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Parabolic Cylinder (Genus 6)

Summary: §126 of the Appendix on Surfaces. Genus 6 — the final and most degenerate second-order surface. Equation : sections perpendicular to the axis are congruent parabolas, all with vertex on the line and axes parallel to one another. Quasi-species degenerates to two parallel planes — analogous to how two parallel lines arise as a degenerate parabola in conic classification. Corresponds to the borderline case in bounded-quadric-criterion where the asymptotic cone factors into a double linear factor.

Sources: appendix5, §126.

Last updated: 2026-05-12.

§126 — The equation

When two coefficients in the canonical form vanish (so the surface has effectively one quadratic term):

This is a parabolic cylinder: every cross-section perpendicular to the axis is a parabola congruent to every other cross-section, with vertex on and axis parallel to the -direction.

Which gives a parabolic cylinder, all of whose sections perpendicular to the axis are congruent parabolas, each with its vertex on the straight line and with axes parallel to each other. (source: appendix5, §126)

The surface extends infinitely along the -direction without varying — a cylinder in the sense of cylindrical-and-prismatic-surfaces with a parabola as its base curve.

Quasi-species: two parallel planes

Setting in gives — i.e., , two parallel planes equidistant from the -axis.

If , so that , this equation gives two parallel planes, which constitutes a quasi species of this genus. We have here an analogy to the second order curves, where we saw that two intersecting straight lines are a species of hyperbola, while two parallel lines are a species of parabola. (source: appendix5, §126)

This is parallel to how:

  • Two intersecting lines degenerate hyperbola (the asymptotic cone of a hyperbola). Lift: two intersecting planes degenerate parabolic hyperboloid (genus 5 at , see paraboloids-and-parabolic-hyperboloid).
  • Two parallel lines degenerate parabola. Lift: two parallel planes degenerate parabolic cylinder (genus 6 at ).

Six genera complete

We can reduce all second order surfaces to these six genera, so that there is no second order surface which does not belong to one of these six genera. (source: appendix5, §126)

Euler’s six-fold classification is complete. The breakdown:

  1. Ellipsoid (bounded)
  2. Elliptic hyperboloid (circumscribes real cone)
  3. Hyperbolic hyperboloid (inscribed in real cone)
  4. Elliptic paraboloid (two complex asymptotic planes)
  5. Parabolic hyperboloid (two real asymptotic planes)
  6. Parabolic cylinder (double asymptotic plane)

Cross-references

  • The borderline of bounded-quadric-criterion §112: when the asymptotic-cone factorization has equal linear factors. The “cone” collapses entirely to a single double plane.
  • 3D lift of two-parallel-lines-as-degenerate-parabola — the conic analogue in classification-of-conics.
  • Genus 6 is a cylindrical-and-prismatic-surfaces with a parabolic profile — the most degenerate second-order surface, equivalent to a 2D parabola extruded along a third axis.

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