• Points

Suppose an image point on the normalized image plane has coordinates as $(x,y)$ (in pixels or meters). Then its homogeneous coordinate is

$m=[x,y,1]^T$

Note the third coordinate is $1$. No need to explain.

• Lines

Given an image point $m=[x,y,1]^T$, what is the line passing through this point? Let’s consider the line equation in a 2D plane. For point $(x,y)$, the line equation is:

$ax+by+c=0$

Write $l=[a,b,c]^T$, then the above equation is

$l^Tm=0$

So given an image point $m$ with homogeneous coordinates, we can use $l$ to denote the line.Then we can say $l$ is a line on the image plane. Note
1. there are infinite number of lines pass through one point. This is obviously.
2. Once $a,b,c$ is determined, the line is determine.
3. $l$ and $kl$ are the same line.

• Is a point on a line?

Given an image point $m$ and an image line $l$, it is clear that the point locates on the line if and only if

$l^T m=0$

• Point determined by two lines

Let $l_1$ and $l_2$ be two image lines. What is the intersection points of these two lines? It is easy to derive. If the intersection point is $m$, then $l_1^Tm=0$ and $l_2^Tm=0$. So

$m\sim l_1\times l_2$

$\sim$ denotes equality up to a scalar factor. Note the third element of $l_1\times l_2$ should be normalized to 1.

• Line determined by two points

Given two image points $m_1$ and $m_2$, the line pass the two points is

$l=m_1 \times m_2$

Note here is = instead of ~ since we need not to normalize.

• Distance between a line and a point

Given an image point $m$ and an image line $l$, what is the distance between the line and point? I guess the distance is

$d=\frac{l^T m}{l^Tl}$

When the point is on the line, the above equation d=0; whether it is correct when point is not on the line. I have not proved yet. Edit: Wrong! Consider a line in 2D $ax+by+c=0$, then the distance between a point $(x_0,y_0)$ and this line is $d= \frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$. Obviously this is not the same as the above equation.

Finally, it is worthwhile to note all the points and lines discussed here are limited to the normalized image plane!!!