General Quadric Surface

Summary: §51 of the Appendix on Surfaces. Closing remark of chapter 2. The most general second-degree surface

$$\alpha x^2+\beta y^2+\gamma z^2+\delta xy+ϵxz+\zeta yz+ax+by+cz+e^2=0$$

has the property that every plane section is a conic — i.e. a (possibly degenerate) curve of order at most 2. Linear surfaces ($\alpha =\cdots =\zeta =0$) are planes; every section of a plane by another plane is a straight line (Euclid). The §51 theorem is the master motivation for chapter 3’s case-by-case study of cylinder, cone, and sphere sections.

Sources: appendix2, §51.

Last updated: 2026-05-12.

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§51 — The two-part statement

Linear case. If the equation has total degree 1, $\alpha x+\beta y+\gamma z=a$, then each section is a straight line, and the surface itself is a plane (source: appendix2, §51):

From Euclid we know that the intersection of two planes must be a straight line.

Quadratic case. If the equation is the general quadric

$$\alpha x^2+\beta y^2+\gamma z^2+\delta xy+ϵxz+\zeta yz+ax+by+cz+e^2=0,$$

then

[the surface] will have for every section an equation which is either a straight line or at most a second order curve. There is absolutely no section whose nature cannot be expressed by a second degree equation. (source: appendix2, §51)

Why it works — via §50

The §50 substitution from oblique-plane-section-method replaces $x,y,z$ with linear forms in the section coordinates $t,v$:

$$x=t\cos \theta +v\cos \varphi \sin \theta ,y=v\cos \varphi \cos \theta -t\sin \theta -f,z=v\sin \varphi .$$

Each of $x,y,z$ is of degree 1 in $(t,v)$. Substituting into a quadric in $x,y,z$ yields an equation whose every term has total degree $\le 2$ in $(t,v)$ — i.e. a quadratic in $(t,v)$. The §50 corollary (degree preserved or dropped) does the work.

Where the case-by-case proof lives

Chapter 3 of the Appendix verifies the §51 theorem instance by instance for the three classical quadrics:

Quadric	Equation	Sections	Page
Right/scalene cylinder	$a^2{c}^2=a^2{y}^2+c^2{x}^2$	All ellipses (or pairs of parallel lines / empty)	cylinder-sections
Right/scalene cone	$m^2{n}^2{z}^2=m^2{y}^2+n^2{x}^2$	All conics — parabola, ellipse, hyperbola according to $\tan \varphi$ vs. $1/m$	cone-sections
Sphere	$x^2+y^2+z^2=a^2$	All circles	sphere-sections

The cone case is the deepest — recovering Apollonius’s conic-section trichotomy from first principles inside the algebraic substitution method.

Plane-curve analogue

The §51 theorem is the surface companion of chapter-5-on-second-order-lines: there the general quadratic in $x,y$ was shown to represent every conic, with chapter 6’s classification-of-conics giving the trichotomy by sign of the discriminant. Here the general quadratic in $x,y,z$ is shown to cut out conics under every plane section. The full surface-side classification of quadrics into ellipsoid, hyperboloid, paraboloid, etc. is not developed in this Appendix.

Cross-references

• Substitution method: oblique-plane-section-method (§§47–50).
• Plane-conic theorems: chapter-5-on-second-order-lines, chapter-6-on-the-subdivision-of-second-order-lines-into-genera, classification-of-conics.
• Worked instances: cylinder-sections, cone-sections, sphere-sections.

Related pages

• appendix-2-on-the-intersection-of-a-surface-and-an-arbitrary-plane
• oblique-plane-section-method
• chapter-5-on-second-order-lines
• classification-of-conics
• appendix-3-on-sections-of-cylinders-cones-and-spheres