Conic (Proper & Degenerated Conic)

General Point conics

can be represented by 6 parameters,

it can also represented with matrix form like this:

$\begin{pmatrix} a &b/2 &d/2 \\ b/2 &c &e/2 \\ d/2 &e/2 &f \end{pmatrix}$

The relationship between point and conic is like this:

$\\$

Dual conic $C^*$:

composed of a bunch of lines which is a tangent line.

Characteristics of conic

• Degenerate conic: a rank of the matrix C is lower than 3

(Point) Conic: it can be represented by the line.

two lines(rank: 2), one line(rank: 1)

Degenerated (point) conic
Dual conic: can be represented by the point.

two points(rank: 2), one point(rank: 1)