (1)
7/18
Leone Battista Alberti
Since we are free to move our eye and the
position of the screen, we have many different
twodimensional representations of the three
dimensional object. An interesting problem,
raised by Alberti himself, is to recognize thecommon properties of all these different
representations.

7/27/2019 Chap20 (1)
8/18
Gerard Desargues
15931662
WroteRough 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, andthen moves on to conics, showing that they can bediscussed in terms of properties that are invariantunder projection.

7/27/2019 Chap20 (1)
9/18
Desargues Famou
Theorem
DESARGUES THEOREM: If corresponding sidesof two triangles meet in three points lying on a
straight line, then corresponding vertices lie onthree concurrent lines
Desargues Theorem, Three Circles Theoremusing Desargues Theorem
http://www.cuttheknot.org/Curriculum/Geometry/Desargues.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/Desargues.shtml 
7/27/2019 Chap20 (1)
10/18
Gaspard Monge
17461818
Invented descriptivegeometry (aka
representing three
dimensional objects in atwodimensional plane)

7/27/2019 Chap20 (1)
11/18
JeanVictor
Poncelet
17881867
studied conic sections anddeveloped the principle ofduality independently ofJoseph Gergonne
Student of Monge (ThreeCircle Theorem)

7/27/2019 Chap20 (1)
12/18
Whats a duality?
How it came about?
Euclidean geometry vs. projective geometry
Train tracks
Euclideantwo points determine a line
Projectivetwo lines determine a point

7/27/2019 Chap20 (1)
13/18
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 aboutvertices.

7/27/2019 Chap20 (1)
14/18
Dual it yourself!
Through every pair of distinct points there isexactly 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

7/27/2019 Chap20 (1)
15/18
Pascals Theorem
Discovered by Pascal in1640 when he was only 16years old.
Basic idea of the theorem
was given a (notnecessarily regular, oreven convex) hexagoninscribed in a conicsection, the three pairs ofthe continuations ofopposite sides meet on astraight line, called thePascal line
http://www.anth.org.uk/NCT/images/Pascal.gif 
7/27/2019 Chap20 (1)
16/18
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
http://www.anth.org.uk/NCT/images/brianchn.gif 
7/27/2019 Chap20 (1)
17/18
Time Line
14th century Artists studied math properties of spatial figures
Leone Alberti thought of screen images to be projections
15th century Gerard Desargues wroteRough 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

7/27/2019 Chap20 (1)
18/18
Bibliography Leone Battista Alberti http://wwwgroups.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.cuttheknot.org/Curriculum/Geometry/Desargues.shtml
Monge via Desargues http://www.cuttheknot.org/Curriculum/Geometry/MongeTheorem.shtml
Intro to Projective Geometryhttp://www.math.poly.edu/courses/projective_geometry/InauguralLecture/inaugural.html
Conic Section
http://wwwgroups.dcs.stand.ac.uk/~history/Mathematicians/Alberti.htmlhttp://wwwgroups.dcs.stand.ac.uk/~history/Mathematicians/Alberti.htmlhttp://wwwgroups.dcs.stand.ac.uk/~history/Mathematicians/Alberti.htmlhttp://www.anth.org.uk/NCT/http://mathworld.wolfram.com/http://www.cuttheknot.org/Curriculum/Geometry/Desargues.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/Desargues.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.math.poly.edu/courses/projective_geometry/InauguralLecture/inaugural.htmlhttp://www.math.poly.edu/courses/projective_geometry/InauguralLecture/inaugural.htmlhttp://www.math.poly.edu/courses/projective_geometry/InauguralLecture/inaugural.htmlhttp://www.math.poly.edu/courses/projective_geometry/InauguralLecture/inaugural.htmlhttp://www.math.poly.edu/courses/projective_geometry/InauguralLecture/inaugural.htmlhttp://www.math.poly.edu/courses/projective_geometry/InauguralLecture/inaugural.htmlhttp://www.cuttheknot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/Desargues.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/Desargues.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/Desargues.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/Desargues.shtmlhttp://www.cuttheknot.org/Curriculum/Geometry/Desargues.shtmlhttp://mathworld.wolfram.com/http://www.anth.org.uk/NCT/http://wwwgroups.dcs.stand.ac.uk/~history/Mathematicians/Alberti.htmlhttp://wwwgroups.dcs.stand.ac.uk/~history/Mathematicians/Alberti.htmlhttp://wwwgroups.dcs.stand.ac.uk/~history/Mathematicians/Alberti.htmlhttp://wwwgroups.dcs.stand.ac.uk/~history/Mathematicians/Alberti.htmlhttp://wwwgroups.dcs.stand.ac.uk/~history/Mathematicians/Alberti.html