Conic sections are usually introduced as the first curves (as opposed to lines) and are given as the locus of points satisfying an equation of the form

@font

picture(6774,5871)(2239,-5965)

In order to understand this statement let us a fix a projective plane
and work within it. The collection of all lines through a fixed point
*O* are in natural 1-1 correspondence with the points on a line *l*
not containing *O*; any point *B* on *l* determines a unique line *b*
joining *O* and *B* and conversely, any line *b* through *O* meets *l*
in exactly one point *B*. The locus of lines through a fixed point is
called a *pencil*.

picture(6432,5758)(1036,-5639)
(1291,-5101)(0,0)[lb]*O*
(1246,-556)(0,0)[lb]*l*
(2296,-1216)(0,0)[lb]*B*
(2611,-46)(0,0)[lb]*b*
(1036,-5581)(0,0)[lb]The pencil of lines through a point *O* is in 1-1 correspondence with the points of *l*

If
Similarly, if a pair of points *A* and *B* are such that neither lies
on a line *l*, then there is a natural 1-1 correspondence between the
pencils through *A* and *B*, since both the pencils have a 1-1
correspondence with the points of *l*. Explicitly, if *p* is a line
containing *A* which meets *l* in *P*, we consider the line *q*
joining *B* with *P*. This correspondence is called the *perspectivity* between the pencils through *A* and *B* with axis *l*.

picture(9414,6261)(154,-5650)
(6316,-5596)(0,0)[lb]Axial perspectivity between pencils
(5101,-3661)(0,0)[lb]*A*
(6166,-4876)(0,0)[lb]*l*
(9346,-3616)(0,0)[lb]*B*
(8506,-1576)(0,0)[lb]*q*
(6196,-1366)(0,0)[lb]*p*
(7471,-46)(0,0)[lb]*P*
(1501,-5581)(0,0)[lb]Central perspectivity between lines
(2161,-4921)(0,0)[lb]*O*
(211,-1186)(0,0)[lb]*l*
(286,-2446)(0,0)[lb]*m*
(1321,-766)(0,0)[lb]*A*
(1816,-1576)(0,0)[lb]*a*
(1711,-2536)(0,0)[lb]*B*

In both case a *projective* correspondence or *projectivity*
is defined as a composition of perspectivities. Thus, the
definition of a conic says, take a pair of points *A* and *B* and a
projectivity between the pencils of lines through *A* and *B*.
Let *C* be the locus of points of the form
*l* (*l* ) where *l* is
a line through *A* and (*l* ) the corresponding line through *B*,
then *C* is a conic.

In order to understand projectivities better we note the following

picture(6729,5335)(1519,-4649)
(4756,-871)(0,0)[lb]*I*
(7456,-316)(0,0)[lb]*Z*
(2131,-3496)(0,0)[lb]*O'*
(2506,-1186)(0,0)[lb]*O*
(7111,-3931)(0,0)[lb]*Z'*
(4261,-3631)(0,0)[lb]*I'*
(3496,-2086)(0,0)[lb]*I''*
(7426,359)(0,0)[lb]*A*
(3031,-661)(0,0)[lb]*B*
(2011,-4591)(0,0)[lb]Central perspectivity from *A* and then *B* gives a general projectivity

The study of conics in projective geometry is a fascinating one but we
leave it here with a pointer to the suggested readings on projective
geometry. At the same time we note that it would be rather difficult
to study more complicated curves such as the locus of points
satisfying *x*^{3} + *y*^{3} = 1, using only the incidences between points and
lines and not algebra; this is possible *in principle* since the
addition and multiplication operations have been defined in terms of
the incidence relations. From now on we will use all the familiar
notions from algebra and deal with coordinate geometry.