Theory

Conics and quadrics

How to write the matrix of a conic or a quadric, how to classify it with the invariants, and how to reduce it to canonical form with a rotation and a translation, using eigenvalues and eigenvectors.

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Ellipse, hyperbola and parabola
The matrix associated with a conic

Ellipse, hyperbola and parabola

Conic

A conic is the set of points $(x, y)$ of the plane that satisfy a second-degree equation

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

with real coefficients and $a_{11}$ , $a_{12}$ , $a_{22}$ not all zero. The factors $2$ in front of the mixed and linear terms make it possible to write the associated matrix without fractions. The aim of this topic is to recognize from the equation which curve one is dealing with and where it lies, bringing it to one of the following canonical forms by a change of coordinates.

The canonical forms

Non-degenerate conics, in a suitable reference frame, have one of the following equations, already known from high-school analytic geometry.

Ellipse: $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ with $a \geq b > 0$ ; it is the locus of points whose sum of distances from the foci $(\pm c, 0)$ equals $2 a$ , where $c^{2} = a^{2} - b^{2}$ . The numbers $a$ and $b$ are the semi-axes; for $a = b$ one obtains the circle.
Hyperbola: $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ with $a, b > 0$ ; it is the locus of points whose difference of distances from the foci $(\pm c, 0)$ equals $2 a$ in absolute value, where $c^{2} = a^{2} + b^{2}$ . The asymptotes are the lines $y = \pm \frac{b}{a} x$ .
Parabola: $y^{2} = 2 p x$ with $p > 0$ ; it is the locus of points equidistant from the focus $(\frac{p}{2}, 0)$ and from the directrix $x = - \frac{p}{2}$ .

In canonical form the coordinate axes are axes of symmetry of the curve and, for the ellipse and the hyperbola, the origin is a center of symmetry.

The ellipse

\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1

, with semi-axes

3

and

2

and foci

(\pm 5, 0)

The hyperbola

\frac{x ^{2}}{4} - y^{2} = 1

, with foci

(\pm 5, 0)

and asymptotes

y = \pm \frac{x}{2}

(dashed)

The parabola

y^{2} = 4 x

, that is,

p = 2

: focus

(1, 0)

and directrix

x = - 1

(dashed)

The matrix associated with a conic

Matrix associated with a conic

The associated matrix of the conic with equation $a_{11} x^{2} + 2 a_{12} x y + a_{22} y^{2} + 2 a_{13} x + 2 a_{23} y + a_{33} = 0$ is the symmetric matrix of order $3$

A = a_{11} a_{12} a_{13} a_{12} a_{22} a_{23} a_{13} a_{23} a_{33}, A_{0} = (a_{11} a_{12} a_{12} a_{22}),

where $A_{0}$ is the upper-left $2 \times 2$ block, which contains only the second-degree terms. With the row-by-column product the equation of the conic is written

(x y 1) A x y 1 = 0,

and this notation makes it possible to treat changes of coordinates as products of matrices. The matrix contains half the coefficients of $x y$ , of $x$ and of $y$ .

Example

For the conic $2 x^{2} - 4 x y + 5 y^{2} + 2 x - 6 y + 1 = 0$ we have $a_{12} = - 2$ , $a_{13} = 1$ , $a_{23} = - 3$ , hence

A = 2 - 2 1 - 2 5 - 3 1 - 3 1, A_{0} = (2 - 2 - 2 5) .

Expanding the product $(x y 1) A (x y 1)^{T}$ returns the original equation, and this is the quickest check on the matrix written.

The theory continues with 8 more sections (rigid motions and invariants, reduction to canonical form, classification of conics…), plus all worked examples, the solutions and the PDF.

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