Line in Polar Coordinates

The lines presented so far tend to have an oversized representation concerning the degrees of freedom. The line in the plane $\mathbb{R}^2$ indeed has only 2 degrees of freedom, while the line expressed in implicit form depends on as many as 3 parameters, known up to a multiplicative factor and without a clearly visible geometric meaning. On the other hand, the explicit equation of the line with two parameters $y=mx + q$ exhibits the singularity of vertical lines.

Figure 1.4: Line expressed in polar coordinates.

A solution to the problem is to change the parametrization and exploit polar coordinates. Using polar coordinates, it is possible to express a line in a two-dimensional space without singularities and using only 2 parameters:

$$x \cos \theta + y \sin \theta = \rho$$ (1.46)

where $\rho$ is the distance between the line and the point $(0,0)$ and $\theta$ is the angle formed by this distance segment (orthogonal to the line) and the x-axis (figure 1.4). One should compare this representation with that expressed in equation (1.49). Under this formulation, the relationship between these two parameters and the equation of the line becomes non-linear.

This equation is commonly used in the Hough transform for lines (section 3.11) in order to exploit a two-dimensional and bounded parameter space.

With this particular form, the distance between a point in space $(x_i,y_i)$ and the line is written in a very compact way as

$$d = \vert x_i \cos \theta + y_i \sin \theta - \rho \vert$$ (1.47)