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Final Answers
© 2000-2017   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.
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)   |   Arthur Cayley (1821-1895)
Gaston Darboux (1842-1917)   |   Felix Klein (1849-1925)   |   Elie Cartan (1869-1951)
Oswald Veblen (1880-1960)
Wikipedia :   Reciprocal polars   |   Projective differential geometry

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

(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.

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 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 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 form a space of dimension 1 commonly called the  line at infinity  (it consists of all projective points whose third homogeneous coordinate is zero).

Homogeneous coordinates   |   Plücker coordinates

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.

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.

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 still working on this one...

Blaise Pascal  (1623-1662)

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

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 still working on this one...

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...

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 still working on this one...


(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 infinity.  Edmond 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)

(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

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