Editing Projective geometry (section)

== Perspectivity and projectivity ==
Given three non-[[collinear]] points, there are three lines connecting them, but with four points, no three collinear, there are six connecting lines and three additional "diagonal points" determined by their intersections. The science of projective geometry captures this surplus determined by four points through a quaternary relation and the projectivities which preserve the [[complete quadrangle]] configuration.

An [[harmonic quadruple]] of points on a line occurs when there is a complete quadrangle two of whose diagonal points are in the first and third  position of the quadruple, and the other two positions are points on the lines joining two quadrangle points through the third diagonal point.{{sfn|Halsted|1906|pp=15,16}}

A spatial [[perspectivity]] of a [[projective configuration]] in one plane yields such a configuration in another, and this applies to the configuration of the complete quadrangle. Thus harmonic quadruples are preserved by perspectivity. If one perspectivity follows  another the configurations follow along. The composition of two perspectivities is no longer a perspectivity, but a '''projectivity'''.

While corresponding points of a perspectivity all converge at a point, this convergence  is ''not'' true for a  projectivity that is ''not'' a perspectivity. In projective geometry the intersection of lines formed by  corresponding points of a projectivity in a plane are of particular interest. The set of such intersections is called a '''projective conic''', and in acknowledgement of the work of [[Jakob Steiner]], it is  referred to as a [[Steiner conic]].

Suppose a projectivity  is formed by two perspectivities centered on points ''A'' and ''B'', relating ''x'' to ''X'' by an intermediary ''p'':
: <math>x \ \overset{A}{\doublebarwedge}\ p \ \overset{B}{\doublebarwedge} \ X.</math>
The  projectivity is then <math>x \ \barwedge \ X .</math> Then given the projectivity <math>\barwedge</math> the induced conic is
: <math>C(\barwedge) \ = \ \bigcup\{xX \cdot yY : x \barwedge X \ \ \land \ \  y \barwedge Y \} .</math>

Given a conic ''C'' and a point ''P'' not on it, two distinct [[secant line]]s through ''P'' intersect ''C'' in four points. These four points determine a quadrangle of which ''P'' is a diagonal point. The line through the other two diagonal points is called the [[pole and polar|polar of ''P'']] and ''P ''is the '''pole''' of this line.{{sfn|Halsted|1906|p=25}} Alternatively, the polar line of ''P'' is the set of [[projective harmonic conjugate]]s of ''P'' on a variable secant line passing through ''P'' and ''C''.