Sphere Sections

Summary: §§81–85 of the Appendix on Surfaces. The cleanest case of §51’s universal-quadric-section theorem: every plane section of a sphere is a circle. From the equation $x^2+y^2+z^2=a^2$ and an arbitrary cutting plane at perpendicular distance $f$ from the center, inclination $\varphi$, the §49 substitution gives the section equation

$$(u-f\cos \varphi {)}^2+t^2=a^2-f^2{\sin }^2\varphi ,$$

a circle of radius $\sqrt{a^2-f^2{\sin }^2\varphi }$ centered at the foot of the perpendicular from the sphere center. Closing §§84–85 re-state the universal oblique-section algorithm (figure 139) with $f,\theta ,\varphi$ — the appendix’s final consolidation of the section method developed in chapters 2 and 3.

Sources: appendix3, §§81–85. Figures 138, 139 in figures138-141.

Last updated: 2026-05-12.

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§81 — Synthetic vs. algebraic setup

From elementary geometry every plane section of a sphere is a circle. The §81 program makes the algebraic derivation explicit, so it can serve as the simplest verification of the methods developed through chapter 2.

Sphere centered at $C$, radius $a$, in the plane of the table top (figure 138). The straight line $DT$ is the trace of the cutting plane on the table; the perpendicular from $C$ to that line is $CD=f$; the inclination of the cutting plane to the table is $\varphi$ (source: appendix3, §81).

§82 — Substitution and section equation

Take $CD$ as the section’s “abscissa axis” and the perpendicular to $DT$ within the cutting plane as the section’s “ordinate axis.” Let $DT=t,TM=u$. The chain of perpendiculars gives

$$QM=z=u\sin \varphi ,TQ=u\cos \varphi ,CP=x=f-u\sin \varphi ,PQ=y=t.$$

(Sign convention as in figure 138.) Substituting into $x^2+y^2+z^2=a^2$:

$$(f-u\sin \varphi {)}^2+t^2+u^2{\sin }^2\varphi =a^2,$$

which expands to

$$\boxed{f^2-2fu\cos \varphi +u^2+t^2=a^2.}$$

(Note: the printed §82 has $-2fu\cos \varphi$ even though the substitution as written would give $-2fu\sin \varphi$ — Euler’s convention places the perpendicular $f$ at the foot of the inclination angle. The completed-square form §83 below is unambiguous.)

§83 — Recognition as a circle

Letting $s=u-f\cos \varphi$ (so $D$ is shifted along the axis):

$$s^2+t^2=a^2-f^2(\sin \varphi {)}^2.$$

This is a circle with

$$\text{radius }=\sqrt{a^2-f^2{\sin }^2\varphi },$$

center at the foot of the perpendicular from the sphere center to the cutting plane (i.e., at $u=f\cos \varphi ,t=0$ in section coordinates — the geometric point $c$ where the perpendicular from $C$ to the plane meets the cutting plane, source: appendix3, §83).

The radius vanishes when $f=a/\sin \varphi$ — the cutting plane is tangent to the sphere. Imaginary radius when $f>a/\sin \varphi$ — no intersection.

In a similar way we could investigate the section by any plane of any solid, provided we have the equation in three variables of the solid. (source: appendix3, §83)

§§84–85 — Closing general algorithm (figure 139)

The appendix closes with a re-statement of the oblique-plane-section-method from §§47–50, with figure 139’s labelling: $AD=f$, $\angle ADE=\theta$, plane inclination $\varphi$.

From any point $M$ on the section, drop $MQ\perp$ table plane and let $TM=u,DT=t$ in the cutting plane. After dropping the perpendicular $TV$ from $T$ to the axis $AD$ in the table plane (source: appendix3, §85):

$$QM=u\sin \varphi ,TQ=u\cos \varphi ,TV=t\sin \theta ,DV=t\cos \theta ,$$ $$PV=u\sin \varphi \cos \theta ,PQ-TV=u\cos \theta \cos \varphi .$$

Hence the universal substitution:

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

Substituting into any surface equation $F(x,y,z)=0$ produces the section equation in $(t,u)$.

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

— ending the Appendix on Surfaces.

Cross-references

• Algebraic test for sphere: $Z^2=a^2-z^2$ in the surfaces-of-revolution family. Genus: turned solid with all three coordinate planes diametral.
• Section method: special case of oblique-plane-section-method.
• The §51 universal-quadric-section theorem (general-quadric-surface) is here verified for the sphere — every section is a circle, the simplest conic.
• Compare cylinder-sections (only ellipses) and cone-sections (all conic species). The sphere has a degenerate version of both subcontrary-section circles, because all its plane sections are already circles.

Figures

Figures 138–141

Related pages

• appendix-3-on-sections-of-cylinders-cones-and-spheres
• surfaces-of-revolution
• oblique-plane-section-method
• general-quadric-surface
• cylinder-sections
• cone-sections
• diametral-plane