Repository: UIHistories Project: Mathematical Models by Arnold Emch - Series 4 (1928) [PAGE 5]

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Mathematical Models

a point P (y) in (x) corresponds the vertex {(ay) (ax) = 0, (by) (bx) = 0}, or P' (3/'), of the pencil of polars (ay) (ax) + X (by) (bx) = 0 of (y) with respect to the pencil of conies. P and Pr correspond to each other in an involutorial quadratic Cremona transformation, which has the base-points AiA2A3A4 of the pencil of conies as invariant points and the diagonal triangle BiB2B3 of the quadrangle AiA2 A3A4 as the fundamental triangle. If w e choose the latter as the coordinate triangle and A± as the unit point, such that A\A± and A2Az, A2A± and A $ A U A%A± and A\A2 intersect in BiB2B3 respectively, the transformation m a y be put in the standard form. / / /

pX\ = X2Xz, pX2 = X3X1, pXz = X\X2.

When A 4 is chosen as the orthocenter of the triangle AiA2A3 the construction of the transformation becomes particularly simple. Let P be a generic point. Join P to B\ and B2 and construct the line h symmetric to P B \ with respect to B\A±, l2 symmetric to P B 2 with respect to B2A± as an axis. Then l\ and l2 intersect in the point P'f which corresponds to P in the transformation. A s a check for the construction the symmetric h to P B Z with respect to BzA± must also pass through Pf, Fig. 1. This transformation was known to Poncelet,a and also to Steinerb w h o defined it as the correspondence between points which are simultaneously harmonic with respect to the three pairs of lines of a quadrangle, which, with B^BJBz as the vertices of these pairs, are x2 — Xz = 0, x2 + #3 = 0; x3 — xi = 0, Xs + Xi = 0; xi — x2 = 0, Xi + x2 = 0. T h e designation "Steinerian" for this transformation was introduced by Duregec and adopted by Schroter,cl and others. T h e procedure by which the theory of cubics m a y be connected with the Steinerian transformation is as follows: T o a line / corresponds a conic L which cuts / in a couple of corresponding points P, P \ These are also the double points of the involution cut out on / by the pencil pf conies through A\A2AZA^ or the points of contact with / of two definite conies of the pencil. Conversely, with every couple P (xi, x2, xz), Pr (x2X3, Xz xi, XiX2) is, in general, uniquely associated a line / whose coordinates are pui = xi (x22— x32), pu2 = x2

(xz2 — Xi2), puz = x3 (xi2 — x22). T h e locus of couples of corresponding points o n the tangents of a curve K m (ui, u2, Uz) = 0 is in general a curve of order 3 m , w h i c h is invariant in the transformation. W h e n K m is rational then the invariant Czm b e c o m e s hyperelliptic. T h e theory of these invariaTraite des proprietes projectives desfigures(1822), 2nd ed. 1865, vol. 1, pp. 185-248. bJ. f. Math. (Crelle's), vol. 3 (1828), pp. 207-212; Werke, vol. 1, pp. 173-180. cDie ebenen Curven dritter Ordnung (1871) pp. 121-128. dSteiner - Schroter; Vorlesungen iiber syn'thetische Geometrie, vol.2,2nd ed. (1876), pp. 301-304.