Cone Sections

Summary: §§68–80 of the Appendix on Surfaces. The algebraic Apollonius. Working from the canonical scalene cone $m^2{n}^2{z}^2=m^2{y}^2+n^2{x}^2$ with vertex at the origin (figure 134), every plane section is derived by direct substitution and classified by sign of $(\cos \varphi {)}^2-m^2(\sin \varphi {)}^2$: parabola at the borderline $\tan \varphi =1/m$ (§73), ellipse when $\tan \varphi <1/m$ (§74) with a subcontrary-circle condition $\sin \varphi =\sqrt{n^2-m^2}/(n\sqrt{1+m^2})$, hyperbola when $\tan \varphi >1/m$ (§75). The right cone is the $m=n$ specialization; the equilateral hyperbola appears at the boundary $m^2+n^2=m^2{n}^2$.

Sources: appendix3, §§68–80. Figures 134–137 in figures133-134 and figures135-137.

Last updated: 2026-05-12.

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§68 — Equation of the cone (figure 134)

Cone with vertex $O$ at the origin and axis $Oc$ perpendicular to the base. Let $ab,ef$ be parallel to the principal axes of any section perpendicular to the axis with $OP=z,PQ=y,QM=z$ (relabelling). With $ac=bc=mz$ and $ec=fc=nz$ (proportional semiaxes), the surface satisfies (source: appendix3, §68):

$$m^2{n}^2{z}^2=m^2{y}^2+n^2{x}^2.$$

Every term has joint degree 2 — confirming the §34 conical-and-pyramidal-surfaces homogeneity property.

The right cone is $m=n$ (sections perpendicular to axis are circles).

§69 — Sections perpendicular to a base axis

Substituting $z=$ const gives an ellipse — already known.

Substituting $x=a$ (perpendicular to one base axis) gives

$$m^2{n}^2{z}^2=m^2{y}^2+n^2{a}^2,$$

a hyperbola with center on $Pp$, transverse semiaxis $a/m$, conjugate semiaxis $na/m$. The same for $y=$ const (source: appendix3, §69).

§§70–71 — Section perpendicular to the base, oblique to both base axes (figure 135)

The cutting plane meets the base $AEBF$ in $BE$ with $OB=a,OE=b$. Following the §49 oblique-plane-section-method substitution and reducing:

$$m^2{n}^2{c}^2{z}^2=(m^2{b}^2+n^2{a}^2)u^2+\frac{m^2{n}^2{a}^2{b}^2{c}^2}{m^2{b}^2+n^2{a}^2},c=\sqrt{a^2+b^2}.$$

Hyperbola with center $G$, transverse semiaxis $Ga=ab/\sqrt{m^2{b}^2+n^2{a}^2}$, conjugate semiaxis $mnabc/(m^2{b}^2+n^2{a}^2)$, asymptote angle $\arctan (mnc/\sqrt{n^2{b}^2+m^2{a}^2})$ (source: appendix3, §71). Equilateral hyperbola when $m^2{n}^2=m^2{b}^2+n^2{a}^2$, i.e. $b/a=\tan \angle OBE=(n\sqrt{m^2-1})/(m\sqrt{1-n^2})$.

§§72–75 — Section oblique to the axis, perpendicular to $AB$ (figure 136)

Cutting plane intersects base in $BT\perp AB$ at $OB=f$, inclined to base at $\varphi$. After the §49 substitution and reduction:

$$m^2{t}^2+n^2((\cos \varphi {)}^2-m^2(\sin \varphi {)}^2)s^2-\frac{m^2{n}^2{f}^2(\sin \varphi {)}^2}{(\cos \varphi {)}^2-m^2(\sin \varphi {)}^2}=0.$$

The coefficient of $s^2$ determines the type:

Sign of $(\cos \varphi {)}^2-m^2(\sin \varphi {)}^2$	Cutting line $BC$ in figure 134	Section
$>0$, i.e. $\tan \varphi <1/m$	intersects $Oa$ on opposite side of cone	ellipse (§74)
$=0$, i.e. $\tan \varphi =1/m$	parallel to $Oa$ (cone’s slant)	parabola (§73)
$<0$, i.e. $\tan \varphi >1/m$	diverges from $Oa$ on opposite side	hyperbola (§75)

This is the algebraic Apollonius trichotomy. The cone’s half-angle is $\arctan m$ at the base intersection of $Oa$ — so the trichotomy is whether the cutting plane is shallower, parallel, or steeper than the cone’s own slant, exactly the synthetic Greek classification.

