[alogo] Good parametrization inverse

Good parametrizations generalize the idea of stereographic projection and establish a bijection between the points of a projective line and those of a conic. Their inverses are of the form x1 = P1(s,t), x2 = P2(s,t), x3 = P3(s,t), where the functions Pi are quadratic polynomials. This is most easily seen by considering a coordinate system (projective) {A,B,C,D} in which the equation of the conic obtains the form x12 = x2x3 (see ProjectiveCoordinates.html ). In such a system, B, C, D are on the conic and A is the intersection point of the tangents at B and C.

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Consider the good parametrization, which to each point E on the conic associates the point E' on BC, E' being the intersection of BC with DE. If E' has the coordinates E' = sB + tC, then E = D + kE' = D + k(sB + tC). Since D = A+B+C, this implies that E = (A+B+C) + k(sB + tC) = A + (1+ks)B + (1+kt)C. Since E is on the conic its coordinates must satify x12 = x2x3, i.e. 1 = (1+ks)(1+kt) ==> ks+kt+k2ts = 0 ==> k = -(s+t)/ts.
Substitution into the equation for E, gives E = D -(s+t)((1/t)B + (1/s)C) = A - (s/t)B - (t/s)C. Equivalently, by multiplying with st ==> E = stA - s2B - t2C.
This shows that the inverse of the good parametrization considered is as stated (x1 = st, x2 = s2, x3 = t2).

The proof for the general case of good parametrization (s',t') results from the fact that two such parametrizations are related through a linear relation (s'=as+bt, t'=cs+dt). This, in turn follows also by an argument similar to the previous one, by first looking at two good parametrisations for the same conic, defined as above. Should the parameters (s,t) and (s',t') represent the same point E in the two systems, then sB+tC = s'B'+t'C'. Writing B' = uA+vB+wC, C'=u'A+v'B+c'C and equating we get the desired relation.

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If the good parametrization uses another line e different from AB, then the correspondence E'<-->E'' is described also through a linear relation between the projective coordinates of the two lines (see RectHypeRelation.html , where the relation is equivalently derived for the quotiens s/t, s'/t' of the projective coordinates).

See Also



Berger, M. Geometry I, II Paris, Springer Verlag 1987, vol. II, par. 16.2, p. 173.
Berger Marcel, Pansu Pierre, Berry Jean-Pic, Saint-Raymond Xavier Problems in Geometry Paris, Springer Verlag 1984, p. 94.

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