Projective Geometry
Pam Todd
Shayla Wesley
Summary
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Conic Sections
Define Projective Geometry
Important Figures in Projective Geometry
Desargues’ Theorem
Principle of Duality
Brianchon’s Theorem
Pascal’s Theorem
Conic Sections
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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
What is Projective Geometry?
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Projective geometry is concerned with where
elements such as lines planes and points either
coincide or not.
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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.
Beginnings
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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
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 twodimensional, will give a faithful impression of the
three-dimensional object.
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Leone Battista Alberti
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Since we are free to move our eye and the
position of the screen, we have many different
two-dimensional representations of the threedimensional object. An interesting problem,
raised by Alberti himself, is to recognize the
common properties of all these different
representations.
Gerard Desargues
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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.
Desargues’ Famous
Theorem
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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
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Desargues Theorem, Three Circles Theorem
using Desargues Theorem
Gaspard Monge
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1746-1818
Invented descriptive
geometry (aka
representing threedimensional objects in a
two-dimensional plane)
Jean-Victor
Poncelet
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1788-1867
studied conic sections and
developed the principle of
duality independently of
Joseph Gergonne
Student of Monge (Three
Circle Theorem)
What’s a duality?
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How it came about?
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Euclidean geometry vs. projective geometry
Train tracks
Euclidean-two points determine a line
Projective-two lines determine a point
Principle of Duality
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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.
Dual it yourself!
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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…
Pascal’s Theorem
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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
Brianchon’s Theorem
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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
Time Line
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14th century
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15th century
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Gerard Desargues wrote Rough draft for an essay on the results of
taking plane sections of a cone
16th Century
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Artists studied math properties of spatial figures
Leone Alberti thought of screen images to be “projections”
Pascal came up with theorem for “Pascal's line” based on a hexagon
inscribed in a conic section.
17th Century
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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
Bibliography
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Leone Battista Alberti http://www-groups.dcs.stand.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-theknot.org/Curriculum/Geometry/Desargues.shtml
Monge via Desargues http://www.cut-theknot.org/Curriculum/Geometry/MongeTheorem.s
html
Intro to Projective Geometry
http://www.math.poly.edu/courses/projective_geo
metry/Inaugural-Lecture/inaugural.html
Conic Section