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conic section

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Definition

Let kk be a field. A conic section over kk is the zero set of a degree 2 polynomial p(x,y)k[x,y]p(x, y) \in k[x, y] in the affine plane 𝔸 2(k)\mathbb{A}^2(k), or better yet the zero set of a homogeneous polynomial p(x,y,z)p(x, y, z) of degree 2 in the projective plane 2(k)\mathbb{P}^2(k).

Over the real numbers

In the classical case k=k = \mathbb{R} of real numbers, conic sections may be pictured in terms of intersections of a standard cone x 2+y 2z 2=0x^2 + y^2 - z^2 = 0 in affine 3-space with various affine hyperplanes (hence the name, “conic section”). In this picture, nonsingular conic sections are classified (up to automorphisms of the affine plane) by the sign of the discriminant of pp. In other words, if we write p(x,y)=ax 2+bxy+cy 2+λ(x,y)p(x, y) = a x^2 + b x y + c y^2 + \lambda(x, y) where deg(λ)=1deg(\lambda) = 1 and put D=b 24acD = b^2 - 4 a c, then an isomorphism class is one of three types: ellipses (when D<0D \lt 0), parabolas (D=0D = 0), and hyperbolas (D>0D \gt 0). Of course when we admit possibly singular conic sections, we get further isomorphism classes involving some degree of degeneracy (e.g., two lines, a double line, etc.).

The distinctions between ellipse, parabola, and hyperbola are artifacts of affine geometry: if we instead consider conic sections as projective subvarieties of 2()\mathbb{P}^2(\mathbb{R}), then considered up to projective transformations (automorphisms of the projective plane), these distinctions evaporate and there is really only one kind of nonsingular conic section. Put differently: if we fix a representation 2()=𝔸 2()L\mathbb{P}^2(\mathbb{R}) = \mathbb{A}^2(\mathbb{R}) \sqcup L where LL is a chosen “line at infinity”, then in the original classification up to affine transformations, i.e., the subgroup of projective transformations which take LL to itself, ellipses are those conic sections which do not intersect LL, parabolas are those which intersect LL in a double point, and hyperbolas are those which intersect LL in two points. By enlarging to the group of all projective transformations, we can move LL to a line which does intersect an “ellipse” in two points, making it a “hyperbola” with respect to the new coordinate system, etc.

Stereographic projection

Considered in terms of projective geometry, all pointed1 nonsingular conic sections C 2(k)C \subset \mathbb{P}^2(k) are isomorphic and can be identified explicitly with a projective line 1(k)\mathbb{P}^1(k) by means of a stereographic projection.

Geometrically, if pp is the chosen basepoint of CC and L 2(k)L \subset \mathbb{P}^2(k) is a line not incident to pp, then for any other point qq of CC the unique line L(p,q)L(p, q) incident to pp and qq intersects LL in exactly one point, denoted ϕ(q)\phi(q). (Here ϕ(p)\phi(p) to be the intersection of the tangent to pp at CC with LL; this can be considered the basepoint of LL.) In the opposite direction, to each point xx of LL, the line L(p,x)L(p, x) intersects CC in pp and (since a quadratic with one root will also have another root) another point qq (which might be the same as pp; this happens precisely when L(p,x)L(p, x) is the tangent at pp); this gives the inverse ϕ 1(x)=q\phi^{-1}(x) = q. In this way we obtain an isomorphism ϕ:CL\phi: C \to L of subvarieties.

Working over an algebraically closed field kk, where every nonsingular conic CC has a point, we may conclude that CC is isomorphic (as a projective variety) to 1(k)\mathbb{P}^1(k). Hence CC is a curve of genus 00.

Projective duality

Working over an algebraically closed field kk (let us assume the characteristic is not 22), all nondegenerate quadratic forms on a vector space VV are isomorphic and we may fix one as standard. For example, for V=k 3V = k^3, we may fix attention on the quadratic form q(x,y,z)=x 2+y 2+z 2q(x, y, z) = x^2 + y^2 + z^2, which determines a conic section C 2(k)C \subset \mathbb{P}^2(k) and an accompanying nondegenerate symmetric bilinear form , q:V×Vk\langle-, -\rangle_q: V \times V \to k.

Projective duality relative to CC is the projectivization of linear duality with respect to ,\langle -, - \rangle, which takes a linear subspace LL to its orthogonal dual L ={vV:(wL)v,w=0}L^\perp = \{v \in V: (\forall w \in L)\; \langle v, w \rangle = 0\}. We note that the orthogonal dual is an involution that takes joins of subspaces to meets and vice-versa. This construction descends through the quotient k 3\{0} 2(k)k^3 \backslash \{0\} \to \mathbb{P}^2(k) to give an operation that takes points in 2(k)\mathbb{P}^2(k) (lines in k 3k^3) to lines in 2(k)\mathbb{P}^2(k) (hyperplanes in k 3k^3), and vice-versa, and moreover takes a join of two distinct points (the line incident to them) to the meet of their dual lines (the point of their intersection).

This duality may be visualized thus: given a nondegenerate conic CC and a point pp off of CC, draw the two lines incident to pp that are tangent to CC, and pass to the line incident to the tangent points. (This is easier to visualize by imagining k=k = \mathbb{R} and considering a point pp exterior to say an ellipse CC.) This defines the line that is projectively dual to pp (dual with respect to the conic CC); if pp is on CC, then the same procedure works by considering the two tangent points as infinitesimally close to CC, so that the line between them is the tangent line at pp: the projective dual of pp on CC is its tangent line (this is the case where the line in k 3k^3 corresponding to pp is isotropic with respect to the bilinear form).

The entire procedure can be reversed and gives an anti-involution on the poset of flats of 2(k)\mathbb{P}^2(k), interchanging points and lines and interchanging meets and joins.

More generally, projective duality can be described in terms of an orthogonality map (V)(V *)\mathbb{P}(V) \to \mathbb{P}(V^\ast), where V *V^\ast is the linear dual of VV, which maps a subvariety to a corresponding dual “envelope” subvariety. This is explored in GKZ. The role of the conic section (or generally a conic hypersurface) is simply to set up an explicit self-duality (V *)(V)\mathbb{P}(V^\ast) \cong \mathbb{P}(V).

Pascal’s “mystic hexagon”

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References

  • I.M. Gel’fand, M. Kapranov, A. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser 2008 (paperback edition).

  1. We took care to say pointed: note that depending on the field kk, there might not even be a solution point on the conic CC, even if CC is by all rights nonsingular. For example, consider p(x,y)=x 2+y 2+1p(x, y) = x^2 + y^2 + 1 over \mathbb{R}. Of course we can relax again if kk is algebraically closed.

Last revised on January 3, 2015 at 21:51:44. See the history of this page for a list of all contributions to it.