Intersection of two lines

Linear equation 1：
$$a_{0}x+b_{0}y+c_0=0$$
Linear equation 1 Homogeneous representation of ：
$$I_0:(a_0,b_0,c_0)$$
Linear equation 2：
$$a_{1}x+b_{1}y+c_1=0$$
Linear equation 2 Homogeneous representation of ：
$$I_1:(a_1,b_1,c_1)$$
According to the relevant contents of projective geometry , The form of homogeneous coordinates of the intersection of two straight lines is ：
$$p=I_0\times I_1$$
$$p:(b_1\ast c_2-c_1\ast b_2,c_1\ast a_2-a_1\ast c_2,a_1\ast b_2-b_1\ast a_2)$$
Short for ：
$$p:(x_0,y_0,z_0)$$

When $z_0=0$ when , If the lines coincide, there are infinite intersections , Straight lines do not coincide , Then parallel , The intersection is at infinity , In the framework of European geometry , Solution does not exist ;
When $z_0!=0$ when , Intersection point for $(\frac{x_0}{z_0},\frac{y_0}{z_0})$, namely
$$(\frac{b_1\ast c_2-c_1\ast b_2}{a_1\ast b_2-b_1\ast a_2},\frac{c_1\ast a_2-a_1\ast c_2}{a_1\ast b_2-b_1\ast a_2})$$

reference ：
《Multiple View Geometry in Computer Vision》