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### Transcript of Chap20 (1)

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Projective Geometry

Pam Todd

Shayla Wesley

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Summary

Conic Sections

Define Projective Geometry

Important Figures in Projective Geometry Desargues Theorem

Principle of Duality

Brianchons Theorem Pascals Theorem

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Conic Sections

a conic section is acurved locus ofpoints, formed by

intersecting a conewith a plane

Two well-known 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
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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.

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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

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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 thecommon properties of all these different

representations.

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Gerard Desargues

1593-1662

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.

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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.cut-the-knot.org/Curriculum/Geometry/Desargues.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/Desargues.shtml
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Gaspard Monge

1746-1818

Invented descriptivegeometry (aka

representing three-

dimensional objects in atwo-dimensional plane)

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Jean-Victor

Poncelet

1788-1867

studied conic sections anddeveloped the principle ofduality independently ofJoseph Gergonne

Student of Monge (ThreeCircle Theorem)

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Whats a duality?

Euclidean geometry vs. projective geometry

Train tracks

Euclidean-two points determine a line

Projective-two lines determine a point

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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.

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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

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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
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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
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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

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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 Geometryhttp://www.math.poly.edu/courses/projective_geometry/Inaugural-Lecture/inaugural.html

Conic Section

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Alberti.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Alberti.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Alberti.htmlhttp://www.anth.org.uk/NCT/http://mathworld.wolfram.com/http://www.cut-the-knot.org/Curriculum/Geometry/Desargues.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/Desargues.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.math.poly.edu/courses/projective_geometry/Inaugural-Lecture/inaugural.htmlhttp://www.math.poly.edu/courses/projective_geometry/Inaugural-Lecture/inaugural.htmlhttp://www.math.poly.edu/courses/projective_geometry/Inaugural-Lecture/inaugural.htmlhttp://www.math.poly.edu/courses/projective_geometry/Inaugural-Lecture/inaugural.htmlhttp://www.math.poly.edu/courses/projective_geometry/Inaugural-Lecture/inaugural.htmlhttp://www.math.poly.edu/courses/projective_geometry/Inaugural-Lecture/inaugural.htmlhttp://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/Desargues.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/Desargues.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/Desargues.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/Desargues.shtmlhttp://www.cut-the-knot.org/Curriculum/Geometry/Desargues.shtmlhttp://mathworld.wolfram.com/http://www.anth.org.uk/NCT/http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Alberti.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Alberti.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Alberti.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Alberti.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Alberti.html