A QUADRATIC TRANSFORMATION OF THE THREE-DIMENSIONAL SPACE OF ALL CONICS THROUGH TWO POINTS OF A PROJECTIVE PLANE

. Using the Steiner's method of projective generation of conics and its dual we define two projective mappings of a double contact pencil of conics into itself and we prove that one is the inverse of the other. We show that these projective mappings are induced by quadratic transformations of the three-dimensional projective space of all conics through two distinct points of a projective plane.


Introduction
In 1832 Jacob Steiner proved that conics may be generated using two projectively related pencils of lines. Let VA an d VB be the pencils of lines with vertices two distinct points A, B of a projective plane over a field K. Let $ be a projectivity between VA and VB that does not map the line A\J B onto itself. In [5] (see also [2], [3] or [4]) it is proved that the set of points of intesection of corresponding lines under $ is a non-singular conic 7 containing A and B. The projectivity $ maps the tangent line to 7 at A onto the line A VB and the line A\J B onto the tangent line to 7 at B. Also due to Steiner is the following dual construction. Let a and b be two distinct lines and let ' J* be a projectivity between a and b that does not map the point adb onto itself. By the principle of duality it follows that the lines joining corresponding points under \I/ are tangent lines to a non-singular conic 7' with the property that a and b are tangent lines to 7'.
A conic is a self-dual concept only if the characteristic of K is different from two. In fact if the characteristic is two all tangent lines to a conic contain a special point called nucleus. Moreover two projectively related pencils of lines generate a conic, but two projectively related lines fail to generate tangent lines to a conic.
Let R and S be two distinct points and let ÍR and ts be two lines containing R and S (respectively), both different from the line i?V5. The points R, S and the lines ts define a double contact pencil of conics TR&tntsi formed by all the non-singular conics through R and S which have ta, ts as tangent lines at R and S (respectively) and by the two singular conics tu U ts and RVS (counted twice). The points R and S are called the base of the pencil and ta, ts are called the tangent lines of the pencil at R and S.
Let if be a field of charcteristic different from two. A conic of the projective plane PG(2, K) over K may be represented by a point of the projective space PG (5,K), and vice versa. Let 7 be a conic of PG(2,K) and let A = (aij)i¿=1,2,3 be a 3 x 3 symmetric matrix over K that represents 7. The six elements {dij)i<j completely determine A and the conic 7 may be represented by the point ((an, a\2,ai3,022,023,033)) of PG (5, K), as the elements defined up to scalar multiplication. Finally we recall that a pencil of conics of PG(2,K) is represented by a line of PG (5, K).
A quadratic transformation of a 3-dimensional projective space PG(3, K) into itself is a birational mapping / : PG(3,K) -> PG (3,K), such that / and are given by homogeneous polynomials of degree two. The points of a plane of PG (3, K)  where /¿, i = 1,2,3,4, is a homogeneous polynomial of degree two. Take S = f*(0( 1)). There exist three different types of quadratic transformations of a three-dimensional projective space (see [1]): • Type I. The linear system S is formed by all quadrics containing a conic 7 and a point Pi. Lines of PG (3,K) are mapped onto conics through Pi intersecting 7 in two points. • Type II. The linear system S is formed by all quadrics containing a line £ and three points Pi, P2, P3. Lines of PG(3, K) are mapped onto twisted cubics with £ as a chord and containing Pi, P2, P3. • Type III. The linear system S is formed by all quadrics containing four points Pi, P2, P3, P4 with the same tangent plane at Pi. Lines of PG(3, K) are mapped onto quartics containing P2, P3, P4 and with Pi as a double point.

A projectivity of the double contact pencil of conics
Let PG{2, K) be a projective plane over a field K of characteristic different from two. Let T be a non-singular conic of PG(2,K) with induced

&(t A ) = B\/t A L = B\J A,\t follows that t A = $ /_1 (BV A) = t' A . Moreover t' B = V{A\/ B) = t B . -
Let T = J-A,B-,t A ,t B be a double contact pencil of conics of PG(2, K) with A,B as base and îa^B as tangent lines at A and B. Using the previous results we may define a map a of the pencil T into itself in the following way: to every non-singular conic r of T, a corresponds the conic T'. The two singular conics of the pencil T are assumed to be both fixed by a.  ((2a 2 fi~l, -b 2 X~1,abX~1)), and this gives a parametric representation of the conic T'. It follows that T' may be represented by the equation 2Ax3 2 + fj,x\x2 = 0. Hence a maps the non-singular conic of T with equation AX3 2 + fix 1X2 = 0 onto the non-singular conic with equation 2AX3 2 + fix 1X2 -0. The map a is then a projectivity of the line T into itself. • Observe that the set of all conics of PG(2,K) containing two distinct points A and B form a linear system of dimension three, so it is represented by a three-dimensional subspace S\ B of PG(5,K).  K) with A, B as base and tA, ts as tangent lines at A and B. We may define a map u of the pencil T into itself in the following way: to every non-singular conic r of !F, <jü corresponds the non-singular conic T". The two singular conics of T are assumed to be both fixed by u.
A quadratic transformation of .
The conic T" is the locus of PG(2, K) whose tangent lines have projective coordinates satisfying (*), hence this locus has an equation of the form Ax3 2 + 2¡1X\X2 = 0. Therefore u> maps the non-singular conic of T with equation \x$ 2 + HX\X2 = 0, onto the non-singular conic with equation \x^2 + 2^iX\X2 = 0. This proves that a is a projectivity of the line T into itself. •