Quadric Classification Algorithm

Summary: §§127–128 of the Appendix on Surfaces. The closing decision procedure for assigning a second-order surface to its genus directly from the original equation, without first reducing to canonical form. Look only at the six second-degree terms: the same four inequalities (and one equality) from bounded-quadric-criterion partition the surface into genus 1 (ellipsoid), 2 or 3 (hyperboloid circumscribing or inscribed in its cone), and the three borderline genera 4, 5, 6 according to whether the homogeneous part factors with complex, real, or coincident linear factors.

Sources: appendix5, §§127–128.

Last updated: 2026-05-12.

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§127 — Algorithm input

Although we have arrived at these six genera of second order surfaces from the simplest equation, still we can now easily assign the genus for any second order surface from any given second degree equation. (source: appendix5, §127)

Given the surface in its general form

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

throw away the linear and constant terms and look only at

$$H(x,y,z):=\alpha z^2+\beta yz+\gamma xz+\delta y^2+ϵxy+\zeta x^2.$$

This is the homogeneous part — equivalently the asymptotic cone of asymptotic-cone.

Decision tree

The decision splits into the following cases:

Case A — Bounded (Genus 1, ellipsoid)

If all four conditions hold:

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

and

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

then the surface is closed — Genus 1. See ellipsoid.

Case B — Hyperboloid (Genera 2, 3)

If one or more of the three discriminant inequalities fails, and

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

then the surface is a hyperboloid surface with an asymptotic cone. Specifically:

• If $\alpha {ϵ}^2+\delta {\gamma }^2+\zeta {\beta }^2>\beta \gamma ϵ+4\alpha \delta \zeta$: Genus 2 (elliptic hyperboloid, circumscribes the cone).
• The complementary inequality with discriminant inequalities reversed: Genus 3 (hyperbolic hyperboloid, inscribed).

See elliptic-and-hyperbolic-hyperboloids. (Euler notes that distinguishing 2 from 3 from a generic equation is harder than the rest of the algorithm; reduction to canonical form via quadric-canonical-form settles it cleanly.)

Case C — Borderline (Genera 4, 5, 6)

If the equality holds:

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

then $H(x,y,z)$ factors as the product of two linear forms (the asymptotic cone degenerates to two planes). Sub-cases:

• Complex linear factors: Genus 4 (elliptic paraboloid). See paraboloids-and-parabolic-hyperboloid.
• Real linear factors (distinct): Genus 5 (parabolic hyperboloid).
• Equal linear factors (perfect square): Genus 6 (parabolic cylinder). See parabolic-cylinder-quadric.

§128 — Practical caveat

It is more difficult to make a decision about the second and third genera since one can change into the other. (source: appendix5, §128)

The mutual transformation: a hyperboloid of one sheet (genus 2) can morph into a hyperboloid of two sheets (genus 3) as a parameter passes through zero — at the moment of transition the surface becomes the asymptotic cone itself, which is genus 2’s limit species $a=0$. So whether to call a generic equation “genus 2” or “genus 3” requires inspecting the right-hand side after canonical reduction.

The other three borderline genera are sharply distinguished by the linear-factor structure of $H$.

Worked sanity check: sphere

$x^2+y^2+z^2=a^2$: $\alpha =\delta =\zeta =1$, $\beta =\gamma =ϵ=0$. The three inequalities are $4>0$ (trivially); the fourth is $0<4$. All four hold — Genus 1, as expected.

Cross-references

• The algorithm is the operational counterpart of bounded-quadric-criterion: the same four expressions are interpreted as a partition of ${\mathbb{R}}^9$-coefficient-space into the six genera.
• All four conditions tie back to asymptotic-cone — the underlying invariant is the cone’s signature.
• 3D analogue of the classification-of-conics $\gamma \gtrless 0$ trichotomy. There, one number partitions; here, four expressions partition.

Related pages

• appendix-5-on-second-order-surfaces
• bounded-quadric-criterion
• asymptotic-cone
• quadric-canonical-form
• ellipsoid
• elliptic-and-hyperbolic-hyperboloids
• paraboloids-and-parabolic-hyperboloid
• parabolic-cylinder-quadric
• classification-of-conics