Conics

A conic is an algebraic curve consisting of the locus of points obtained as the intersection of a circular cone and a plane. The implicit equation of a conic is

$$a x^2 + b xy + c y^2 + d x + e y + f = 0$$ (1.92)

It should be noted that the parameters of a conic are known only up to a multiplicative factor.

Equation (1.92) shows the equation of the conic written in traditional, nonhomogeneous Cartesian coordinates. The use of homogeneous coordinates allows quadratic equations to be written in matrix form.

If homogeneous coordinates are used instead of Cartesian coordinates, applying the substitutions $x = x_1 / x_3$ and $y = x_2 / x_3$ yields the equation of the conic in homogeneous form:

$$a x^2_1 + b x_1 x_2 + c x^2_2 + d x_1 x_3 + e x_2 x_3 + f x^2_3 = 0$$ (1.93)

In this way, equation (1.92) can be represented in matrix form:

$$\mathbf{x}^{\top} \mathbf{C} \mathbf{x}=0$$ (1.94)

where $\mathbf{C}$ is the symmetric $3 \times 3$ matrix of the parameters and $\mathbf {x}$ is the locus of points (expressed in homogeneous coordinates) of the conic. Since it is expressed through homogeneous ratios, this matrix is defined only up to a multiplicative factor. The conic has 5 degrees of freedom, namely the 6 elements of the symmetric matrix minus the scale factor.

By point-line duality, the line $\mathbf{l}$ tangent to a conic $\mathbf{C}$ at the point $\mathbf {x}$ is simply $\mathbf{l}= \mathbf{C}\mathbf{x}$.

The representation of the conic in equation (1.94) has the form of a curve defined as a locus of points and is therefore also called a point conic, because it defines the equation of the conic using points in space. Using the duality theorem, it is also possible to express a conic $\mathbf{C}^* \propto \mathbf{C}^{-1}$, dual to $\mathbf{C}$, this time in terms of lines: a tangent line $\mathbf{l}$ to the conic $\mathbf{C}$ satisfies $\mathbf{l}^{\top} \mathbf{C}^{*} \mathbf{l} = 0$.

Section 4.6.7 will present techniques for estimating the parameters encoding a conic from given points.