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

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

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:

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

Solution.

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

let's identify the coefficients \(a_{ij}\):

Calculate the determinant \(\Delta:\)

Now let's find the \(\delta-\)invariant:

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

Solution.

Let's write down the coefficients of this equation:

Calculate the determinant \(\Delta:\)

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

Thus we see that

It can be a real or imaginary ellipse. Since

So, this equation describes a real ellipse.

Solution.

The coefficients in this equation look like this:

Then \(\Delta-\)invariant is equal to

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

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

So we have here the following signs of invariants:

This corresponds to an imaginary ellipse.

Solution.

We have the following coefficients \(a_{ij}:\)

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

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

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

Solution.

It is easy to write coefficients \(a_{ij}:\)

The main determinant \(\Delta\) is equal:

This means that the curve is degenerate.

Find the \(\delta-\)invariant:

Consider the \(A-\)invariant:

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