Conics

The conic is an algebraic curve that is the locus of points obtainable as the intersection between a circular cone and a plane. The equation of a conic written in implicit form is

$$a x^2 + b xy + c y^2 + d x + e y + f = 0$$ (1.56)

It is worth noting that the parameters of the conic are known up to a multiplicative factor.

Equation (1.56) shows the equation of the conic written in traditional, non-homogeneous Cartesian coordinates. The use of homogeneous coordinates allows the writing of quadratic equations in matrix form.

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

$$a x^2_1 + b x_1 x_2 + c x^2_2 + d x_1 x_3 + e x_2 x_3 + f x^2_3 = 0$$ (1.57)

In this way, it is possible to represent equation (1.56) in matrix form

$$\mathbf{x}^{\top} \mathbf{C} \mathbf{x}=0$$ (1.58)

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

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

Writing the conic in equation (1.58) takes the form of a curve defined by 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 $\mathbf{C}$, in terms of lines this time: a tangent line $\mathbf{l}$ to the conic $\mathbf{C}$ satisfies $\mathbf{l}^{\top} \mathbf{C}^{*} \mathbf{l} = 0$.

In section 3.6.7, techniques for estimating the parameters that encode a conic given the points will be presented.