Conical and Pyramidal Surfaces

Summary: §§34–38 of the Appendix on Surfaces. The second surface genus, characterized by a homogeneous equation in $x,y,z$. Sections by planes parallel to a principal plane are then all similar and grow proportionally with distance from the origin (figure 123). Curve base → cone (right or scalene), polygonal base → pyramid; vertex always at the origin $A$. The unifying generalization (§§37–38): replace $z$ by an arbitrary single-valued $Z(z)$ and require homogeneity in $x,y,Z$. With $Z=z$ this is the cone; with $Z=(b-z)/b$ the cone has vertex displaced; with $Z=1$ — i.e., $z$ absent — the surface degenerates to a cylindrical-and-prismatic-surfaces one. So cones and cylinders form a single homogeneous-equation family.

Sources: appendix2, §§34–38. Figure 123 in figures121-123.

Last updated: 2026-05-12.

---

§34 — Homogeneous equation ↔ similar sections

The next [genus] most worthy of notice is that which arises from a homogeneous equation in the three variables $x,y$, and $z$. That is, in the equation the joint degree in each term is the same. (source: appendix2, §34)

Example: $z^2=mxz+x^2+y^2$ (each term joint degree 2). Substituting a constant $z=h$ gives

$$h^2=mhx+x^2+y^2,$$

and the same shape rescales as $h$ varies. Sections parallel to $APQ$ are not just similar: they grow proportionally to $h$, so corresponding points lie on straight lines through the origin $A$ (source: appendix2, §34).

§35 — The cone-and-pyramid construction (figure 123)

Pick the section $TSsMm$ at $z=h=AR$, parallel to $APQ$. Pass an indefinite straight line through the origin $A$ and through any point of $TSsMm$. As that point sweeps the section, the line sweeps the surface (source: appendix2, §35):

• $TSsMm$ a circle centered at $R$ → right circular cone;
• $TSsMm$ a non-centered ellipse → scalene cone;
• $TSsMm$ a polygon → pyramid.

Vertex always at $A$. Hence the genus name conical or pyramidal.

§36 — All-direction similarity

The same homogeneity gives similarity-and-proportional-scaling for sections parallel to any of the three coordinate planes — and indeed to any plane through $A$ (anticipated here, fully derived in chapter 3 §§68–80, see cone-sections):

All sections by planes parallel to any plane through the point $A$, will be similar to each other and proportional to the distance of the plane from the vertex $A$. (source: appendix2, §36)

The cone and the pyramid are the two surface genera that have this scaling-with-distance property.

§§37–38 — The unified $Z(z)$ generalization

The conical genus generalizes by replacing $z$ with an arbitrary single-valued function $Z(z)$ and requiring homogeneity in $x,y,Z$. Setting $z=h$ then gives $Z=H=Z(h)$ and the section is similar with scaling proportional to $H$ (not $h$) — so the lines through corresponding section-points are no longer straight but are curves following $H$ as a function of $h$ (source: appendix2, §37).

This $Z(z)$-generalization contains both elementary genera as special cases (source: appendix2, §38):

Choice of $Z(z)$	Surface
$Z=z$	cone with vertex at origin
$Z=\alpha z$	cone with vertex at origin (scalar)
$Z=\alpha +\beta z$	cone with vertex at $(0,0,-\alpha /\beta )$
$Z=(b-z)/b$	cone with vertex at $(0,0,b)$
$Z\to 1$ (limit $b\to \infty$)	$z$ absent → cylinder

So a cylinder is a cone with vertex at infinity — algebraically, the limit of the homogeneous-in-$(x,y,Z)$ family as the $z$-dependence of $Z$ degenerates to a constant.

Worked instance: the canonical scalene cone

In the next chapter (§68, figure 134) Euler studies the canonical scalene cone with vertex at the origin and elliptical base:

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

Each term is joint degree 2 — the §34 homogeneity condition. Sections perpendicular to the axis are ellipses with semiaxes proportional to $z$. The full conic-section trichotomy of oblique cuts (parabola at $\tan \varphi =1/m$, ellipse below, hyperbola above) is developed in cone-sections.

Cross-references

• Cylinder limit $b\to \infty$ recovers cylindrical-and-prismatic-surfaces — both genera are unified §38.
• The scaling-with-distance property (§§35–36) is the surface analogue of similar-curves (Chapter 18) for plane curves with a single thread-degree-homogeneous parameter.
• Worked-out section behaviour for the canonical cone: cone-sections (§§68–80).
• The “cylinder as cone with infinite vertex” trick mirrors the “parabola as ellipse with infinite axis” framing in parabola (Chapter 6).

Figures

Figures 121–123

Figures 133–134

Related pages

• appendix-2-on-the-intersection-of-a-surface-and-an-arbitrary-plane
• cylindrical-and-prismatic-surfaces
• surfaces-of-revolution
• cone-sections
• similar-curves