conic sections

Matrix representation of conic sections

In mathematics, the matrix representation of conic sections is one way of studying a conic section, its axis, vertices, foci, tangents, and the relative position of a given point. We can also study conic sections whose axes aren't parallel to our coordinate system.

Conic sections have the form of a second-degree polynomial:

Q stackrel{mathrm{def}}{=} Ax^2+By^2+Cx+Dy+Exy+F=0. ,

That can be written as:

mathbf{x}^T A_Qmathbf{x}=0

Where mathbf{x} is the vector:

begin{pmatrix} 1 x y end{pmatrix}

And A_Q a matrix:

A_Q = begin{pmatrix} F & C/2 & D/2 C/2 & A & E/2 D/2 & E/2 & B end{pmatrix}.

Classification

Regular and degenerated conic sections can be distinguished based on the determinant of AQ.

If |A_Q| = 0 ,, the conic is degenerate.

If Q isn't degenerate, we can see what type of conic section it is by computing the subdeterminant resulting from removing the first row and the first column of AQ (ie the minor A11).

A_{11} =

begin{pmatrix}

A & E/2

E/2 & B

end{pmatrix}

  • If and only if |A_{11}| < 0 , it is a hyperbola.
  • If and only if |A_{11}| = 0 , it is a parabola.
  • If and only if |A_{11}| > 0 , it is an ellipse.

In the case of an ellipse, we can make a further distinction between an ellipse and a circle by comparing the last two diagonal elements corresponding to x2 and y2.

  • If a_{11} = a_{22}, it is a circle.

If the conic section is degenerate (|A_Q| = 0), |A_{11}| still allows us to distinguish its form:

  • If and only if |A_{11}| < 0 , it is two intersecting lines.
  • If and only if |A_{11}| = 0 , it is two (possibly coincident) parallel straight lines.
  • If and only if |A_{11}| > 0 , it is empty.

Center

We can calculate the center by taking the last two rows of the associated matrix, set them equal to 0 and solve the system.

S stackrel{mathrm{def}}{=} left{ begin{matrix} a_{21} + a_{22}x + a_{23}y & = & 0 a_{31} + a_{32}x + a_{33}y & = & 0 end{matrix} right. stackrel{mathrm{def}}{=} left{begin{matrix}
   C/2  + Ax  + (E/2)y & = & 0 
   D/2  + (E/2)x +  By & = & 0
end{matrix} right.

This becomes,

begin{pmatrix} x_c y_c end{pmatrix} = begin{pmatrix} A & E/2 E/2 & B end{pmatrix}^{-1} begin{pmatrix} -C/2 -D/2 end{pmatrix} = begin{pmatrix} (ED-2BC)/(4AB-E^2) (CE-2AD)/(4AB-E^2) end{pmatrix}

Axes

The major and minor axes are two lines determined by the center of the conic as a point and eigenvectors of the associated matrix as vectors of direction.

a_{1,2} stackrel{mathrm{def}}{=} left{begin{matrix} S(x_0,y_0) &qquad mbox{(center of the conic)} vec u(u_x,u_y) &qquad mbox{(eigenvector of A)} end{matrix} right.

So we can write a canonical equation:

a_{1,2} stackrel{mathrm{def}}{=} frac{x-x_0}{u_x} = frac{y-y_0}{u_y}

Because a 2x2 matrix has 2 eigenvectors, we obtain 2 axes.

Vertices

For a general conic we can determine its vertices by calculating the intersection of the conic and its axes — in other words, by solving the system:

V stackrel{mathrm{def}}{=} left{begin{matrix} & e &qquad mbox{(axis)} & Q &qquad mbox{(the general equation of the conic)} end{matrix} right.

Tangents

Through a given point, P, there are generally two lines tangent to a conic. Expressing P as a column vector, p, the two points of tangency are the intersections of the conic with the line whose equation is

mathbf{p}^T A_Qmathbf{x}=0

When P is on the conic, the line is the tangent there. When P is inside an ellipse, the line is the set of all points whose own associated line passes through P. This line is called the polar of the pole P with respect to the conic.

Just as P uniquely determines its polar line (with respect to a given conic), so each line determines a unique P. This is thus an expression of geometric duality between points and lines in the plane.

As special cases, the center of a conic is the pole of the line at infinity, and each asymptote of a hyperbola is a polar (a tangent) to one of its points at infinity.

Using the theory of poles and polars, the problem of finding the four mutual tangents of two conics reduces to finding the intersection of two conics.

Reduced equation

The reduced equation of a conic section is the equation of a conic section translated and rotated so that its center lies in the center of the coordinate system and its axes are parallel to the coordinate axes. This is equivalent to saying that the coordinates are moved to satisfy these properties. See the figure.

If lambda_1 and lambda_2 are the eigenvalues of the matrix A11, the reduced equation can be written as:

lambda_1 x'^2 + lambda_2 y'^2 + frac
{|A_{11} = 0>

Dividing by -frac

{|A_{11}> we obtain a reduced canonical equation. For example, for an ellipse:

frac{{x'}^2}{a^2} + frac{{y'}^2}{b^2} = 1

From here we get 'a' and 'b'.

The transformation of coordinates is given by:

T: RS(O,X,Y) mapsto (O'=S,X',Y') stackrel{mathrm{def}}{=} left{begin{matrix}
   vec t &=&  vec OO' = S
alpha &=& operatorname{arccos} frac{vec a_1 cdot {1 choose 0}}> end{matrix} right.

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