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© 2000-2023   Gérard P. Michon, Ph.D.

Projective Geometry

Geometry is the gate to Science.  This gate is
so small that one can only enter it as a child
.
William Clifford   (1845-1879)

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Related Links (Outside this Site)

The Geometry Center (University of Minesota).
The Geometry Junkyard by David Eppstein (UC Irvine).
Geometry from the Land of the Incas  by  Antonio Gutierez.
 
Bruck-Ryser-Chowla Theorem  by  Robin Whitty  (Theorem of the Day #3).
 
Gérard Desargues (1591-1661)   |   Blaise Pascal (1623-1662)   |   Philippe de La Hire (1640-1718)
 
Charles Julien Brianchon (1783-1864; X1803)   |   Jean-Victor Poncelet (1788-1867; X1807)
August Möbius (1790-1868)   |   Michel Chasles (1793-1880; X1812)   |   Jakob Steiner (1796-1863)
Karl von Staudt (1798-1867)   |   Julius Plücker (1801-1868)   |   Ernest de Jonquières (1820-1901)
Arthur Cayley (1821-1895)   |   Edmond Laguerre (1834-1886; X1853)   |   Gaston Darboux (1842-1917)
Felix Klein (1849-1925)   |   Elie Cartan (1869-1951)   |   Oswald Veblen (1880-1960)
 
Wikipedia :   Reciprocal polars   |   Projective differential geometry   |   Incidence geometry   |   Moulton plane (1902)

Videos :

Quantum Physics and Universal Beauty  by  Frank Wilczek (1951-)  (RI, 2015).
 
[ History of ]  Projective geometry  by  Norman J. Wildberger  (UNSW, Sidney).
WT31 | WT32 | WT33 | WT34 | WT35 | WT36 | WT37 | WT38 | WT39 | WT40 | WT41

Projective geometry | Math History (1:09:40)  by  N.J. Wildberger  (2011-05-10).
Elementary projective geometry, for K-6 (35:40)  N.J. Wildberger  (2012-04-15).
 
Projective line (16:05)  by  Daniel Chan  (2017-05-12).
Projective plane (12:29)  by  Daniel Chan  (2017-05-12).
Projective Varieties (23:26)  by  Daniel Chan  (2017-05-13).
Extraordinary Conics (16:45)  by  CodeParade  (2020-03-15).

 
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Projective Geometry


(2014-10-23)   Polarity of Apollonius  
A dual relationship between points and lines, with respect to a circle.

Following Apollonius of Perga (262-190 BC)  we introduce polarity with respect to a circle, but the concept can be generalized to any conic.

 Come back later, we're
 still working on this one...


(2014-09-27)   Perspective   (Filippo Brunelleschi,  c. 1413)
The rules discovered and exploited by Renaissance artists.

When painters became concerned with realistic representations of extended backgrounds, it became important to understand the basic laws of  perspective.

When the points in an  horizontal plane  are observed, sets of parallel lines always meet at a point on the  horizon.  The horizon itself is a special straight line  augmented by a single point  (which may be viewed as infinitely far away to the left or to the right of the viewer).  This point "at infinity" is just what's required to prevent an exception for the above statement in the case of lines parallel to the horizon.

The horizontal plane so depicted is an example of a two-dimensional projective space, which can be naively described as a distorted Euclidean plane  ("squeezed" into the half-plane below the horizon)  and a "line at infinity"  (the horizon).  The horizon itself isn't a Euclidean line but a projective line  (a projective space of dimension 1)  namely a Euclidean line with the addition of the  single  point at infinity introduced above.

The basic rules of perspective which transform the actual Euclidean space of two or three dimension into a two-dimensional projection are simple enough for artists to master.  Their mathematical exploration by Gérard Desargues led to an entire branch of mathematics known as  projective geometry  with many intriguing and surprising results like  Pascal's hexagram theorem.

Remarkably, the rules of perspective transform Euclidean space into a very different kind of beast whose abstract definition can be made utterly simple, as introduced in the  next section.

Filippo Brunelleschi (1377-1446)   |   De pictura (1435) by Leon Battista Alberti (1404-1472)


(2014-09-26)   Projective spaces
Projective line,  projective plane,  etc.

projective space  of dimension  n  consists of all subspaces of dimension 1  in a vector space of dimension n+1.

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 still working on this one...

Projective plane   |   Projective space   |   Homography (or projective transformation)


(2014-09-27)   Homogeneous coordinates   (1827 & 1828)

In practice,  an element of an n-dimensional projective space is represented by  n+1  coordinates which aren't  all  zero,  with the understanding that multiplying all of those by a nonzero factor gives the same element  (projective point).  Such coordinates are called  either  projective coordinates  or  homogeneous coordinates.

Homogeneous coordinates  were introduced independently by  Karl Feuerbach (1827)  August Möebius (also 1827)  and Julius Plücker (1828).

