Chapter 3: The Algebraic Description of Conic Sections

From cones to algebra

Apollonius of ancient Greece (3rd century BC) treated the circle, ellipse, hyperbola, and parabola uniformly as sections of a cone cut by a plane. Changing the tilt of the cut yields different curves, as follows:

• Parallel to the base → circle
• Tilted, cutting both sides → ellipse
• Parallel to a generator line → parabola
• Parallel to the axis (cutting both nappes of the cone) → hyperbola (with two branches)

Circle — cut perpendicular to the axis

Ellipse — cut at an angle gentler than a generator

Parabola — cut parallel to a generator

Hyperbola — cut parallel to the axis (both sides)

After Descartes introduced the coordinate plane in the 17th century, it became clear that all of these can be written as the zero set of a two-variable second-degree polynomial:

$$ax^2 + bxy + cy^2 + dx + ey + g = 0$$

That is, the correspondence "a geometric section of a cone" = "an algebraic second-degree curve" is the first great achievement of algebraic geometry.

Canonical form

By a suitable coordinate transformation (translation + rotation), a second-degree curve can be rewritten in a concise form called its canonical form. The result is one of the following three kinds (apart from degenerate cases):

Ellipse (centered at the origin, axes along the coordinate axes)

$$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \quad (a, b > 0)$$

If $a = b$ it is a circle. If $a > b$ it is an ellipse with major axis along the $x$-axis and minor axis along the $y$-axis.

Parabola

$$y^2 = 4px \quad (p > 0)$$

Or the form $y = ax^2$ (this just swaps the roles of $x$ and $y$). A symmetric figure defined by one focus and one directrix.

Hyperbola

$$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \quad (a, b > 0)$$

It has two branches. It approaches the asymptotes $y = \pm \dfrac{b}{a} x$.

Classification by the discriminant

Which of ellipse, hyperbola, or parabola the general form $ax^2 + bxy + cy^2 + dx + ey + g = 0$ corresponds to is determined by the sign of the discriminant

$$\Delta = b^2 - 4ac$$

as follows:

This naming is the same as the classification of second-order linear PDEs (elliptic, parabolic, hyperbolic), using the same discriminant. This is no coincidence: it comes from the fact that the characteristic equation of a PDE is a second-degree curve.

Worked checks

(1) $x^2 + y^2 = 1$ (circle): $a = c = 1, b = 0$, $\Delta = 0 - 4 = -4 < 0$ → elliptic ✓

(2) $y - x^2 = 0$ (parabola $y = x^2$): the second-degree term is only $-x^2$, so $a = -1, b = c = 0$, $\Delta = 0 - 0 = 0$ → parabolic ✓

(3) $x^2 - y^2 = 1$ (hyperbola): $a = 1, c = -1, b = 0$, $\Delta = 0 + 4 = 4 > 0$ → hyperbolic ✓

(4) $xy = 1$ (rectangular hyperbola): $a = c = 0, b = 1$, $\Delta = 1 > 0$ → hyperbolic ✓

Degenerate conics

The "zero set of a second-degree equation" is not necessarily an ellipse, hyperbola, or parabola in the ordinary sense. Degenerate forms:

• Union of two lines: $x^2 - y^2 = 0$ ⇔ $(x - y)(x + y) = 0$ → the union of the two lines $y = x, y = -x$. The discriminant is $\Delta = 0 - 0 = 0$ but it is not a parabola
• A single (double) line: $x^2 = 0$ ⇔ the $y$-axis drawn "doubly"
• A single point: $x^2 + y^2 = 0$ → the only real solution is $(0, 0)$. The discriminant is $\Delta = -4 < 0$ ("elliptic"), but it is a single point as a limit of an ellipse
• The empty set: $x^2 + y^2 + 1 = 0$ → no real solutions

In the context of classical algebraic geometry, these degenerate forms arise naturally as "limiting cases of ellipses, hyperbolas, and parabolas". Moreover, passing to projective geometry changes how they look; for example, the difference between "two parallel lines" and "two intersecting lines" disappears. Details are covered at the Intermediate level.

Five points determine a conic

A second-degree curve $ax^2 + bxy + cy^2 + dx + ey + g = 0$ has six coefficients $a, b, c, d, e, g$, but scaling everything by a constant gives the same curve, so the number of degrees of freedom is 5. Therefore, specifying five points in general position determines (in general) a unique conic through them.

This reduces to a problem in linear algebra: the condition of passing through five points $(x_i, y_i)$ ($i = 1, \ldots, 5$) is the system of five linear equations in the coefficients

$$a x_i^2 + b x_i y_i + c y_i^2 + d x_i + e y_i + g = 0 \quad (i = 1, \ldots, 5)$$

Since this is five equations in six unknowns, the solution space is in general one-dimensional (a single solution up to scaling).

This fact that "five points determine a conic" is applied, for example, in Pascal's theorem in projective geometry and in Steiner's construction problems.

Next step

So far we can describe first-degree curves (lines) and second-degree curves (conics) in the language of polynomials. That is the end of the Intro level. At the Intermediate level we proceed to:

• The multivariable case (affine algebraic varieties in three or more dimensions)
• Introduction of projective space and projective varieties
• The correspondence between sets of polynomials and ideals (Hilbert's basis theorem)
• The Zariski topology and irreducible decomposition

The Basic-level article Classical Algebraic Geometry can also be read as a continuation of this article.