Image Formation

2D points

$$\mathbf x=(x,y)\in\textit R^2$$ $$\mathbf x=\begin{bmatrix} x \\ y \end{bmatrix}$$

homogeneous vector $\mathbf{\tilde x}$

$$\mathbf{\tilde x}=(\tilde x,\tilde y,\tilde w)\in\textit P^2$$

2D projective space

$$\textit P^2=\textit R^3-(0,0,0)$$

inhomogeneous vector $\mathbf{\overline x}$

$$\mathbf{\tilde x}=\tilde w(x,y,1)=\tilde w\mathbf{\overline x}$$

2D lines

homogeneous vector

$$\tilde{\mathbf{\textit l}}=(a,b,c)$$

line equation

$$\mathbf{\overline{x}}\cdot\tilde{\mathbf{\textit l}}=ax+by+c=0$$

$$\mathbf{\textit l}=(\hat n_x,\hat n_y,d)=(\hat{\mathbf n},d)with\Vert{\hat{\mathbf n}}\Vert=1$$

Reference Book
Computer Vision: Algorithms and Applications by Richard Szeliski