For example,  the Euclidean plane can be considered to be part of the real projective plane by mapping the point of cartesian coordinates  (x,y)  to the projective point of homogeneous coordinates  [x:y:1].  Colons (:) are traditionally used to separate homogeneous coordinates and square brackets are popular to enclose them  (but neither is compulsory).

The only projective points which are not so obtained have homogeneous coordinates  [x:y:0]  (the  same  object is also denoted by   [kx:ky:0]  for any nonzero number k).  They belong to the  line at infinity  which has no counterpart in the Euclidean plane.

Homogeneous coordinates   |   Plücker coordinates
 
Homogeneous coordinates (7:56)  by  Norman Wildberger  (WildTrig33, 2009-01-28).
 
Homogeneous coordinates in photogrammetry (1:20:01)  by  Cyrill Stachniss  (WildTrig33, 2015-07-09).


Gaspard Monge (2013-01-05)   Projective Duality
In the axioms of planar projective geometry, "points" and "lines" are interchangeable.

Gaspard Monge.  Come back later, we're
 still working on this one...

Wikipedia :   Duality (projective geometry).

 Theorem of Pappus
(2013-01-05)   The Theorem of Pappus
Pappus of Alexandria  lived in the  4th  century  (AD).

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 still working on this one...

Pappus of Alexandria  (c. AD 290-350)   |   Cut-the-Knot  by  Alexander Bogomolny.
 
Démontrer en dessinant sur sa fenêtre (54:43, French)  by  Cécile Gachet,  Ulm 2016  (2017-10-02).


Blaise Pascal (2013-01-05)   Pascal's Theorem   (Pascal, 1639)
Proven by  Blaise Pascal  (1623-1662)  when he was 16.
 Pascal's Theorem  

Alternate sides of an hexagon inscribed in a conic intersect on three collinear points.

This is a proper generalization of  Pappus's theorem  because two straight lines form a degenerate conic.

Blaise Pascal  (1623-1662)


(2013-01-05)   Brianchon's Theorem   (Brianchon, 1810)
The  dual  of  Pascal's theorem.

Brianchon's theorem states:   The three  principal diagonals  of an hexagon  circumscribed to a conic  are  concurrent.

By definition,  the  principal diagonals  of an hexagon are the lines which join two opposite vertices.  In the geometry of the projective plane,  the locution  polygon circumscribed to a conic  replaces end generalizes what's called a  circumscribed polygon  in planar Euclidean geometry.  Indeed,  a circle is a special case of a  conic  but the former is  undefined  in projective geometry,  since the notion of distance is deliberately shunned.

Brianchon's theorem   |   Charles Julien Brianchon (1783-1864, X1803)


(2013-01-05)   Desargues' Theorem
Gérard Desargues  was the founder of modern projective geometry.

Two triangles are in perspective  axially  iff  they're in perspective  centrally.

 Theorem of Desargues

Desargues' theorem   |   Gérard Desargues (1591-1661)


(2014-09-24)   Cross-Ratio   (double-ratio, anharmonic ratio)
The only projective invariant of a quadruple of points.

A pencil of lines is...

 Come back later, we're
 still working on this one...

Cross-ratio.
 
The Cross-Ratio (16:18)  by  Federico Ardila  (Numberphile, 2018-07-06).


(2014-09-24)   Chasles' theorem
Cross-ratio of four points on a conic section.

The cross-ratio of four lines from any base point on a nondegenerate conic to four given points on that same conic doesn't depend on the base point.

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 still working on this one...

Chasles' theorem  by  Hubert Shutrick   |   Michel Chasles (1793-1880; X1812)


(2014-09-27)   The two cyclic points   (Jean-Victor Poncelet)
I and J have homogeneous coordinates  (1:i:0)  and  (1:-i:0)  respectively.

Also called  isotropic points  or  circular points at infinityEdmond Laguerre  called them  ombilics  (of the complex projective plane).

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 still working on this one...

Complex projective plane   |   Cyclic points   |   Jean-Victor Poncelet (1788-1867; X1807)


(2015-10-20)   Laguerre formula
Planar angle relative to a conic.

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 still working on this one...

Wikipedia :   Laguerre formula   |   Edmond Laguerre (1834-1886; X1853)


(2015-10-20)   Distance relative to a conic  (Laguerre, Cayley)
Cayley's projective definition of length.

All geometry is projective geometry.
Arthur Cayley  (1821-1895)

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 still working on this one...

Wikipedia :   Cayley-Klein metrics


(2015-10-19)   Bézout's theorem   (1779)
Two planar curves of degrees m and n  normally  have mn intersections.

Arguably, this is the first result in  algebraic geometry.

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 still working on this one...

Wikipedia :   Bézout's theorem


(2019-06-21)   Fano Plane   PG(2,2).  7 points and 7 lines.
Finite  projective geometry  of dimension  2  and order  2.

Fano described finite projective spaces of arbitrary dimensions and prime orders.

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 still working on this one...

Wikipedia :   Fano plane   |   Gino Fano (1871-1952)

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