Paraboloids and Parabolic Hyperboloid

Summary: §§123–125 of the Appendix on Surfaces. The two paraboloid genera in which one coefficient of the canonical form vanishes. Genus 4 = elliptic paraboloid $A{p}^2+B{q}^2=ar$ (figure 146, both $A,B>0$); sections perpendicular to the axis $AD$ are ellipses, sections containing it are parabolas. Genus 5 = parabolic hyperboloid $A{p}^2-B{q}^2=ar$ (figure 147, opposite signs); first principal section is two intersecting straight lines $Ee,Ff$, the two perpendicular planes through which are asymptotic planes. Species at the boundary include the parabolic cone ($A=B$, round) and the elliptic / hyperbolic cylinder ($a=0$).

Sources: appendix5, §§123–125. Figures 146, 147 in figures145-147.

Last updated: 2026-05-12.

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§123 — Center at infinity from vanishing coefficient

If one coefficient in the canonical form $A{p}^2+B{q}^2+C{r}^2+Gp+Hq+Ir+K=0$ vanishes — say $C=0$ — then the translation step in quadric-canonical-form can no longer kill the $Ir$ term. The general equation becomes

$$A{p}^2+B{q}^2+Gp+Hq+Ir+K=0,$$

and by shifting $p,q$ further:

$$A{p}^2+B{q}^2=ar$$

(absorbing $K,G,H$ into $r$). The center now lies at $r=\infty$.

In either case the center of the surface will lie on the axis $CD$ but at an infinite distance. (source: appendix5, §123)

§124 — Genus 4: elliptic paraboloid (figure 146)

$$A{p}^2+B{q}^2=ar,A,B>0.$$

Principal sections:

• $r=0$: single point $A$ at the origin.
• $q=0$: parabola $MAm$ with axis $AD$, equation $A{p}^2=ar$.
• $p=0$: parabola $NAn$ with axis $AD$, equation $B{q}^2=ar$.
• Section by $r=$ const: ellipse $A{p}^2+B{q}^2=$ const — every level perpendicular to axis $AD$ is an ellipse.

Since every section perpendicular to the axis $AD$ is an ellipse and every section containing this axis is a parabola, the solid of this genus is called an elliptic paraboloid. (source: appendix5, §124)

Species:

• Parabolic cone ($A=B$, “round”): the ellipses become circles. Generated by revolving a parabola about its axis.
• Cylinder ($a=0$): equation degenerates to $A{p}^2+B{q}^2=b^2$ (an ellipse independent of $r$) — an elliptic cylinder. Right if $A=B$, scalene otherwise.

§125 — Genus 5: parabolic hyperboloid (figure 147)

$$A{p}^2-B{q}^2=ar,A,B>0.$$

First principal section ($r=0$): $A{p}^2-B{q}^2=0$, i.e., $p\sqrt{A}=\pm q\sqrt{B}$ — two intersecting straight lines $Ee,Ff$ in figure 147. Each section parallel to the plane $ABC$ (i.e., $r=$ const) is a hyperbola with center on the axis $AD$ and asymptotes parallel to these two lines.

The two perpendicular planes through $Ee$ and $Ff$ approach the surface at infinity — they are asymptotic planes. The surface has two asymptotic planes rather than an asymptotic cone — this is the genus-5 signature.

The surfaces belonging to this genus are called parabolic hyperboloids with two planes for an asymptote. (source: appendix5, §125)

The other two principal sections ($p=0$ and $q=0$) are parabolas as in genus 4.

Species:

• Hyperbolic cylinder ($a=0$): equation becomes $A{p}^2-B{q}^2=b^2$ — sections perpendicular to $AD$ are congruent hyperbolas.
• Two asymptotic planes ($a=b=0$): the equation reduces to $A{p}^2=B{q}^2$, i.e., $p\sqrt{A}=\pm q\sqrt{B}$ — two planes intersecting in the $r$-axis.

Modern note: the parabolic hyperboloid as a saddle

The parabolic hyperboloid $z=(x^2-y^2)/c$ is the modern “saddle surface” — every horizontal section is a hyperbola, every vertical section is a parabola. The two intersecting lines at $z=0$ are the two straight rulings through the saddle point. Euler’s genus 5 is the first description of this surface in the analytic-geometry literature.

Cross-references

• The asymptotic-cone-degenerates-to-planes case: the §111–112 intermediate equality. The two planes are real (Genus 5) or complex (Genus 4) according to whether the homogeneous linear factors are real or complex; coincident factors give Genus 6 parabolic-cylinder-quadric.
• 3D analogue of the parabola: where the parabola is the limit of the ellipse with $b\to \infty$, the paraboloid is the limit of the ellipsoid as one axis goes to infinity. Where the hyperbola has asymptotes-of-hyperbola, the parabolic hyperboloid has asymptotic planes.
• The cylinder species at $a=0$ are pure-cylindrical-and-prismatic-surfaces examples in Genus 4 (elliptic) and Genus 5 (hyperbolic).

Figures

Figures 145–147

Related pages

• appendix-5-on-second-order-surfaces
• asymptotic-cone
• quadric-canonical-form
• parabolic-cylinder-quadric
• parabola
• asymptotes-of-hyperbola
• cylindrical-and-prismatic-surfaces
• surfaces-of-revolution
• elliptic-and-hyperbolic-hyperboloids