Bounded Quadric Criterion

Summary: §§108–112 of the Appendix on Surfaces. The conditions for a second-order surface to lie in a bounded region — i.e., its asymptotic cone to be complex (reduced to the vertex). Three inequalities $4\alpha \zeta >{\gamma }^2$, $4\alpha \delta >{\beta }^2$, $4\delta \zeta >{ϵ}^2$ together with $\alpha {ϵ}^2+\delta {\gamma }^2+\zeta {\beta }^2<\beta \gamma ϵ+4\alpha \delta \zeta$. The borderline equality $=$ in the fourth condition gives the intermediate case (§112), the threshold between bounded and going-to-infinity, where the cone degenerates to two asymptotic planes — the source of Genera 4, 5, 6.

Sources: appendix5, §§108–112.

Last updated: 2026-05-12.

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§108 — Three sectional discriminant inequalities

For the homogeneous cone

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

to reduce to the single point at the origin, each of its three coordinate sections (set $x=0$, $y=0$, or $z=0$) must have a negative discriminant:

$$4\delta \zeta >{ϵ}^2(\text{from }z=0),$$ $$4\alpha \delta >{\beta }^2(\text{from }x=0),$$ $$4\alpha \zeta >{\gamma }^2(\text{from }y=0).$$

Otherwise the cone has real points on one of the coordinate planes, and the surface goes to infinity.

§109 — The missing fourth condition

The three inequalities are necessary but not sufficient. Even with all three, the asymptotic expression

$$({\beta }^2-4\alpha \delta )y^2+(2\beta \gamma -4\alpha ϵ)xy+({\gamma }^2-4\alpha \zeta )x^2$$

inside the $z$-quadratic of §105 (see asymptotic-cone) must itself be negative for all $(x,y)\ne (0,0)$. Since ${\beta }^2-4\alpha \delta <0$ and ${\gamma }^2-4\alpha \zeta <0$ by the §108 conditions, this further requirement reduces to the discriminant condition

$$(\beta \gamma -2\alpha ϵ{)}^2<({\beta }^2-4\alpha \delta )({\gamma }^2-4\alpha \zeta ).$$

§110 — Expanded form

Expanding and simplifying — the cross terms cancel some, leaving — Euler arrives at

$$\boxed{\alpha {ϵ}^2+\delta {\gamma }^2+\zeta {\beta }^2<\beta \gamma ϵ+4\alpha \delta \zeta .}$$

This is the fourth, and crucial, condition. Assuming $\alpha >0$ for normalization (no loss of generality), the other three conditions automatically imply $\delta ,\zeta >0$ as well.

A surface of the second order is confined to a bounded region if in its equation the four following conditions are satisfied. (source: appendix5, §110)

The sphere $x^2+y^2+z^2=a^2$ has $\alpha =\delta =\zeta =1$ and $\beta =\gamma =ϵ=0$, so all four conditions hold. More generally any equation $\alpha z^2+\delta y^2+\zeta x^2=a^2$ with all three coefficients positive is bounded.

§§111–112 — The intermediate case

If the strict inequality of §110 becomes equality:

$$\alpha {ϵ}^2+\delta {\gamma }^2+\zeta {\beta }^2=\beta \gamma ϵ+4\alpha \delta \zeta ,$$

then the asymptotic expression

$$\alpha z=-\beta y-\gamma x+y\sqrt{{\beta }^2-4\alpha \delta }+x\sqrt{{\gamma }^2-4\alpha \zeta }$$

factors — the homogeneous cone equation now has two linear factors, which can be:

• Complex (when underlying discriminants are negative): the surface is Genus 4 (elliptic paraboloid, see paraboloids-and-parabolic-hyperboloid).
• Real: the surface is Genus 5 (parabolic hyperboloid with two intersecting asymptotic planes).
• Equal (a perfect square): the surface is Genus 6 (parabolic cylinder, see parabolic-cylinder-quadric).

This threefold diversity gives the three genera of surfaces which go to infinity. Thus we have all together five genera of second order surfaces. (source: appendix5, §112)

(Plus the bounded ellipsoid family for a total of six — Euler’s count.)

The strict-inequality unbounded case

If one or more of the three §108 inequalities fails and the §110 inequality is not borderline, then the cone is genuinely 3D (a real 2-sheeted cone), and the surface either circumscribes it (Genus 2, elliptic hyperboloid) or is inscribed in it (Genus 3, hyperbolic hyperboloid). The first case occurs when

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

the diametrically opposite case to the bounded one (§111).

Cross-references

• The criterion uses asymptotic-cone as the master invariant — the inequalities ensure the cone reduces to a point.
• These same four expressions reappear in quadric-classification-algorithm §§127–128 as the read-off table from any given equation.
• 3D analogue of the conic discriminant ${\beta }^2-4\alpha \gamma$ in classification-of-conics — but where the conic case has one discriminant, the quadric case has four coupled inequalities.

Related pages

• appendix-5-on-second-order-surfaces
• asymptotic-cone
• quadric-classification-algorithm
• ellipsoid
• elliptic-and-hyperbolic-hyperboloids
• paraboloids-and-parabolic-hyperboloid
• classification-of-conics