Cylinder Sections

Summary: §§52–67 of the Appendix on Surfaces. Complete section analysis of an elliptical-base cylinder $a^2{c}^2=a^2{y}^2+c^2{x}^2$ (figure 130). Sections parallel to base = ellipses (or circles if the cylinder is right); sections parallel to axis = pairs of parallel lines or empty; sections oblique to axis = ellipses with semiaxes $a/\cos \varphi$ and $c$ when the cutting line is parallel to a conjugate axis (§56), with subcontrary-circle condition when $\cos \varphi =a/c$ (§57). The general oblique section (§§58–63) gives the headline §64 theorem: for any cylinder section, the product of the conjugate semiaxes is to the product of the base semiaxes as $1$ is to $\cos \varphi$.

Sources: appendix3, §§52–67. Figures 130–133 in figures128-130, figures131-132, figures133-134.

Last updated: 2026-05-12.

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§52 — Right vs. scalene (Euclid)

Euclid distinguishes two cylinder species:

• Right cylinder: all sections perpendicular to the axis are congruent circles whose centers lie on one straight line.
• Scalene (oblique) cylinder: all sections perpendicular to the axis are congruent ellipses whose centers lie on one straight line — but the axis itself is not perpendicular to those sections.

Euler reformulates: a cylinder is scalene if its perpendicular-to-axis sections are congruent ellipses (source: appendix3, §52).

§§53–54 — Equation and parallel-plane sections

Set up coordinates as in figure 130: ellipse base $AEBF$ in the table plane, semiaxes $AC=BC=a$, $CE=CF=c$, axis $CD$ perpendicular to the base. The cylinder satisfies (source: appendix3, §53):

$$a^2{c}^2=a^2{y}^2+c^2{x}^2.$$

The variable $z$ is absent — confirming the §31/§33 cylindrical-and-prismatic-surfaces genus.

Sections parallel to the base = the base ellipse (or circle if right) by §31 (source: appendix3, §54). Sections parallel to the axis, with line of intersection $x=$ const or $y=$ const in the base, give pairs of parallel straight lines (or empty if the line misses the cylinder).

§§55–57 — Oblique section parallel to one conjugate axis (figure 130)

Let the cutting plane meet the base plane in a line parallel to $EF$, perpendicular to $AB$ at $G$ with $CG=f$. The plane is inclined at $\varphi$; let $D$ be where it meets the axis. Then $DG=f/\cos \varphi$ and $CD=f\sin \varphi /\cos \varphi$.

Take new coordinates $DS=t$, $SM=u$ in the cutting plane. After computing $y=u,x=t\cos \varphi$ and $z=f\tan \varphi -t\sin \varphi$ and substituting into the cylinder equation:

$$a^2{c}^2=a^2{u}^2+c^2{t}^2{\cos }^2\varphi .$$

This is the equation of an ellipse centered at $D$, with semiaxes

$$aD=\frac{a}{\cos \varphi },cE=c.$$

(Source: appendix3, §56.)

Subcontrary-section circle (§57). The section is a circle when both semiaxes coincide:

$$\frac{a}{\cos \varphi }=c⟺\cos \varphi =\frac{a}{c}⟺\cos \varphi =\frac{BC}{BH},$$

which holds in two planes (one above the base, one below — both oblique to the axis $CD$). The scalene cylinder is therefore characterized by having two oblique circular sections distinct from the base.

§§58–63 — General oblique section

The cutting line $GT$ in the base is now oblique to both conjugate axes. Let $GC=f$, $\angle BCG=\theta$, $\angle CGD=\varphi$. After the full substitution (with $DS=t,SM=u$):

$$a^2{c}^2=\alpha u^2+2\beta tu+\gamma t^2,$$

where

$$\alpha =a^2{\cos }^2\theta +c^2{\sin }^2\theta ,\beta =(a^2-c^2)\sin \theta \cos \theta \cos \varphi ,\gamma =a^2{\sin }^2\theta {\cos }^2\varphi +c^2{\cos }^2\theta {\cos }^2\varphi .$$

Rotation by $\zeta$ with $\tan 2\zeta =2\beta /(\gamma -\alpha )$ diagonalizes the form; the principal semiaxes are then read off (§§61–62). After algebraic manipulation:

$$aD\cdot eD=\frac{a^2{c}^2}{\sqrt{\alpha \gamma -{\beta }^2}}=\frac{ac}{\cos \varphi }.$$

($\sqrt{\alpha \gamma -{\beta }^2}=ac\cos \varphi$ is the key identity, §63.)

§64 — Theorem

THEOREM. If any cylinder is cut by any plane, then the product of the principal axes of the section is to the product of the principal axes of the base of the cylinder as $1$ is to the cosine of the angle made by the plane of the section with the plane of the base. (source: appendix3, §64)

In symbols:

$$\boxed{aD\cdot eD=\frac{ac}{\cos \varphi }.}$$

Equivalent statement in terms of inscribed parallelograms: every parallelogram on a pair of conjugate axes of any section has area ${\pi }^{-1}$ times the area of the section’s ellipse, and that area scales as $1/\cos \varphi$ relative to the base.

This generalizes the §56 special case (cutting line parallel to one conjugate axis, $\theta =0$ or $\pi /2$) to every oblique cutting plane.

§§65–67 — Reformulation along the cutting line (figure 133)

A cleaner derivation using the perpendicular $CH$ from the center $C$ of the base to the cutting line $TH$, with $\angle GCH=\theta$ and $\angle CHD=\varphi$. The result is the same; this presentation makes the geometry of the oblique cut more transparent for §§68–80 cone-section calculations (figure 134 onward).

Cross-references

• Algebraic test for cylinder: $z$ absent. Genus: cylindrical-and-prismatic-surfaces.
• Section method: every formula is a special case of oblique-plane-section-method.
• The §51 universal-quadric-section theorem (general-quadric-surface) is here verified for the cylinder — every section is a conic.
• Conjugate-diameter algebra parallels conjugate-diameters (Chapter 5 §§111–120) for the plane case.

Figures

Figures 128–130

Figures 131–132

Figures 133–134

Related pages

• appendix-3-on-sections-of-cylinders-cones-and-spheres
• cylindrical-and-prismatic-surfaces
• oblique-plane-section-method
• general-quadric-surface
• conjugate-diameters
• cone-sections
• sphere-sections