Axial-Section Solids

Summary: §§45–46 of the Appendix on Surfaces. A surface for which every section containing the axis $AP$ is a straight line has equation $z=T+S$, where $T$ depends on $y$ and $z$ to the first power only and $S$ is independent of $y,z$. Cylinders and cones are both special cases (figure 127). §46 gives the general formula for the section through the axis at angle $\varphi$: substitute $y=v\cos \varphi$, $z=v\sin \varphi$ to obtain a curve in the meridian plane.

Sources: appendix2, §§45–46. Figure 127 in figures124-127.

Last updated: 2026-05-12.

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§45 — Equation of an axial-section solid (figure 127)

In figure 127 a section $AKMP$ through the axis $AP$ meets the line $PM$ in the plane $APQ$ at angle $MPV=\varphi$. With $AP=x,PQ=y,QM=z$, we have

$$\frac{z}{y}=\tan \varphi ⟹PM=\frac{z}{\sin \varphi }.$$

If we require the axial line $KM$ to be straight,

$$\frac{z}{\sin \varphi }=\alpha x+\beta ,$$

where $\alpha ,\beta$ are constants depending only on the angle $\varphi$ — that is, functions of $y,z$ but not of $x$. Let $R,S$ be functions of $y$ or $z$ such that $z=Rx+S$ — but with $R$ depending on $y$ or $z$ only to the first power. Generalizing, all surfaces of this kind satisfy

$$z=T+S,$$

where $T$ is linear in $y,z$ and $S$ is independent of $y,z$ (source: appendix2, §45).

Cylinders ($T=0$, $S$ arbitrary in $x$) and cones (homogeneous case, $S=0$, $T$ linear) are both instances.

§46 — Section through the axis at angle $\varphi$

For any surface (not just axial-section ones), the section by the plane $AKMP$ containing the axis $AP$ and making angle $\varphi$ with the base plane $APQ$ has a standard parametrization. Let $PM=v$ be the ordinate in that section. Then

$$QM=z=v\sin \varphi ,PQ=y=v\cos \varphi .$$

Substitute these into the surface equation in $x,y,z$ to obtain a section equation in $x$ and $v$ — the equation of the meridian curve in the axial plane.

By the §10–§11 coordinate-permutation symmetry, the same formula applies (with relabelling) to sections through each of the other two principal axes $AQ,AR$.

Application: cone and cylinder meridian profiles

For the right circular cone $z^2=x^2+y^2$, substitute $z=v\sin \varphi$, $y=v\cos \varphi$:

$$v^2{\sin }^2\varphi =x^2+v^2{\cos }^2\varphi ⟹x^2=v^2({\sin }^2\varphi -{\cos }^2\varphi )=-v^2\cos 2\varphi .$$

The axial sections are pairs of straight lines through the origin: $x=\pm v\sqrt{-\cos 2\varphi }$ when $\varphi >\pi /4$, axis itself when $\varphi =\pi /4$, empty when $\varphi <\pi /4$. (The cone’s interior is $\varphi >\pi /4$, exterior $\varphi <\pi /4$.)

For the right cylinder $x^2+y^2=a^2$, substitute $y=v\cos \varphi$:

$$x^2+v^2{\cos }^2\varphi =a^2⟹x^2=a^2-v^2{\cos }^2\varphi ,$$

a pair of parallel straight lines or empty according to whether $|v\cos \varphi |\le a$.

Both confirm the §45 characterization.

Cross-references

• The §46 substitution is a special case of the general oblique-plane section formula in oblique-plane-section-method (§§47–50), restricted to planes through a fixed axis.
• The axial-line condition $z=T+S$ generalizes both cylindrical-and-prismatic-surfaces (where $T=0$) and the right-cone (where $S=0$ and the equation is homogeneous, see conical-and-pyramidal-surfaces).

Figures

Figures 124–127

Related pages

• appendix-2-on-the-intersection-of-a-surface-and-an-arbitrary-plane
• conical-and-pyramidal-surfaces
• cylindrical-and-prismatic-surfaces
• oblique-plane-section-method