classification-of-conics

Classification of Conics into Genera

Summary: Every second-order line, written about its orthogonal diameter as $y^2=\alpha +\beta x+\gamma x^2$, falls into exactly one of three genera according to the sign of $\gamma$: hyperbola ($\gamma >0$, four infinite branches), ellipse ($\gamma <0$, fully bounded), parabola ($\gamma =0$, the limit case with two infinite branches). The criterion is behavior at infinity: how many branches escape to $|x|=\infty$.

Sources: chapter6 §§131-137

Last updated: 2026-04-26

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Why a classification?

chapter-5-on-second-order-lines developed every property that all conics share — diameter, center, conjugate diameters, tangent rectangles, principal axes, focal polar equation — without distinguishing one species from another. But conics nevertheless “can be quite different from each other” as curves (source: chapter6, §131), so it is convenient to assign them to genera and study what is proper to each.

Euler is careful (§133) to note that not every change in the constants $\alpha ,\beta ,\gamma$ produces a change in genus or species. Translation of the origin shifts $\alpha ,\beta$ without changing the curve’s shape; magnification (as for circles of different radii) changes magnitude but not shape. So the classification must look past these inessential variations.

The set-up

Choose an orthogonal diameter as the axis of abscissas (always possible for a conic). Then because each abscissa $x$ has two ordinates $\pm y$, the equation simplifies to (source: chapter6, §132):

$y^2=\alpha +\beta x+\gamma x^2.$

Three free constants $\alpha ,\beta ,\gamma$ remain.

The criterion: sign of γ

§134 — Hyperbola ($\gamma >0$). As $|x|\to \infty$, the term $\gamma x^2$ dominates and $\alpha +\beta x+\gamma x^2\to +\infty$, so $y$ has two real values (one positive, one negative) growing without bound. This happens at both $x=+\infty$ and $x=-\infty$. Hence four branches escape to infinity. Curves of this kind constitute a single genus, the hyperbolas.

§135 — Ellipse ($\gamma <0$). Now as $|x|\to \infty$, the expression $\alpha +\beta x+\gamma x^2$ becomes negative, so $y$ becomes imaginary. Neither abscissa nor ordinate can be infinite; the whole curve is bounded. This species is the ellipse, with equation $y^2=\alpha +\beta x+\gamma x^2$ where $\gamma$ is negative.

§136 — Parabola ($\gamma =0$). The boundary case. The equation reduces to $y^2=\alpha +\beta x$, lying between hyperbolas and ellipses. With $\beta >0$ (the sign is inessential since one can negate $x$), as $x\to +\infty$ the ordinate grows without bound in both signs — two infinite branches. For $x\to -\infty$ the ordinate is imaginary, so the curve cannot have more than two infinite branches. This is the parabola, $y^2=\alpha +\beta x$.

Summary of the trichotomy

Genus	Sign of $\gamma$	Branches to infinity	Bounded?
Hyperbola	$\gamma >0$	4	no
Parabola	$\gamma =0$	2	no
Ellipse	$\gamma <0$	0	yes

(source: chapter6, §137)

Notable points

• The discriminant in disguise. In the orthogonal-diameter form $y^2=\alpha +\beta x+\gamma x^2$, the sign of $\gamma$ is the discriminant of the species. In the original general equation $\alpha +\beta x+\gamma y+\delta x^2+ϵxy+\zeta y^2=0$, this corresponds to the sign of ${ϵ}^2-4\delta \zeta$ (the modern discriminant), since rotation to principal axes diagonalises the quadratic form. Euler does not state this projectively-invariant version, but his procedure of first reducing to the orthogonal axis (§132) implicitly performs the diagonalisation.
• Why “behavior at infinity” is the right invariant. Translation, scaling, and rotation can change every coefficient of the equation, but they cannot turn a bounded curve into an unbounded one or change the number of escaping branches. Sign-of-$\gamma$ captures exactly the topological feature that survives all coordinate transformations.
• Parabola as boundary. The parabola is not a separate species so much as the unique transitional curve between ellipses and hyperbolas. This perspective is exploited in parabola (§149), where every property of the parabola is read off from the corresponding ellipse property by sending the major semiaxis to infinity.
• Three genera, infinitely many species inside each. Within each genus, the further free parameters generate infinitely many shapes (more or less elongated ellipses, wider or narrower parabolas, hyperbolas with different asymptote angles). The classification is coarse: it captures only the most fundamental difference.

Related pages

• chapter-6-on-the-subdivision-of-second-order-lines-into-genera
• ellipse
• parabola
• hyperbola
• principal-axes-and-foci
• chapter-5-on-second-order-lines