Quadric Canonical Form

Summary: §§113–115 of the Appendix on Surfaces. Every second-order surface can be reduced to $A{p}^2+B{q}^2+C{r}^2+K=0$ by composing a rotation (kills the three cross-terms $pq,pr,qr$) with a translation (kills the three linear terms $Gp,Hq,Ir$). The reduced form makes the center and the three mutually perpendicular diametral planes visible — the 3D lift of center-of-conic + principal-axes-and-foci.

Sources: appendix5, §§113–115.

Last updated: 2026-05-12.

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§113 — Rotation kills cross-terms

Starting from the general equation

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

apply the change-of-coordinates-3d §92 substitution with three angles $k,m,n$ (Euler’s notation here):

$$x=p(\cos k\cos m-\sin k\sin m\cos n)+q(\cos k\sin m\cos n)-r\sin k\sin n,$$

etc. The general equation in $(p,q,r)$ takes the form

$$A{p}^2+B{q}^2+C{r}^2+Dpq+Epr+Fqr+Gp+Hq+Ir+K=0.$$

Although the calculations are rather long, we can find the actual angles which bring this result. (source: appendix5, §114)

The three angles can be chosen so that $D=E=F=0$ — i.e., the three cross-products vanish. The result:

$$A{p}^2+B{q}^2+C{r}^2+Gp+Hq+Ir+K=0.$$

This is the analogue of the principal-axes diagonalization for a 3D quadratic form.

§115 — Translation kills linear terms

The coordinates $p,q$ and $r$ can be increased or decreased in such a way that the coefficients $G,H$ and $I$ also vanish. This is accomplished simply by changing the position of the origin. (source: appendix5, §115)

A translation $p\mapsto p-G/(2A)$ etc. (the standard completing-the-square trick) eliminates the linear terms — provided all three $A,B,C$ are nonzero. The final canonical form:

$$\boxed{A{p}^2+B{q}^2+C{r}^2+K=0.}$$

§115 — Three diametral planes and a center

In the canonical form:

• Each of the three coordinate planes ($r=0$, $q=0$, $p=0$) bisects the surface — substituting $r\to -r$ leaves the equation unchanged, so the surface is symmetric about $r=0$. These are the three mutually perpendicular diametral planes of the surface.
• Their common point (the new origin) is the center of the surface.

Every second order surface has not only a diametral plane but three mutually perpendicular planes of this kind. The point of intersection of these three planes is the center of the surface, although the center in some cases is at an infinite distance. (source: appendix5, §115)

Degenerate cases — center at infinity

The translation step assumes $A,B,C$ are all nonzero. If one of them vanishes — say $C=0$ — then $Ir$ cannot be killed by translation, and the canonical form becomes

$$A{p}^2+B{q}^2+Ir=K$$

($K$ removable by shifting $r$), giving the paraboloid family (§123, paraboloids-and-parabolic-hyperboloid). The “center” of such a surface lies at infinity along the axis $r$ — analogous to a parabola whose center lies at infinity along the axis.

If two coefficients vanish — say $B=C=0$ — only one quadratic term remains, $A{p}^2=$ linear, giving the parabolic cylinder (§126, parabolic-cylinder-quadric).

Six canonical-form representatives (genera)

Form	Genus
$A{p}^2+B{q}^2+C{r}^2=a^2$, all signs $+$	1 (ellipsoid)
$A{p}^2+B{q}^2-C{r}^2=a^2$	2 (elliptic hyperboloid)
$A{p}^2-B{q}^2-C{r}^2=a^2$	3 (hyperbolic hyperboloid)
$A{p}^2+B{q}^2=ar$	4 (elliptic paraboloid)
$A{p}^2-B{q}^2=ar$	5 (parabolic hyperboloid)
$A{p}^2=aq$	6 (parabolic cylinder)

Plus degenerate quasi-species at each level (sphere, parabolic cone, two parallel planes, etc.).

Cross-references

• 3D analogue of center-of-conic (every conic has a center, possibly at infinity for the parabola) and principal-axes-and-foci (every conic has a unique orthogonal conjugate diameter pair = principal axes).
• The rotation step §113 uses the same Euler-angle substitution as change-of-coordinates-3d §92, instantiated to make $D=E=F=0$.
• The canonical equations $A{p}^2+B{q}^2+C{r}^2+K=0$ are then enumerated genus by genus in appendix-5-on-second-order-surfaces §§116–126.

Related pages

• appendix-5-on-second-order-surfaces
• change-of-coordinates-3d
• ellipsoid
• elliptic-and-hyperbolic-hyperboloids
• paraboloids-and-parabolic-hyperboloid
• parabolic-cylinder-quadric
• center-of-conic
• principal-axes-and-foci
• diametral-plane