Parabola case (§73)

$\tan \varphi =1/m$: section is parabola with vertex at $G$ where $BG=f/(2\cos \varphi )$, latus rectum $2n^{2}f\cos \varphi /m^2$.

Ellipse case (§74)

Semiaxis along $BC$: $mf\sin \varphi /((\cos \varphi {)}^2-m^2(\sin \varphi {)}^2)$. Conjugate semiaxis: $nf\sin \varphi /((\cos \varphi {)}^2-m^2(\sin \varphi {)}^2{)}^{1/2}$. Semilatus rectum: $n^{2}f\sin \varphi /m$.

Subcontrary-section circle (§74). Circle when $m=n\sqrt{(\cos \varphi {)}^2-m^2(\sin \varphi {)}^2}$, i.e.,

$$\sin \varphi =\frac{\sqrt{n^2-m^2}}{n\sqrt{1+m^2}},\cos \varphi =\frac{m\sqrt{1+n^2}}{n\sqrt{1+m^2}}.$$

Requires $n>m$. So an oblique cone admits a non-axial circular section iff its base ellipse has $c>a$ (the “wider” semiaxis is the one perpendicular to the cutting line’s projection).

Hyperbola case (§75)

Transverse semiaxis $mf\sin \varphi /(-(\cos \varphi {)}^2+m^2(\sin \varphi {)}^2)$. Conjugate semiaxis $nf\sin \varphi /((m^2(\sin \varphi {)}^2-(\cos \varphi {)}^2{)}^{1/2})$. Asymptote-axis angle has tangent $(n/m)\sqrt{m^2(\sin \varphi {)}^2-(\cos \varphi {)}^2}$.

Equilateral hyperbola when $m^2{n}^2(\sin \varphi {)}^2-n^2(\cos \varphi {)}^2=m^2$, i.e.,

$$\sin \varphi =\frac{\sqrt{m^2+n^2}}{n\sqrt{1+m^2}},\cos \varphi =\frac{m\sqrt{n^2-1}}{n\sqrt{1+m^2}}.$$

Requires $n>1$.

§§76–80 — General oblique section (figure 137)

The fully general cutting plane is perpendicular to neither base axis. The base intersection $BR$ makes angle $\theta$ with $AB$, $OB=f$, $\angle OBR=\theta$, plane inclination $\varphi$.

After the §49 substitution and a long reduction (§§77–80), the section equation is

$${\mu }^2\cdot C{b}^2\cdot t^2+{\nu }^2(O{b}^2-{\mu }^{2}O{c}^2)s^2=\frac{{\mu }^2{\nu }^2\cdot O{b}^2\cdot O{c}^2\cdot C{b}^2}{O{b}^2-{\mu }^{2}O{c}^2},$$

where $\mu ,\nu$ are auxiliary semidiameters depending on $\theta$ and $m,n$. The trichotomy condition collapses to

$$\tan \varphi ?\frac{\nu }{mn}.$$

Ellipse for $\tan \varphi <\nu /(mn)$, parabola for $\tan \varphi =\nu /(mn)$, hyperbola for $\tan \varphi >\nu /(mn)$.

The §73 perpendicular-to-$AB$ formula is recovered by setting $\theta =\pi /2$.

Cross-references

• Algebraic test for cone: homogeneous in $(x,y,z)$. Genus: conical-and-pyramidal-surfaces.
• Section method: every formula is a special case of oblique-plane-section-method.
• Plane-side classification of conics: classification-of-conics, ellipse, parabola, hyperbola — every species of conic arises here as a section of a cone.
• The §51 universal-quadric-section theorem (general-quadric-surface) is here verified for the cone — every section is a conic, and every conic species is realized, in contrast to the cylinder which only gives ellipses (and degenerate lines).

Figures

Figures 133–134

Figures 135–137

Related pages

• appendix-3-on-sections-of-cylinders-cones-and-spheres
• conical-and-pyramidal-surfaces
• oblique-plane-section-method
• general-quadric-surface
• classification-of-conics
• ellipse
• parabola
• hyperbola
• cylinder-sections
• sphere-sections