Confocal conics and 4-webs of maximal rank
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Journal of Geometry
Confocal conics and 4-webs of maximal rank Sergey I. Agafonov Abstract. Confocal conics form an orthogonal net. Supplementing this net with one of the following: (1) the net of Cartesian coordinate lines aligned along the principal axes of conics, (2) the net of Apollonian pencils of circles whose foci coincide with the foci of conics, (3) the net of tangents to a conic of the confocal family, we get a planar 4-web. We show that each of these 4-webs is of maximal rank and characterize confocal conics from the web theory viewpoint. Mathematics Subject Classification. 53A60, 51N20. Keywords. Webs of maximal rank, Confocal conics, Linearizable webs.
1. Introduction Confocal conics and quadrics, being objects of mathematical study from antiquity till modern times, are apparently well known and understood in both their most elementary (see [8,9]) and the most profound aspects (see [7]). So much surprising is a novel characterization [4] thereof as coordinate surfaces (or lines) of curvilinear orthogonal coordinate system s1 , s2 , . . . , sn such that the Cartesian coordinates x1 , x2 , . . . , xn , aligned along the principal axes of quadrics/conics, are factored into n functions each depending only on one variable: xi = fi,1 (s1 )fi,2 (s2 ) . . . fi,n (sn ), i = 1, . . . , n. Considering foliations of Rn by Cartesian coordinate hyperplanes xi = const and curvilinear coordinate surfaces sk = const, we obtain a (singular) 2n-web. Taking logarithm of both sides of the above factorization formulae, we get n Abelian relations of this 2n-web. Recall that an Abelian relation of a d-web is a functional identity of the simplest possible form d k=1
uk = 0
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among some first integrals ui of the web foliations. For n = 2, we have 2 Abelian relations for the planar 4-web of confocal conics and Cartesian coordinate lines. It turns out that there is a third Abelian relation, independent of the above two and therefore the rank of the web (the number of non-trivial independent Abelian relations) is 3. This is maximal possible for a planar 4-web. It is well known that any planar 4-web of maximal rank is locally diffeomorphic to one formed by tangents to some algebraic curve of 4th class C. Let us call its dual C ∗ , which is a curve of degree 4, the rank curve or rank quartic of the 4-web of maximal rank. The rank curve can be reducible. If it has rectilinear components then the corresponding foliations of the 4-web linearization are pencils of lines. We describe confocal conics in terms of web theory as follows. Theorem 1. A planar orthogonal net and coordinate lines of a Cartesian coordinate system are a confocal family of conics and coordinate lines of the Cartesian coordinate system aligned along the principal axes of conics if and only if the following two conditions are satisfied: 1. the 4-web formed by the net and coordinate lines of Cartesian coordinate system is of maximal rank and 2. the rank quartic of the 4-web splits into a conic and 2 lines meeting at this
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