Projective geometry is a non-euclidean geometry
with applications in the theory of perspective
. It can be described as the branch of geometry that deals with properties of shapes that are invariant
. It can be modelled in Euclidean 3-space as follows: Let "point
" mean a line through the origin and "line
" mean a plane through the origin. This doesn't satisfy the parallel postulate
because there are no parallel
lines - any two planes through the origin have a line in common, so any two projective lines have a projective point in common.
To make the projective plane look more like a plane, all you need to do is project it. I mean this in pretty much the same sense as a film projector projects. Light passing through a lens travels in a straight line, and ultimately hits a screen. Similarly, we can choose a plane that doesn't pass through the origin (called the image plane of the projection). Every "point" in the model above will hit it at one point, and every "line" will hit it along one line... unless it's parallel to the image plane. We can think of this as modelling the projective plane as a Euclidean plane plus one line, corresponding to the plane through the origin parallel to the image plane. This is the ideal "line at infinity," where parallel lines meet. Since every pair of lines meets in exactly one point, this line at infinity must have one point for every complete set of lines that would be mutually parallel without it - which is to say, a point for every possible slope.
It should be emphasized that the line at infinity is just like all the other lines, and choosing a different image plane will put a different line at infinity.
Projective geometry is generally practiced with three-dimensional coordinates, corresponding to vectors contained in the lines through the origin in the 3-D model. When applied to the plane-plus-line-at-infinity model, it's customary to use the image plane defined by the equation z=1. When z is not equal to 0, the coordinates (x, y, z) correspond to the point (x/z, y/z). When z=0, (x, y, z) corresponds to the point at infinity with slope y/x.
The act of switching from one image plane to another is called projective transformation. Any invertable 3x3 matrix defines a projective transformation; the new coordinates of a point are obtained by multiplying its old coordinate vector by the matrix. Projective transformations preserve lines, but not angles or Euclidean distance. Circles will not necessarily remain circles after a projective transformation, but they will remain quadratic curves of some sort. After all, what is a circle in projective geometry but a cone with its vertex at the origin? And are not quadratic curves also known as conic sections?
Perhaps the neatest and most surprising thing about projective geometry is the duality principle, which states that all true propositions have a dual form, also true, in which the words "point" and "line" are interchanged. For example, the dual of "Given any two points, there exists exactly one line that lies on both" is "Given any two lines, there is exactly one point that lies on both." To get an intuitive handle on the duality principle, think of the 3D model: for each line through the origin, there is one plane through the origin perpendicular to it, and vice versa. Thus, points and lines in the projective plane are joined in dual pairs. It's less obvious that the dual of the point of intersection of two lines is the line joining the two lines' dual points, but in fact this turns out to be true.