Classification of Second Order Curves

General Equation of a Second Order Curve

The general equation of a second-order curve in Cartesian coordinates can be expressed as

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

where A, B, C, D, E, and F are constants such that the coefficients A, B, C are not equal to zero at the same time, and x and y are the variables.

For classification purposes, it is more convenient to rewrite this equation in this form

$a_{11}x^2 + 2a_{12}xy + a_{22}y^2 + 2a_{13}x + 2a_{23}y + a_{33} = 0$

in which at least one of the coefficients a11, a12, a22 is different from zero. Otherwise, if all three first coefficients are equal to zero, this equation becomes linear. The coefficient 2 for some terms is used to simplify a number of formulas.

Invariants of a Second Order Curve

The shape of the curve depends on four quantities also known as invariants:

${\Delta} = \left| {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\ {{a_{12}}}&{{a_{22}}}&{{a_{23}}}\\ {{a_{13}}}&{{a_{23}}}&{{a_{33}}} \end{array}} \right|$
$\delta = \left| {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}\\ {{a_{12}}}&{{a_{22}}} \end{array}} \right| = {a_{11}}{a_{22}} - a_{12}^2$
$S = \text{tr}\left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}\\ {{a_{12}}}&{{a_{22}}} \end{array}} \right] = {a_{11}} + {a_{22}}$
$A = \left| {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{13}}}\\ {{a_{13}}}&{{a_{33}}} \end{array}} \right| + \left| {\begin{array}{*{20}{c}} {{a_{22}}}&{{a_{23}}}\\ {{a_{23}}}&{{a_{33}}} \end{array}} \right|$

The first three quantities Δ, δ, S are called invariants because their values do not change under parallel displacement and rotation of the coordinate system. The last quantity A is preserved only when the coordinate system is rotated, so it is sometimes called semi-invariant.

Depending on the values of these parameters, the second order equation can represent 9 different types of lines.

Non-Degenerate Curves

If $$\Delta \ne 0,$$ then the curve is called non-degenerate. The following cases may occur:

1. If $$\delta \gt 0$$ and $$\Delta \cdot S \lt 0,$$ the curve represents an ellipse. A circle is a special case of an ellipse under the condition $$S^2 = 4\delta,$$ which is equivalent to $$a_{11} = a_{22}, a_{12} = 0.$$
2. If $$\delta \gt 0$$ and $$\Delta \cdot S \gt 0,$$ the curve represents an imaginary ellipse;
3. If $$\delta \lt 0,$$ the curve represents a hyperbola;
4. If $$\delta = 0,$$ the curve represents a parabola.

Ellipse, hyperbola and parabola (cases 1, 3, 4) are often called conic sections because they occur as intersections of planes with cones in space.

Degenerate Curves

If $$\Delta = 0,$$ then the second order curve is called degenerate. There may be the following cases:

1. If $$\delta \gt 0$$, the curve degenerates into a pair of imaginary intersecting straight lines (a real point);
2. If $$\delta \lt 0$$, we get a pair of real intersecting straight lines;
3. If $$\delta = 0$$ and $$A \lt 0$$, we have a pair of real parallel lines;
4. If $$\delta = 0$$ and $$A = 0$$, we have a pair of coincident lines;
5. If $$\delta = 0$$ and $$A \gt 0$$, we get a pair of imaginary parallel lines.

Solved Problems

Example 1.

Determine the type of conic section represented by the equation $${4x^2 - y^2 - 6y + 3 = 0.}$$

Solution.

Assuming that the equation of the curve is written in the form

$a_{11}x^2 + 2a_{12}xy + a_{22}y^2 + 2a_{13}x + 2a_{23}y + a_{33} = 0,$

let's identify the coefficients $$a_{ij}$$:

$a_{11} = 4,\;a_{12} = 0,\;a_{22} = -1,$
$a_{13} = 0,\;a_{23} = -3,\;a_{33} = 3.$

Calculate the determinant $$\Delta:$$

${\Delta} = \left| {\begin{array}{*{20}{r}} 4 & 0 & 0\\ 0 & -1 & -3\\ 0 & -3 & 3 \end{array}} \right| = 4\left| {\begin{array}{*{20}{r}} -1 & -3\\ -3 & 3 \end{array}} \right| = 4\left({-3-9}\right) = -48 \ne 0.$

Now let's find the $$\delta-$$invariant:

${\delta} = \left| {\begin{array}{*{20}{r}} 4 & 0\\ 0 & -1 \end{array}} \right| = -4 \lt 0.$

Since $$\delta \lt 0,$$ this equation represents the graph of a hyperbola.

Example 2.

What type of conic section is represented by the equation $${25x^2 + 9y^2 - 100x + 54y - 44 = 0?}$$

Solution.

