surfaces-of-revolution

Surfaces of Revolution (Turned Solids)

Summary: §39 of the Appendix on Surfaces. The genus produced when sections perpendicular to one axis are concentric circles whose centers lie on that axis. Algebraic test: the equation has the form

$$Z(z{)}^2=x^2+y^2,$$

i.e. $x$ and $y$ enter only through $x^2+y^2$ — invariance under rotation about the $z$-axis. Euler calls these turned solids (the lathe analogy). Right circular cone $Z^2=z^2$, cylinder $Z^2=a^2$, sphere $Z^2=a^2-z^2$ are all in this family.

Sources: appendix2, §39.

Last updated: 2026-05-12.

---

§39 — The lathe construction and equation

Among the solids whose sections by planes parallel to one of the principal planes are all similar, those most worthy of notice are those in which these sections are all circles whose centers lie on the same straight line $AR$, perpendicular to the plane $APQ$. These solids are turned on a lathe, and so they are called turned. (source: appendix2, §39)

For solids of this kind the general equation is

$$Z^2=x^2+y^2,$$

where $Z=Z(z)$ is some function of $z$ alone. Substituting $z=h$ gives $Z=H$ and the section equation $H^2=x^2+y^2$ — a circle of radius $H$ centered on the $z$-axis $AR$.

Three canonical instances

$Z$	Surface	Equation
$Z^2=z^2$	right circular cone	$z^2=x^2+y^2$
$Z^2=a^2$	right circular cylinder	$a^2=x^2+y^2$
$Z^2=a^2-z^2$	sphere of radius $a$	$x^2+y^2+z^2=a^2$

(Source: appendix2, §39.)

The cylinder case has $z$ absent — and is also a cylindrical-and-prismatic-surfaces instance with circular base. The cone case is homogeneous of degree 2 — also a conical-and-pyramidal-surfaces instance. The sphere is in neither of those genera but is in this one — and gives the cleanest illustration of why “turned” is its own genus.

Geometric content

A surface is turned iff it is invariant under all rotations about a fixed axis (here, the $z$-axis). The function $Z(z)$ is the profile of the surface — the curve in any meridian plane (any plane containing the axis). The surface is generated by rotating the profile $r=Z(z)$ around the axis.

This is the cleanest case of the §§37–38 unified $Z(z)$-generalization: there homogeneity in $(x,y,Z)$ allowed any base shape; here circular base shape is enforced and $Z(z)$ is unconstrained.

Where the rest of the appendix uses this

• §82 derives the sphere section $\sqrt{a^2-f^2{\sin }^2\varphi }$ by direct substitution into $Z^2=a^2-z^2$, see sphere-sections.
• §83 confirms that every plane section of a sphere is a circle — the strongest instance of the turned-solid symmetry.

Cross-references

• Algebraic intersection: turned solids that are also cylinders are right circular cylinders; turned solids that are also cones are right circular cones. The intersection of all three genera is a single equation $z^2=x^2+y^2$.
• A turned solid is a 3D analogue of a diameter-and-center-from-equation curve with infinitely many diameters — i.e. a plane curve invariant under all rotations about a point. By §346 in concurrence-of-diameters, the only such plane curve is the circle. By the lifted argument here, the only turned solid invariant under arbitrary axis is the sphere.

Related pages

• appendix-2-on-the-intersection-of-a-surface-and-an-arbitrary-plane
• cylindrical-and-prismatic-surfaces
• conical-and-pyramidal-surfaces
• sphere-sections
• concurrence-of-diameters