triangular-section-solids

Triangular-Section Solids (Wedge-Cone, Affine Sections)

Summary: §§40–44 of the Appendix on Surfaces. Three more genera characterized by the shape of cross-sections perpendicular to a coordinate axis. Wallis’s wedge-cone (§40, figure 124): every section perpendicular to the $x$-axis is a triangle whose vertex slides along a fixed straight line $DT$ parallel to $AP$. Right-triangle sections (§41, figure 125): vertex slides on a curve $AT$. Affine-section solids (§§42–44, figure 126): adjacent sections related by an affine scaling rather than a similarity. Each is to the cone what the next dimension up of “shape constancy” allows.

Sources: appendix2, §§40–44. Figures 124–126 in figures124-127.

Last updated: 2026-05-12.

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§40 — Wallis’s wedge-cone (figure 124)

Setup: every section perpendicular to the $x$-axis is a triangle $PTV$ with vertex $T$ on a fixed straight line $DT$ parallel to the axis $AP$, at perpendicular distance $AD=c$. Let $AVB$ be the base curve in the plane $APQ$, and let $PV$ at abscissa $x$ have length $P(x)$. From the similar triangles $VQM\sim VPT$,

$$\frac{P}{c}=\frac{P-y}{z}⟹z=c-\frac{cy}{P},$$

so the surface equation is

$$\frac{cy}{c-z}=P(x).$$

This solid differs from a cone in that it is defined on the straight edge $DT$, while the cone is defined on a single point. If the base $AVB$ happens to be a circle, then the resulting solid is that which Wallis discussed and called a wedge-cone. (source: appendix2, §40)

The wedge-cone is the cone with its vertex stretched into an edge — the simplest non-conical, non-cylindrical, axis-aligned solid.

§41 — Right-triangle sections on a curved guide (figure 125)

Generalization: the vertex now slides on a curve $AT$ in the plane perpendicular to $AB$, with the section right angle at $P$. Let $PT$ at abscissa $x$ be $Q(x)$ and $PV$ be $P(x)$. Similar triangles give $z=Q-Qy/P$, or

$$Pz+Qy=PQ,\frac{z}{Q}+\frac{y}{P}=1.$$

When $y$ and $z$ each appear to first power only, the solid belongs to this genus (source: appendix2, §41).

§§42–44 — Affine-section solids (figure 126)

Cross-sections need not be similar: it suffices that they be affine to one another (corresponding abscissas have proportional ordinates — the affine-curves relation, lifted to one dimension up).

Setup with three principal sections $ABC,ACD,ABD$. Take $AC=a$ (base), $AD=b$ (altitude). Let the principal section $ACD$ have ordinate equation $q=q(p)$. For a parallel section $PTV$ at $AP=x$, let the base length be $P(x)$ and altitude $Q(x)$. Affinity of $PTV$ to $ACD$ gives

$$\frac{a}{p}=\frac{P}{y},\frac{b}{q}=\frac{Q}{z}⟹y=\frac{Pp}{a},z=\frac{Qq}{b}.$$

Substitute $p=ay/P$, $q=bz/Q$ into the relation $q=q(p)$ to get the surface equation in $x,y,z$.

The §44 special case: if $y$ and $z$ have the same exponent in every term of the surface equation, then sections perpendicular to $AP$ are all straight lines — and the surface has two-fold symmetry across the planes $APQ$ and $APR$ in the way set out in octants-and-region-symmetries §20.

Where this genus sits

Genus	Cross-section shape	Equation hallmark
Cylindrical/prismatic	All congruent	one variable absent
Conical/pyramidal	All similar, scaling linearly	homogeneous in $(x,y,z)$
$Z(z)$-generalization	All similar, scaling by $Z(z)$	homogeneous in $(x,y,Z(z))$
Turned (revolution)	All circles, axis-aligned	$Z(z{)}^2=x^2+y^2$
Wedge-cone	All triangles, vertex on a straight line	$z=c-cy/P(x)$
Right-triangle	All right triangles, vertex on a curve	$Pz+Qy=PQ$
Affine-section	All affine to a fixed plane curve	$y=Pp/a$, $z=Qq/b$

Cross-references

• The affine-section construction is the direct lift of affine-curves (Chapter 18 §§442–446) to one higher dimension. There two coordinates were independently scaled; here the cross-section curves are independently scaled with $x$.
• Wallis appears in the historical note attached to the wedge-cone (§40); his name is the only one Euler attributes to a specific surface in this Appendix.
• The Apollonian conic-section trichotomy emerges from the cone genus, not from these triangle-section genera, see cone-sections.

Figures

Figures 124–127

Related pages

• appendix-2-on-the-intersection-of-a-surface-and-an-arbitrary-plane
• conical-and-pyramidal-surfaces
• surfaces-of-revolution
• affine-curves
• octants-and-region-symmetries