sections-by-coordinate-planes

Sections by Coordinate Planes (and Parallels)

Summary: §§26–31 of the Appendix on Surfaces. The first and easiest cuts of a surface $F(x,y,z)=0$: setting one variable to zero gives the section by the corresponding coordinate plane; setting one variable to a constant $h$ gives the section by the parallel plane at distance $h$. Sweeping $h$ over all values traces out the whole surface as a one-parameter family of planar curves. Sections parallel to one coordinate plane are all congruent iff that variable is missing from the equation — the precondition for the cylindrical-and-prismatic-surfaces genus.

Sources: appendix2, §§26–31. Figure 121 in figures121-123.

Last updated: 2026-05-12.

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§§26–27 — Sections by the three principal planes

Just as two lines (curved or straight) meet at points, two surfaces meet along a curve. The intersection of two planes is a straight line (Euclid); of a sphere with a plane, a circle (source: appendix2, §26).

For the three principal planes of figure 121 (source: appendix2, §27):

• $z=0$ in the surface equation gives the curve in plane $APQ$;
• $y=0$ gives the curve in plane $APR$;
• $x=0$ gives the curve in plane $AQR$.

These three sections together capture the surface’s “shadow” against each of the three principal planes.

§28 — The sphere as worked example

For the sphere $x^2+y^2+z^2=a^2$ centered at $A$ with radius $a$, all three principal sections are great circles (source: appendix2, §28):

$$APQ:x^2+y^2=a^2,APR:x^2+z^2=a^2,AQR:y^2+z^2=a^2.$$

§§29–30 — Parallel-plane sections sweep out the surface

If we let $z=h$ instead of $z=0$, the resulting equation in $x,y,h$ is the section by the plane parallel to $APQ$ at perpendicular distance $h$. Letting $h$ run through all positive and negative values produces an infinite family of planar sections that collectively trace out the whole surface (source: appendix2, §30):

$$F(x,y,h)=0\text{(one equation, parameter }h\text{)}.$$

Similarly $y=h$ gives sections parallel to $APR$, and $x=h$ sections parallel to $AQR$.

The same three families of parallel sections give three orthogonal “stacks” of cross-sections — knowing any one stack determines the surface.

§31 — Congruent parallel sections ↔ missing variable

All of the sections by planes parallel to the plane $APQ$ will be congruent (and the same is true if the plane is $APR$ or $AQR$) if the equation in $x$ and $y$ is the same no matter what the value of $h$ might be. This will happen only if the variable $z$, for which $h$ is substituted, does not occur in the equation for the surface. (source: appendix2, §31)

Equivalent dictionary:

• $z$ absent → all sections parallel to $APQ$ congruent;
• $y$ absent → all sections parallel to $APR$ congruent;
• $x$ absent → all sections parallel to $AQR$ congruent.

This is the algebraic gateway to the cylindrical-and-prismatic-surfaces genus of §§32–33: a missing variable means the surface is generated by translating its base curve along the axis of that missing variable.

Cross-references

• The next genera up — sections all similar (not congruent) — characterize the conical-and-pyramidal-surfaces homogeneous case (§§34–38).
• The most general sections — by planes oblique to all three principal planes — require the substitution method of oblique-plane-section-method (§§47–50).
• The §51 universal-quadric-section theorem in general-quadric-surface reduces to the §28 sphere sections when the quadric is a sphere.

Figures

Figures 121–123

Related pages

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