appendix-2-on-the-intersection-of-a-surface-and-an-arbitrary-plane

Appendix Chapter 2 — On the Intersection of a Surface and an Arbitrary Plane

Summary: Cuts $F(x,y,z)=0$ by every flat. Sections by the three principal planes (set one variable to zero) and by parallel planes (set it to a constant) reveal the surface’s basic geometry. Genera defined by what those sections look like: a missing variable gives cylindrical / prismatic surfaces; a homogeneous equation gives conical / pyramidal surfaces; the unified $Z(z)$ form covers both; $Z^2=x^2+y^2$ gives the turned solids (surfaces of revolution); various triangle-section solids include Wallis’s wedge-cone; an axis-containing-section condition yields $z=T+S$; and the chapter closes with the §51 universal observation that every plane section of a quadric is a conic.

Sources: appendix2, §§26–51. Figures 121–129 in figures121-123, figures124-127, and figures128-130.

Last updated: 2026-05-12.

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§§26–31 — Sections by the three coordinate planes and parallels to them

Setting $z=0$ in the surface equation gives the curve where the surface meets the plane $APQ$; $y=0$ gives the curve in $APR$; $x=0$ gives the curve in $AQR$. For the sphere $x^2+y^2+z^2=a^2$ each of these is a great circle (§28).

Setting $z=h$ instead of $z=0$ gives the section by the plane parallel to $APQ$ at distance $h$. As $h$ runs through all reals, these parallel sections sweep out the whole surface, sharing one parametric equation in $x,y,h$ (§§29–30). They are all congruent iff $z$ is absent from the equation (§31). See sections-by-coordinate-planes.

§§32–33 — Cylindrical and prismatic surfaces

If the variable $z$ is missing, the surface is generated by sliding a vertical line along the base curve $BMD$ in the plane $APQ$ (figure 122). Special cases: circle base → right cylinder, ellipse base → scalene cylinder, polygon base → prism. Either of $x$ or $y$ can play the role; the surface is named cylindrical or prismatic accordingly. See cylindrical-and-prismatic-surfaces.

§§34–38 — Conical, pyramidal, and the unified $Z(z)$ generalization

If the equation is homogeneous in $x,y,z$ (every term of equal joint degree), every section by a plane parallel to a principal plane is similar to every other and grows proportionally with distance from the origin (§34). Geometrically this is a cone (curved base) or pyramid (polygonal base), with vertex at the origin (§35). Circle base + central axis = right circular cone; otherwise scalene cone (figure 123).

The unifying generalization (§§37–38): replace $z$ by $Z(z)$ and ask only that the equation be homogeneous in $x,y,Z$. With $Z=z$ this is the cone; with $Z=\alpha +\beta z$ the cone has vertex displaced from the origin; with $Z=1$ — i.e., $z$ absent — the surface is cylindrical. So the cone-and-cylinder genus is one homogeneous family. See conical-and-pyramidal-surfaces.

§39 — Turned solids (surfaces of revolution)

When an equation has the form $Z(z{)}^2=x^2+y^2$ — sections perpendicular to the $z$-axis are concentric circles whose centers lie on the axis $AR$ — Euler calls the solid turned (lathed). Right cone: $Z^2=z^2$. Cylinder: $Z^2=a^2$. Sphere: $Z^2=a^2-z^2$. See surfaces-of-revolution.

§§40–44 — Triangular-section solids

Several genera defined by what triangle every cross-section looks like:

• Wallis’s wedge-cone (§40, figure 124): every section perpendicular to the $x$-axis is a triangle whose vertex slides along a fixed straight line $DT$ parallel to $AP$. Equation $z=c-cy/P$ with $P$ a function of $x$. Reduces to a cone when the line $DT$ collapses to a point.
• Right-triangle sections (§41, figure 125): vertex slides on a curve $AT$.
• Affine-section solids (§§42–44, figure 126): adjacent sections related by an affine scaling rather than a similarity. See triangular-section-solids.

§§45–46 — Axial-section solids

If every section containing the axis $AP$ is a straight line, the surface satisfies $z=T(y,z)+S$ with $T$ linear in $y,z$ (§45, figure 127). General-axis-section formula (§46): substitute $y=v\cos \varphi$, $z=v\sin \varphi$ to read off the section in plane $AKMP$. See axial-section-solids.

§§47–50 — General oblique section

The master substitution that lets you read off the section by any flat. Two warm-ups (figures 128, 129) build to the full formula:

$$z=v\sin \varphi ,x=t\cos \theta +v\cos \varphi \sin \theta ,y=v\cos \varphi \cos \theta -t\sin \theta -f.$$

Substituting these for $x,y,z$ in the surface equation gives the equation $G(t,v)=0$ of the section in coordinates of the cutting plane. Algebraic surfaces have algebraic sections, of equal or smaller total degree (§50). See oblique-plane-section-method.

§51 — Every section of a quadric is a conic

The quadric

$$\alpha x^2+\beta y^2+\gamma z^2+\delta xy+ϵxz+\zeta yz+ax+by+cz+e^2=0$$

has the property that every plane section is a curve of order at most 2 — i.e., a (possibly degenerate) conic. The §51 theorem closes the chapter and motivates the next: the next chapter studies what those conic sections actually look like for the three classical quadrics. See general-quadric-surface.

Cross-references

• Cylindrical/prismatic and conical/pyramidal surfaces import the homogeneity calculus of similar-curves and affine-curves (Book II Chapter 18) one dimension up.
• §51’s universal-conic-section property is the bridge to chapter 5–6 of Book II (chapter-5-on-second-order-lines, chapter-6-on-the-subdivision-of-second-order-lines-into-genera) and is fully cashed out in appendix-3-on-sections-of-cylinders-cones-and-spheres.

Figures

Figures 121–123

Figures 124–127

Figures 128–130

Related pages

• sections-by-coordinate-planes
• cylindrical-and-prismatic-surfaces
• conical-and-pyramidal-surfaces
• surfaces-of-revolution
• triangular-section-solids
• axial-section-solids
• oblique-plane-section-method
• general-quadric-surface
• appendix-1-on-the-surfaces-of-solids
• appendix-3-on-sections-of-cylinders-cones-and-spheres