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    Projective Geometry

    Pam Todd

    Shayla Wesley

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    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
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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


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    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


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    Gerard Desargues


    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


    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
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    Gaspard Monge


    Invented descriptivegeometry (aka

    representing three-

    dimensional objects in atwo-dimensional plane)

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    studied conic sections anddeveloped the principle ofduality independently ofJoseph Gergonne

    Student of Monge (ThreeCircle Theorem)

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    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

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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
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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
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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


    Charles Julien Brianchon came up with the dual of Pascals theorem

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    Bibliography Leone Battista Alberti Projective Geometry

    Math World

    Desargues' theorem

    Monge via Desargues

    Intro to Projective Geometry

    Conic Section