degree-invariance-surfaces

Degree Invariance for Surfaces

Summary: §§93–96 of the Appendix on Surfaces. The total degree of a surface’s equation is invariant under every change of coordinates — origin shift, axis rotation, plane tilt, or the full §92 rigid motion. Equivalently: order is an intrinsic property of the surface, not of the equation. The corollary, §95, recovers the oblique-plane-section-method §50 invariance: every plane section of an order-$n$ surface has order $\le n$. The 3D lift of Book II’s degree-invariance.

Sources: appendix4, §§93–96.

Last updated: 2026-05-12.

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§94 — The theorem

Although the general equation is usually very complicated, still if we consider the total degree of the coordinates, that total degree is always equal to the total degree of the original coordinates $x,y$ and $z$. (source: appendix4, §94)

Sphere example: $x^2+y^2+z^2=a^2$ is second-degree. Substituting change-of-coordinates-3d §92 produces an equation in $p,q,r$ that is also second-degree (and indeed remains a sphere — the substitution is a rigid motion). No matter how the coordinates are repositioned, the degree stays the same.

This justifies assigning an order to a surface: first-order surfaces, second-order surfaces, etc.

§95 — Plane sections have order $\le n$

The order of the section was always equal to the order of the surface. (source: appendix4, §95)

From §85 of the previous appendix chapter (oblique-plane-section-method), the section by an arbitrary plane is obtained by substituting

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

This is a special case of the §92 most-general substitution (with two of the three rotations and one translation set, the rest zero). Hence the section’s degree in $(t,u)$ matches the surface’s degree in $(x,y,z)$ — never higher, possibly lower if a factorization causes the section to split.

§96 — Examples

• First-order surface: every plane section is a line. Therefore the surface itself is a plane. See plane-inclination-angles for the explicit positioning.
• Second-order surface: every plane section is a conic. This is general-quadric-surface §51 — re-derived here from a different angle.
• Conical surface $z^2=\alpha x^2+\beta y^2$ is second-order; section through the vertex factors into two straight lines, but their combined degree is still two.
• $n$th-order surface: every plane section is a curve of order $\le n$.

It can happen that the equation for some section admits a factorization. In this case the section will consist of two or more curves of lower order. (source: appendix4, §96)

Cross-references

• Book II curve analogue: degree-invariance — degree of a curve’s equation is preserved under every rotation/translation/oblique change of axes.
• Implies the §97–98 statement that first-order surfaces are exactly planes (just as first-order curves are exactly lines, straight-line-equation).
• Re-derives oblique-plane-section-method §50 from a higher-level invariance — the section method is one instance of the general-coordinate-change machinery.

Related pages

• appendix-4-on-the-change-of-coordinates
• change-of-coordinates-3d
• degree-invariance
• oblique-plane-section-method
• general-quadric-surface
• plane-inclination-angles