Let's write down the coefficients of this equation:

$a_{11} = 25,\;a_{12} = 0,\;a_{22} = 9,$
$a_{13} = -50,\;a_{23} = 27,\;a_{33} = -44.$

Calculate the determinant $$\Delta:$$

${\Delta} = \left| {\begin{array}{*{20}{r}} 25 & 0 & -50\\ 0 & 9 & 27\\ -50 & 27 & -44 \end{array}} \right| = 25\left| {\begin{array}{*{20}{r}} 9 & 27\\ 27 & -44 \end{array}} \right| + \left({-50}\right)\left| {\begin{array}{*{20}{r}} 0 & 9\\ -50 & 27 \end{array}} \right| = 25\left[{9\cdot\left({-44}\right) - 27^2}\right] - 50\left[{0-9\cdot\left({-50}\right)}\right] = 25\cdot\left({-396-729}\right) - 50\cdot 450 = -28125 -22500 = -50625 \ne 0.$

Let's also determine the invariants $$\delta$$ and $$S:$$

${\delta} = \left| {\begin{array}{*{20}{r}} 25 & 0\\ 0 & 9 \end{array}} \right| = 225 \gt 0.$
$S = 25 + 9 = 34 \gt 0.$

Thus we see that

$\Delta \ne 0\;\text{ and }\;\delta \gt 0.$

It can be a real or imaginary ellipse. Since

$\Delta \lt 0\;\text{ and }\;S \gt 0,\;\Rightarrow \Delta\cdot S \lt 0.$

So, this equation describes a real ellipse.

Example 3.

Determine the type of second order curve $${2x^2 + y^2 + 4x - 6y + 12 = 0}.$$

Solution.

The coefficients in this equation look like this:

$a_{11} = 2,\;a_{12} = 0,\;a_{22} = 1,$
$a_{13} = 2,\;a_{23} = -3,\;a_{33} = 12.$

Then $$\Delta-$$invariant is equal to

${\Delta} = \left| {\begin{array}{*{20}{r}} 2 & 0 & 2\\ 0 & 1 & -3\\ 2 & -3 & 12 \end{array}} \right| = 2\left| {\begin{array}{*{20}{r}} 1 & -3\\ -3 & 12 \end{array}} \right| + 2\left| {\begin{array}{*{20}{r}} 0 & 1\\ 2 & -3 \end{array}} \right| = 2\cdot\left({12-9}\right) + 2\cdot\left({0-2}\right) = 6 - 4 = 2 \ne 0.$

Since determinant $$\Delta$$ is not equal to zero, the curve is non-degenerate. Let's calculate the $$\delta-$$invariant:

${\delta} = \left| {\begin{array}{*{20}{r}} 2 & 0\\ 0 & 1 \end{array}} \right| = 2 \gt 0.$

To clarify whether this curve is an imaginary and a real ellipse, we also calculate $$S-$$invariant:

$S = 2 + 1 = 3 \gt 0.$

So we have here the following signs of invariants:

$\Delta \gt 0,\;\delta \gt 0,\;S \gt 0,\;\Rightarrow \Delta\cdot S \gt 0.$

This corresponds to an imaginary ellipse.

Example 4.

Identify the type of second order curve $${3x^2 - 4\sqrt{5}xy + 4y^2 = 0}.$$

Solution.

We have the following coefficients $$a_{ij}:$$

$a_{11} = 3,\;a_{12} = -2\sqrt{5},\;a_{22} = 4,$
$a_{13} = 0,\;a_{23} = 0,\;a_{33} = 0.$

It is obvious that the determinant $$\Delta$$ is equal to zero:

${\Delta} = \left| {\begin{array}{*{20}{r}} 3 & -2\sqrt{5} & 0\\ -2\sqrt{5} & 4 & 0\\ 0 & 0 & 0 \end{array}} \right| = 0,$

that is, we have here a degenerate case. Determine the $$\delta-$$invariant:

${\delta} = \left| {\begin{array}{*{20}{r}} 3 & -2\sqrt{5}\\ -2\sqrt{5} & 4 \end{array}} \right| = 3\cdot 4 - \left({-2\sqrt{5}}\right)^2 = 12 - 20 = -8 \lt 0.$

Hence, this equation represents a pair of real intersecting straight lines.

Example 5.

Determine the type of second order curve $${9x^2 - 6xy + y^2 - 10 = 0}.$$

Solution.

It is easy to write coefficients $$a_{ij}:$$

$a_{11} = 9,\;a_{12} = -3,\;a_{22} = 1,$
$a_{13} = 0,\;a_{23} = 0,\;a_{33} = -10.$

The main determinant $$\Delta$$ is equal:

${\Delta} = \left| {\begin{array}{*{20}{r}} 9 & -3 & 0\\ -3 & 1 & 0\\ 0 & 0 & -10 \end{array}} \right| = -10 \cdot \left| {\begin{array}{*{20}{r}} 9 & -3\\ -3 & 1 \end{array}} \right| = -10 \cdot \left({9-9}\right) = -10 \cdot 0 = 0.$

This means that the curve is degenerate.

Find the $$\delta-$$invariant:

${\delta} = \left| {\begin{array}{*{20}{r}} 9 & -3\\ -3 & 1 \end{array}} \right| = 9\cdot 1 - \left({-3}\right)^2 = 9 - 9 = 0,$

Consider the $$A-$$invariant:

$A = \left| {\begin{array}{*{20}{r}} 9 & 0\\ 0 & -10 \end{array}} \right| + \left| {\begin{array}{*{20}{r}} 1 & 0\\ 0 & -10 \end{array}} \right| = \left({-90 - 0}\right) + \left({-10 - 0}\right) = -100 \lt 0.$

In this case, we get a pair of real parallel straight lines.