Image Points and image lines
Suppose an image point on the normalized image plane has coordinates as (in pixels or meters). Then its homogeneous coordinate is
Note the third coordinate is . No need to explain.
Given an image point , what is the line passing through this point? Let’s consider the line equation in a 2D plane. For point , the line equation is:
Write , then the above equation is
So given an image point with homogeneous coordinates, we can use to denote the line.Then we can say is a line on the image plane. Note
1. there are infinite number of lines pass through one point. This is obviously.
2. Once is determined, the line is determine.
3. and are the same line.
- Is a point on a line?
Given an image point and an image line , it is clear that the point locates on the line if and only if
- Point determined by two lines
Let and be two image lines. What is the intersection points of these two lines? It is easy to derive. If the intersection point is , then and . So
denotes equality up to a scalar factor. Note the third element of should be normalized to 1.
- Line determined by two points
Given two image points and , the line pass the two points is
Note here is = instead of ~ since we need not to normalize.
- Distance between a line and a point
Given an image point and an image line , what is the distance between the line and point? I guess the distance is
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 , then the distance between a point and this line is . 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!!!