(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.
(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.
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.
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.
(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.
