Projective geometry | Math History (1:09:40)
by N.J. Wildberger (2011年05月10日).
Elementary projective geometry, for K-6 (35:40)
N.J. Wildberger (2012年04月15日).
Projective line (16:05)
by Daniel Chan (2017年05月12日).
Projective plane (12:29)
by Daniel Chan (2017年05月12日).
Projective Varieties (23:26)
by Daniel Chan (2017年05月13日).
Extraordinary Conics (16:45)
by CodeParade (2020年03月15日).
Following Apollonius of Perga (262-190 BC) we introduce polarity with respect to a circle, but the concept can be generalized to any conic.
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)
A projective space of dimension n consists of all subspaces of dimension 1 in a vector space of dimension n+1.
[画像: Come back later, we're still working on this one... ]
Projective plane | Projective space | Homography (or projective transformation)
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年01月28日).
Homogeneous coordinates in photogrammetry (1:20:01)
by Cyrill Stachniss (WildTrig33, 2015年07月09日).
Gaspard Monge.
[画像: Come back later, we're still working on this one... ]
Wikipedia : Duality (projective geometry).
[画像: Come back later, we're 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年10月02日).
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)
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)
Two triangles are in perspective axially iff they're in perspective centrally.
Desargues' theorem | Gérard Desargues (1591-1661)
A pencil of lines is...
[画像: Come back later, we're still working on this one... ]
Cross-ratio.
The Cross-Ratio (16:18) by
Federico
Ardila (Numberphile, 2018年07月06日).
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.
[画像: Come back later, we're still working on this one... ]
Chasles' theorem by Hubert Shutrick | Michel Chasles (1793-1880; X1812)
Also called isotropic points or circular points at infinity. Edmond Laguerre called them ombilics (of the complex projective plane).
[画像: Come back later, we're still working on this one... ]
Complex projective plane | Cyclic points | Jean-Victor Poncelet (1788-1867; X1807)
[画像: Come back later, we're still working on this one... ]
Wikipedia : Laguerre formula | Edmond Laguerre (1834-1886; X1853)
All geometry is projective geometry.
Arthur Cayley (1821-1895)
[画像: Come back later, we're still working on this one... ]
Wikipedia : Cayley-Klein metrics
Arguably, this is the first result in algebraic geometry.
[画像: Come back later, we're still working on this one... ]
Wikipedia : Bézout's theorem
Fano described finite projective spaces of arbitrary dimensions and prime orders.
[画像: Come back later, we're still working on this one... ]
Wikipedia : Fano plane | Gino Fano (1871-1952)
