And this leads to the point and line duality in a homogeneous representation. That the point is representing a three dimensional vector, and so it's a plane. And with this concept we see that the line that goes from two points Is defined as the cross product between the two rays going through p1 and p2. And this is because the line forms a plane, and the plane has a surface normal l, which is both perpendicular to p1 and p2. Similarly, two lines, which again each represented by three dimensional vectors. The surface normal of the plane formed by the line or origin, they intersect at a point and the way to find intersection of that point is again take a cross product between the two vectors. And as again because the intersection point as a ray is perpendicular both to the surface normal one and surface normal two. As such their cross product between the surface normal one and surface normal two give the ray direction, which is intersection point. So this is called duality because the point can be thought of as functioning in exactly the same way a line functions. Everything is three-dimensional. And then, to get from points to lines, you take cross product between them to get two lines to intersect them, get to a point, which again is. Okay, so now we are thinking about we know how to take an intersection between two points, we know how to take intersection between two lines, so what are the points having the coordinate system of x y 0? So we did the computation given two sets of lines, you cross plot on them, you get three numbers back out. And sometimes you can look at them and look at them sometimes you realize this is a very strange number, we have, a number which is X Y 0. As we started remember, every points in an image plain has a coordinates of X Y 1. So what do we do, when we have a point X Y 0? Recall we take a three dimensional ray, convert to a two dimensional points are dividing the last elements out. Now we can just do that, we say x divided by zero, y divided by zero. What do we get? We get two infinities so, what does it say. Intuitively, it says that, that point must be at a point in infinity, but which one? So, it turns out there are many points in infinity, in the image plane. In fact, there are one infinity point associated with every parallel set of lines in an image. Let's go through this exercise. Imagine we have a line which says x equals one. So that's a line going vertical in the image, going up and down in the image, set through x equal to one. We have another line that goes x equal to two, just one distance away, up and down in the image. And we take those two lines and those lines can be representative of homogeneous coordinates of minus one zero one, minus one zero two. Take cross product between them or through them, we'll get the answer of zero one zero. Again we have a. The third elements of this homogenous coordinate has elements of 0. In fact this indicates that this is a vanishing point, a point at infinity in the vertical direction, y. So we can see in fact vanishing points in the image is formed by parallel lines intersecting. And parallel lines in the image physically do not intersect but we can take a cross product nonetheless and obtain the intersection in terms of homogeneous coordinates with last element of zero. And furthermore we can see the first two elements are indicating which direction this vanishing points lies. 01 means vertical. So for two lines, which are parallel to each other, of the form of ABC, ABC prime, we can again take cross product between them. Through a derivation and a simplification we see the answer is going to be the form of b minus a and zero. Again the last element is zero, indicating point at infinity and the first two elements, b minus a, indicating the line direction associated with this set of parallel lines. So now, to summarize, and the projective representation, a point at infinity, is always represented as two elements which are non-zero and the last element is zero. And any line a, b, and c can intersect at a location of (b,-a,0). And we have this one-to-one correspondence between one point at infinity, this homogeneous coordinates, the third element is equal to 0, with a certain parallel lines in the image plane which are parallel to each other. So now what is a line in infinity? We have concept of point of infinity. So you can imagine we're having a plane, the point of infinity sits all around us. One for every direction that we have. So what is the point of align of infinity in that space? So algebraically we can think of define form, a line as a representation A B and C, and A B and C must pass through all the fines of X Y 0. For every possible X, for every possible Y, the third element is zero. Again, every direction we have has a form of X, Y, zero, as we change X Y in a circle, we have different point in infinity, but no matter what point in infinity we're looking at. They always pass through this line, it's a unique line. By simple derivation we can see that line must have the form zero zero one because if it's not zero zero one the two elements will not be zero. So this line at infinity has a simple representation which is zero zero one. And we call this is a surface normal, which point again straight away from origin three to the center of the image, straight out. And that's what we said was a plane which is almost tilting to be frontal parallel to the image plane. And that's the line of infinity. Just to summarize one more time. We call it ideal points and ideal lines. Those ideal points and ideal lines have homogeneous coordinates of x, y, zero. In the image those ideal points. Our point in infinity. Each point in infinity has (x, y, 0) and they corresponding to a set of lines parallel to each other pointing in that direction. All the power lines pointing the same direction has one vanishing point of (x, y, 0). In the line space the lines that form x, y, 0 or a, b, 0, are nothing but a set of physical lines in an image that has a coordinate system such that those lines pass through the origin in the image space. And the only interesting thing about the line space is the line in infinity has a coordinate of zero zero one. So this summarizes an introduction to projected planes in terms of pointed infinity, point a line in infinities and how do you form two lines. How do we find one line from two points, and how do we form intersection of one point on intersecting two lines?
