Diametral Plane

Summary: §§13–14 of the Appendix on Surfaces. The 3D analogue of the diameter-of-conic: a plane that bisects the solid into two similar and equal parts. Read off directly from the parity of an exponent in the surface equation: if the variable $z$ measuring distance to the plane $APQ$ has only even exponents, then $APQ$ is a diametral plane. The sphere has all three coordinate planes diametral; this is the surface analogue of the 1D-parity diagnostic in diameter-and-center-from-equation.

Sources: appendix1, §§13–14.

Last updated: 2026-05-12.

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§13 — The parity criterion

If in the equation in $x,y,z$ the variable $z$ — which measures the perpendicular distance to the plane $APQ$ — has only even exponents, then $z$ always has two equal values, one positive and one negative. The surface therefore has matching parts on either side of $APQ$, so the plane bisects the bounded solid into two similar and equal halves (source: appendix1, §13).

Just as when a plane curve is divided into two similar and equal parts by a straight line, that line is called a diameter, so in the solid case, that plane which divides the solid into two similar parts, is called a diametral plane. (source: appendix1, §13)

The same parity argument run on $y$ shows $APq$ is diametral; run on $x$ shows $Ap\xi$ is diametral. The three diagnostics are independent: zero, one, two, or all three of the coordinate planes can be diametral.

§14 — The sphere

For the sphere of radius $a$ centered at $A$,

$$AM=\sqrt{x^2+y^2+z^2}=a,x^2+y^2+z^2=a^2.$$

All three variables appear with even exponents only — so all three coordinate planes are diametral. This matches the elementary observation that any plane through the center of a sphere bisects it (source: appendix1, §14).

Generalizations from later sections

• §19 strengthens “even-exponent” to a region-counting statement: even $z$ means each of the eight octants has a congruent partner across $APQ$. The full table is in octants-and-region-symmetries (§§19–25).
• Joint-degree (parity of total degree) conditions in octants-and-region-symmetries capture rotational symmetries that no single-variable parity test detects — the surface analogue of concurrence-of-diameters for plane curves with several diameters.
• A center of the surface would be the analogue of center-of-conic / diameter-and-center-from-equation: a point at which all diametral planes meet. The Appendix does not develop this systematically, but the §22–23 condition that the joint degree of all three variables be everywhere even (or everywhere odd) is precisely the central-symmetry test, the surface counterpart of the §340 “all-even or all-odd homogeneous parts” criterion in 2D.

Cross-references

• Plane-curve analogue: diameter-of-conic (Chapter 5) and especially the parity calculus in diameter-and-center-from-equation (Chapter 15).
• Region counting and centred symmetries: octants-and-region-symmetries (§§19–25).
• For a worked example beyond the sphere: any quadric in general-quadric-surface (§51) with $\delta =ϵ=\zeta =0$ and $a=b=c=0$ has all three coordinate planes diametral.

Related pages

• appendix-1-on-the-surfaces-of-solids
• surface-coordinates-and-equation
• octants-and-region-symmetries
• diameter-of-conic
• diameter-and-center-from-equation