Oblique-Plane Section Method

Summary: §§47–50 (and again §§84–85) of the Appendix on Surfaces. The master substitution that converts a surface equation $F(x,y,z)=0$ into the equation of its section by any plane. Two warm-ups (figures 128, 129) build to the full three-parameter rigid-motion substitution involving the perpendicular distance $f$ of the cutting plane from the axis, the angle $\theta$ the trace makes with the $AP$ axis, and the inclination $\varphi$ of the cutting plane to the base plane $APQ$. Algebraic surfaces have algebraic sections, of equal or lower total degree (§50). This is the surface-level analogue of coordinate-transformations and underwrites the §51 universal-quadric-section theorem and every worked section in appendix-3-on-sections-of-cylinders-cones-and-spheres.

Sources: appendix2, §§47–50. Repetition with figure 139 in appendix3, §§84–85. Figures 128, 129 in figures128-130. Figure 139 in figures138-141.

Last updated: 2026-05-12.

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§47 — The plan

We refer all sections of a surface to the base plane $APQ$. Any other plane either is parallel to $APQ$ (in which case substitute $z=$ const, §29), or meets $APQ$ in a straight line. The latter case admits three sub-cases by the position of that intersection line:

1. Parallel to one of the principal axes $AP$ or $AQ$ — §48;
2. General position — §§49–50;
3. Through the origin $A$ — section through an axis, §46.

The three sub-cases are unified by adding more rotations and translations.

§48 — Cutting line parallel to the $AP$ axis (figure 128)

The cutting plane $ESM$ meets the base plane $APQ$ in the straight line $ES\parallel AP$, with $AE=f$. The plane is inclined to $APQ$ at angle $\varphi =\angle QSM$. Take $ES$ as the abscissa of the section, $SM=v$ as the ordinate.

Then

$$ES=x,SQ=y+f.$$

From the inclination angle $\varphi$,

$$QM=z=v\sin \varphi ,SQ=v\cos \varphi ,$$

so $y=v\cos \varphi -f$. Substituting

$$y=v\cos \varphi -f,z=v\sin \varphi$$

into the surface equation gives the section equation in $(x,v)$.

§§49–50 — General oblique cutting plane (figure 129)

Let the cutting plane meet $APQ$ in the line $ES$, with $AE\perp AP$, $AE=f$, and angle $\angle TES=\theta$. The plane is inclined to $APQ$ at $\varphi$. Coordinates in the section plane: $ES=t$ (along the trace), $SM=v$ (perpendicular within the cutting plane).

Drop $QS\perp ES$ within $APQ$. By the chain of perpendiculars (figure 129):

$$QM=z=v\sin \varphi ,QS=v\cos \varphi ,SV=v\cos \varphi \sin \theta ,QV=v\cos \varphi \cos \theta ,$$ $$ST=VX=t\sin \theta ,ET=t\cos \theta .$$

Combining,

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

Substitute these for $x,y,z$ in the surface equation to obtain the equation of the section in $(t,v)$ (source: appendix2, §49).

§50 — Algebraic sections of algebraic surfaces

If the equation for the solid is an algebraic equation in the three coordinates $x,y,z$, then all of its sections will also be algebraic. Furthermore, since the equation for the section in the coordinates $t$ and $v$ is obtained by substituting in the equation for the solid [the formulas above], it is clear that the variables $t$ and $v$ cannot appear with a total degree higher than the total degree of the original equation in $x,y,z$. It can happen, however, that the equation for a section may have a smaller total degree due to cancellations after the substitutions. (source: appendix2, §50)

The total degree is preserved or dropped, never increased — the surface analogue of the degree-invariance theorem for plane curves.

§§84–85 — Repetition in chapter 3 with figure 139

Chapter 3 closes the appendix with a re-derivation of the same algorithm for the general arbitrary section (source: appendix3, §§84–85). With $AD=f$, $\angle ADE=\theta$, and $\varphi$ the inclination of the cutting plane to the base:

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

(The sign conventions differ slightly from §49 — the placement of the perpendicular $f$ relative to the origin moves between the two figures — but the structure is identical.)

This method is almost the same as that which we used before in section 50. (source: appendix3, §85)

Cross-references

• Two-dimensional analogue: coordinate-transformations (Chapter 2 §§25–34) provides the rotation+translation substitutions for plane curves.
• §50 algebraic-section-of-algebraic-surface invariance lifts degree-invariance one dimension up.
• The §51 universal-quadric-section theorem in general-quadric-surface is the first immediate corollary.
• Worked applications: every section computed in cylinder-sections, cone-sections, sphere-sections reduces to a specific instance of this substitution.

Figures

Figures 128–130

Figures 138–141

Related pages

• appendix-2-on-the-intersection-of-a-surface-and-an-arbitrary-plane
• general-quadric-surface
• coordinate-transformations
• degree-invariance
• cylinder-sections
• cone-sections
• sphere-sections