home | index | units | counting | geometry | algebra | trigonometry & functions | calculus
analysis | sets & logic | number theory | recreational | misc | nomenclature & history | physics

Final Answers
© 2000-2016   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)

Related articles on this site:

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 :

[ History of ]  Projective geometry  by  Norman J. Wildberger  (UNSW, Sidney).
WT31 | WT32 | WT33 | WT34 | WT35 | WT36 | WT37 | WT38 | WT39 | WT40 | WT41
 
border
border

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 the 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).  That's to say 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.

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

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.

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

Projective space


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).

 Come back later, we're
 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.

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

Blaise Pascal  (1623-1662)


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

 Come back later, we're
 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...

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

Cross-ratio.


(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.

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

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


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

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

Homogeneous coordinates  were discovered independently by  Karl Feuerbach in 1827, August Möebius, also in 1827, and Julius Plücker in 1828.

Homogeneous coordinates   |   Plücker coordinates


(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).

 Come back later, we're
 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.

 Come back later, we're
 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)

 Come back later, we're
 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.

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

Wikipedia :   Bézout's theorem

border
border
visits since August 26, 2014
 (c) Copyright 2000-2016, Gerard P. Michon, Ph.D.