Conic Section Identifier

What is a Conic Section?

A conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a flat plane. The three major types of conic sections are the ellipse, the parabola, and the hyperbola. The circle is a special case of the ellipse. Historically, conics have been studied since ancient Greek times and play a vital role in physics, astronomy (orbital mechanics), and engineering. For related tools, see the Completing the Square Calculator.

The General Second-Degree Equation

Every conic section can be represented as an implicit equation of the second degree in two variables $x$ and $y$:

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$

Where $A$, $B$, $C$, $D$, $E$, and $F$ are constant coefficients, and at least one of $A$, $B$, or $C$ is non-zero.

How to Classify Conic Sections

Classification relies on two mathematical invariants calculated from the coefficients:

1. The Conic Discriminant

The discriminant is defined as:

$$\text{Discriminant} = B^2 - 4AC$$

This tells us the geometric shape of the conic assuming it is non-degenerate:

• If $B^2 - 4AC < 0$: The curve is an ellipse (or a circle if $A = C$ and $B = 0$).
• If $B^2 - 4AC = 0$: The curve is a parabola.
• If $B^2 - 4AC > 0$: The curve is a hyperbola.

2. The Degeneracy Determinant

To detect if a conic is degenerate (meaning it collapses into intersecting lines, parallel lines, a single point, or no real locus), we evaluate the determinant of the $3 \times 3$ matrix representing the quadratic form:

$$M = \begin{pmatrix} A & B/2 & D/2 \\ B/2 & C & E/2 \\ D/2 & E/2 & F \end{pmatrix}$$

Evaluating this determinant yields the condition:

$$\det(M) = A\left(CF - \frac{E^2}{4}\right) - \frac{B}{2}\left(\frac{BF}{2} - \frac{DE}{4}\right) + \frac{D}{2}\left(\frac{BE}{4} - \frac{CD}{2}\right)$$

Multiplying by 4 to clear fractional divisions gives:

$$4\det(M) = 4ACF - AE^2 - B^2F + BDE - CD^2$$

• If $\det(M) = 0$: The conic is degenerate.
• If $\det(M) \neq 0$: The conic is non-degenerate (a true curve).

Rotation and Center of Conics

If the coefficient $B \neq 0$, the conic section is rotated relative to the standard coordinate axes. The angle of rotation $\theta$ required to align the conic with the coordinate axes is given by:

$$\theta = \frac{1}{2} \operatorname{atan2}(B, A - C)$$

For ellipses and hyperbolas, the center coordinates $(h, k)$ are the solution to the system of linear equations obtained from partial derivatives:

$$h = \frac{2CD - BE}{B^2 - 4AC}, \quad k = \frac{2AE - BD}{B^2 - 4AC}$$

Frequently Asked Questions

What is a degenerate conic?

A degenerate conic is a quadratic equation where the plane intersects the apex of the cone. Rather than forming a curved boundary, it collapses into simpler geometric structures: a single point, two intersecting lines, or two parallel/coincident lines.

How does the xy term affect the shape of the conic?

The $Bxy$ term represents rotation. When $B = 0$, the axes of the ellipse, parabola, or hyperbola are parallel to the $x$ and $y$ axes. When $B \neq 0$, the shape is tilted by an angle $\theta$ relative to the standard axes.

Why is a circle considered a special case of an ellipse?

A circle occurs when the horizontal and vertical semi-axes are equal ($a = b$), which happens when $A = C$ and $B = 0$ in the general quadratic form. Its eccentricity is exactly zero.