A line projected on the image space, is defined by the plane formed by the rays intersecting with the image space, and that plane is defined (mathematically) by a normal(perpendicular line) in projective space.
Line on the image is defined by the normal to the plane:
Basically:
The Formula is in the homogeneous coordinate system and is ax+ by + cz = 0, in other words: The normal(abc) and plane formed by all the rays(xyz) compute the line formula and are equal to zero(their dot product abc*xyz=0).
In more detail:
Perpendicular line to a point in projective geometry means perpendicular to the ray that defines that point.
So that means that all the points(all the rays) that are perpendicular to that normal form the line on the image space.
Lines and points are defined as (3X1) or (1X3) vectors in homogeneous coordinate system,
and that is one of properties of the duality which one can replace points to lines in either formulas:
Basically:
1) To find the line in image space that is spanned by two rays, we just do the cross product between the two rays(or points on the projection image).
And the same goes can be also:
2) To find the point of intersection of two lines in image space, we can just do the cross product between the two lines(or lines on the projection image).
In more detail:
1)To find the line in image space that is spanned by two rays:
Two lines (vectors) from a point in projective space form a plane in projective space and meet the image at points p1 and p2 (where p is point or ray of point), their cross product is the normal to the plane which is the line that is perpendicular to the plane.
2) To find the point of intersection of two lines in image space:
Two lines(vectors) in projective space make a plane that doesn’t meet at the image space but their normal meets at the point of intersection between them at the projection image.