Generalized pencils of conics derived from cubics
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Generalized pencils of conics derived from cubics Lorenz Halbeisen1 · Norbert Hungerbühler1 Received: 10 December 2019 / Accepted: 25 March 2020 © The Managing Editors 2020
Abstract Given a cubic K . Then for each point P there is a conic C P associated to P. The conic C P is called the polar conic of K with respect to the pole P. We investigate the situation when two conics C0 and C1 are polar conics of K with respect to some poles P0 and P1 , respectively. First we show that for any point Q on the line P0 P1 , the polar conic C Q of K with respect to Q belongs to the linear pencil of C0 and C1 , and vice versa. Then we show that two given conics C0 and C1 can always be considered as polar conics of some cubic K with respect to some poles P0 and P1 . Moreover, we show that P1 is determined by P0 , but neither the cubic nor the point P0 is determined by the conics C0 and C1 . Keywords Pencils · Conics · Polars · Polar conics of cubics Mathematics Subject Classification 51A05 · 51A20
1 Terminology We will work in the real projective plane RP2 = R3 \{0}/ ∼, where X ∼ Y ∈ R3 \{0} are equivalent, if X = λY for some λ ∈ R. Points X = (x1 , x2 , x3 )T ∈ R3 \{0} will be denoted by capital letters, the components with the corresponding small letter, and the equivalence class by [X ]. However, since we mostly work with representatives, we often omit the square brackets in the notation. Let f be a non-constant homogeneous polynomial in the variables x1 , x2 , x3 of degree n. Then f defines a projective algebraic curve C f := {[X ] ∈ RP2 | f (X ) = 0}
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Norbert Hungerbühler [email protected] Lorenz Halbeisen [email protected]
1
Department of Mathematics, ETH Zentrum, Rämistrasse 101, 8092 Zurich, Switzerland
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Beitr Algebra Geom
of degree n. For a point P ∈ RP2 , P f (X ) := P, ∇ f (X ) is also a homogeneous polynomial in the variables x1 , x2 , x3 . If the homogeneous polynomial f is of degree n, then C P f is an algebraic curve of degree n − 1. The curve C P f is called the polar curve of C f with respect to the pole P; sometimes we call it the polar curve of P with respect to C f . In particular, when C f is a cubic curve (i.e., f is a homogeneous polynomial of degree 3), then C P f is a conic, which we call the polar conic of C f with respect to the pole P, and when C f is a conic, then C P f is a line, which we call the polar line of C f with respect to the pole P (see, for example, Wieleitner 1939). By construction, the intersections of a curve C f and its polar curve C P f with respect to a point P give the points of contact of the tangents from P to C f , as well as points on C f where ∇ f = 0 (see Examples 3 and 4 ). The geometric interpretation of poles and polar lines (or polar surface in higher dimensions) goes back to Monge, who introduced them in 1795 (see Monge 1809, § 3). The names pole and polar curve (or polar surface) were coined by Bobillier (see Bobillier 1827–1828a, b, c, Bobillier 1828–1829a, b) who also iterated the construction and considered higher polar cur
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