Asymptotic Cone

Summary: §§104–107 of the Appendix on Surfaces. The behaviour at infinity of any algebraic surface is controlled by its homogeneous highest-degree part — set the lower-degree terms to zero, and the resulting equation defines a cone with vertex at the origin whose two-sheeted asymptotic structure the surface approaches. For second-order surfaces this cone is itself second-order; whether it’s real, complex, or degenerate is the master invariant for classification (bounded-quadric-criterion, quadric-classification-algorithm). The 3D analogue of branches-at-infinity for plane curves.

Sources: appendix5, §§104–107.

Last updated: 2026-05-12.

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§104 — Isolating the infinite part

For the general second-order surface

$$\alpha z^2+\beta yz+\gamma xz+\delta y^2+ϵxy+\zeta x^2+\eta z+\theta y+\iota x+\kappa =0,$$

sending one variable (say $z$) to $\infty$ makes the linear and constant terms $\eta z,\theta y,\iota x,\kappa$ vanish in comparison with the second-degree terms. The part of the surface going to infinity satisfies

$$\alpha z^2+\beta yz+\gamma xz+\delta y^2+ϵxy+\zeta x^2=0.$$

All of the terms which are not infinite, or are at least less infinite than $\alpha z^2$ will vanish. (source: appendix5, §104)

§105 — Solving for $z$

Treating the homogeneous equation as quadratic in $z$:

$$z=\frac{-\beta y-\gamma x\pm \sqrt{({\beta }^2-4\alpha \delta )y^2+(2\beta \gamma -4\alpha ϵ)xy+({\gamma }^2-4\alpha \zeta )x^2}}{\alpha }.$$

For each $(x,y)\ne (0,0)$, this gives two values of $z$ — two ruling lines of the cone through the origin in the direction of $(x,y)$.

§106 — The cone as asymptote

The homogeneous equation defines a cone with vertex at the origin: every coordinate set $(\lambda x,\lambda y,\lambda z)$ also satisfies it for any $\lambda$. As the original surface goes to infinity, it approaches this cone in the sense that the bounded distance between the two surfaces shrinks to zero.

Just as we distinguished branches of a curve which go to infinity by straight line asymptotes, so now we can distinguish parts of a surface which go to infinity by asymptotic cones. (source: appendix5, §106)

§107 — Real, complex, or degenerate

Three possibilities for the asymptotic cone:

1. Real cone with multiple ruling lines — the surface really does go to infinity (Genera 2, 3 in appendix-5-on-second-order-surfaces).
2. Complex cone (reduces to the single point at the origin) — the surface is bounded (Genus 1, the ellipsoid).
3. Degenerate — the cone factors into asymptotic planes. This intermediate case happens precisely when $\alpha {ϵ}^2+\delta {\gamma }^2+\zeta {\beta }^2=\beta \gamma ϵ+4\alpha \delta \zeta$ (bounded-quadric-criterion §112); the planes can be real (Genus 5), complex (Genus 4), or coincident (Genus 6).

If the asymptotic cone is complex, then the given surface has no part which goes to infinity, so that it is contained in a bounded region. (source: appendix5, §107)

§108 — The bounded criterion derivation

For the cone $\delta y^2+ϵxy+\zeta x^2=0$ (the $z=0$ section) to have no nontrivial real solution, the discriminant must be negative: $4\delta \zeta >{ϵ}^2$. Similarly from $z=0$ in either of the other two sections:

$$4\alpha \delta >{\beta }^2,4\alpha \zeta >{\gamma }^2.$$

These three conditions are necessary for boundedness but not sufficient — see bounded-quadric-criterion §109.

Cross-references

• 3D analogue of branches-at-infinity: there, a plane curve has infinite branches iff the highest-degree member has a real linear factor; here, a surface has infinite parts iff the highest-degree member (the cone equation) has real points other than the origin.
• The asymptotic-cone perspective is the master tool of appendix-5-on-second-order-surfaces: every genus has a distinctive cone (or pair of asymptotic planes, or none).
• Conceptually similar to a Newton-polygon / leading-form analysis — strip the lower-degree terms, read off the behaviour at infinity.

Related pages

• appendix-5-on-second-order-surfaces
• bounded-quadric-criterion
• quadric-classification-algorithm
• elliptic-and-hyperbolic-hyperboloids
• branches-at-infinity
• higher-order-surface-cones