So we will continue on the concept of vanishing points and vanishing lines. The property of vanishing points is that if you have a set of lines on the ground plane, so imagine we are driving or flying over a ground plane and there are a set of lay markers on the ground. And all the imagined lay markers are parallel to each other in the physical world. Those set of lines will project through the camera into a set of lines which converge to a single point called a vanishing point. So many, many lines in a physical space corresponding to one single point in a major space. Can you imagine, what happens is, there is a ray going from the optical center of the camera going to the image. And this ray is physically going straight out into space and that line is parallel to all the lines on the ground plane. And that's the representation of all the physical lines in the physical space in the image space to that single point. A single point, you're going to form a line between optical center to the signal point, vanishing point to a ray and that's going to be a representation to all of the physical lines out there. And because all the physical points out there is going to eventually converse to each other to the single point. So, this concept is interesting because we have many points in the image. And in those points, every point could be a vanishing point of some direction in the physical space. Now if you were looking the lines situated on the ground plane, every lines in the ground planes could converge to one, unique point in the image, called the vanishing point of that line direction. For example, you have lines going to the left, and those are going to converge into a point in infinity, in a vanishing point. Again, we'll have another set of lines, that go to the right of me to the vanishing points. Physical, they're undergrounds, they're not touching each other, they're whole set of a lines. But in the image space, when you look at it, they are converging to a single point, and that point is representing the direction of a set of lines. In fact, for every direction we have on the ground plane, we have one unique point in the image space. If we connect all the points of the vanishing point together on the ground plane, what do they form? They form what we call the horizon. Horizon is a straight line, such that every point on the line is one unique direction on the ground plane. If I were to draw a line on the ground plane, instead of a ground lines in the same parallel direction, when I take a picture of it, those lines will converge to that single point. So horizon is the set of all directions into the infinity. So for example, if someone asked you direction where to go, you'd simply say go to this landmark, take a right. You can just tell people you walk to the vanishing point of this direction and make a right. What you're saying to him is you want to go in that direction and make a right for the download. Another way to look at vanishing lines on the horizon is imagine you have a whole ground plane, which is lifted up into space, and this ground plane is penetrating through the optical center straight out, perpendicular to the image plane. And this intersection of the ground plane lifted up with the image plane, in fact, it's called a horizon. As such you can deduce if the camera were perpendicular to the ground plane, if I can take a picture of the scene of the horizon, I would know exactly how tall I am relative to other objects. Because this horizon is exactly the height of the optical center, which is the cameraman himself. And this is become a very useful tool in the following exercise. So imagine we want to measure the height of a person. And imagine we are seeing this person standing on the ground. And the ground is conveniently laid out so that we see the vanishing points, and we see the vanishing lines. So if the person were in a different position, you can see, they are going to be at a different height in the image space. We know the person is the same height. He's not getting shorter as he moves into the distance, he just appears shorter. So imagine we can trace a line, one line, just at the bottom of his feet as he walks straight in the direction into the world. That's the red line on the bottom. And we see another line, which is through his hand, as he goes through the space. This two lines as he moves in the space we know they're parallel to each other. If you were looking from a third person perspective you would see this two line stay parallel to each other. But from my personal point of view taking picture of the scene, he's getting shorter and shorter. In fact, the two parallel lines again converge into a unique vanishing point on the horizon. And we will use this fact to help us to measure the height of this person. To do that, we need a reference object. Just having a horizon and object is not sufficient. We need a reference object. Imagine we have this imaginary rulers in the scene, so it could be a ruler itself. It could be a building that you can recognize, or it could be an object that you have a known height for. So imagine we have this vertical stick that has known size information to it, and we would like to measure, given this other vanishing points on horizon, which is the horizon line and this vertical object with no height, a picture of you, we want to measure how tall you are. So how do we do this? Well, we can do this by imagining that we are standing next to this ruler. And we imagine that we can walk straight into the scene, follow the ruler to you. And there's a straight line that we can trace on the ground plane. An image, in fact, that image of that line and it goes to infinity and is going to reach an infinity on the horizon. So where we draw the bottom, we draw a line between the bottom of the ruler to the bottom of your feet. And that line's going to intersect the horizon at a vanishing point for that direction. So how do we measure the height? So we need to measure if the person was standing to the ruler, where he's going to be? So this, we can do it through the process of tracing the vanishing point in the horizon back through your hand intersecting with the ruler. That's exactly where you're going to be if you were standing next to the ruler. So now we can read it out, the height of this person. So how about we have a much shorter object, like a turtle, how tall is a turtle we know it's shorter than me, exactly how short it is. Well, we can do the same thing. We can trace a line from bottom of the rulers to the bottom of the turtle. And their line represented the direction where we're walking from the ruler to the turtle. And this line is going to go to the infinity and intersection between that with the horizon. Again, we traced intersection point to the top of the turtle. We virtually walked the turtle back to the rulers, and the intersection between that line and the vertical ruler tells you how tall is the turtle. How tall is the person? Well we said the last time the horizon, in fact, is showing you how tall the camera man is. If I have a ruler in the scene, I can also measure exactly where the horizontal line, the horizon line intersect with my vertical ruler and that tells me exact height of the cameraman. So we'll come back to this more precisely in the situation when the camera's no longer exactly perpendicular to ground plane, where such is true if the camera is exactly parallel, perpendicular to the ground plane. But if you're taking a picture where the camera is no longer oriented in this way, there's a different set of equations but the basic concept remains. Now let's have a little bit of fun with the vanishing points, and people know this because in arts, there's many illustrations of such phenomena that we visually, mentally when we see a set of vanishing points, we start discounting the size changes. When the size doesn't change, it creates a sort of illusion showing that the object as bigger than what it is. We imagine if the object is so small in the field of view in the distance, it must be a much giant object in the near distance. Your mind plays this trick to you all the time, it's difficult to override. This is in fact a physical picture, a picture of a person standing close to you and a person far away. Now we know the person standing far away is just short, not because he's a midget, just because he's further away. In fact, when you take that picture of a person and place it next to the person next to you, digitally copy it, he looks like a tiny, tiny person in our field of view. Again your mind is playing this trick to you, and it says I can see the vanishing point. I can see the vanishing line so I know the object is shrinking my field of view so, therefore, when I see an object as being small, mentally I'm amplifying it so that mentally is it as bigger than it actually is. There are many illusions like this in the world. I encourage you to look for those, take a picture of them, and share them. I look forward to seeing you next time.