In this lecture, we'll elaborate more on the perspective projection. We're going to give a historic perspective on the perspective projection. This is a drawing of Leonardo da Vinci, trying to create a perspective drawing from an object in front of him, a globe in this case. He uses a pinhole just in front of his eyes and then a glass plate, where he draws on the glass plate exactly what he sees through his eye. There's another way of depicting it here. You see a picture of a cube and how this cube is projected on such a glass plate with all the rays from the eye of the drawer intersecting the glass plate and how the square of the cube appears as a quadrilateral. And you get some idea about the vanishing points that are created in this perspective projection. People in the Renaissance and for the next century after, they have wondered how one can create a mechanical construction from this perspective projection. And in 1752 Lambert, a mathematician, came up with the idea of what is called a perspectograph. And to be more precise he called it the bi-perspectograph. What is this perspectograph? This is a mechanical device where if you have just one plane, you don't have any more glass plate in the world. And you want to recreate a perspective image of a planar figure like this carpet, the green carpet we see there. How would we create a perspective projection in exactly the same plane? And he created this mechanical realization which has two fixed points on the top, and then you always have a perpendicular and the 45 degrees. And exactly where this perpendicular, the 45 degrees are intersecting, this is the point that renders the perspective projection of another point where the two initial roads are intersecting. This is a very beautiful construction, and we're going to explain geometrically how actually this holds and why indeed this drawing on the plane is a perspective projection from the original picture. Let us assume that the image plane is perpendicular to the ground plane where the original figure is. Let P trace the original figure. Let O* be the projection center, and let P* being exactly what we would draw on this image plane if the image plane would be a glass plate. Let's do the following construction. We project O* down to a point S. And we take a segment OS which is equal to SO*. SO* is just the height, so SO will be really representative of the height of the camera or the eye of the person who does the drawing. Then let's connect O and P and take the intersection with the image plane M. Now, because OS is equal to SO*, this is a right triangle which is isosceles. So, this angle here is 45 degrees, the angle SOO*. And because of the construction, the angle L, M, and P* is also 45 degrees. We are going to do the following trick. We are going to draw a line M P prime, which is with a fixed angle 45 degrees to the image plane, and then a perpendicular L, P prime. In this case we're going to create the exact image of the triangle P*LM down to the ground plane L P prime, M. And because these triangles shares a side, LM, and they are both right angles 45 degrees, these two triangles are congruent. So exactly what P* will do on the image plate, P prime will do it on the ground plane. This is the proof how this will work. Now how is this locus of points created which create the perspective projection? We're going to see it in the following animation. We have on the left again the same construction. And on the right we have a circle which is actually the original picture from which we want to have a perpective drawing. Now if you have a circle in the world and you look at it obliquely, the circle looks like an ellipse. Let's see if indeed an ellipse is created. And indeed when this point is running on the circle, the intersection of the perpendicular and the fixed line with 45 degrees, we see that it renders an ellipse. This is really wonderful because there is no other way to render such a curve. We cannot use several points like the way we'd use with say a square or a parallelogram. And this is really the proof that we can render the perspective projection from an image plane which is perpendicular to the ground. Plane is not an arbitrary image plane, but still a very common case for perspective drawings. Now, we took some freedom in choosing our parameters. How high is the camera, is determined by how high O* is. We took SO* and we projected it on the right in this segment. So if the segment now is changing, it means that the camera is going lower and then again higher. And what happens when a camera looks at the circle from a high position? The circle is not so much squeezed as we see right now. When it goes down and the camera looks at the circle obliquely, the circle appears as a squeezed ellipse. So exactly this segment determines of how obliquely we look at the world. And what else did we choose as a parameter? It was the distance between these two vertical lines. This is between these two vertical lines. There's nothing else in the distance of O* from the image plane, which is the focal length. And we see from this effect when we remove the image plane, and we have to take it away from the projection center, the ellipse does not get more squeezed. It does get bigger when the focal length is bigger or smaller. It is the same effect as getting the glass plate out or towards the viewer. So with this very simple construction with two parameters, the height of the camera and the focal length. We have been able to create the perspective drawing of anything lying on the ground plane on exactly the same ground plane without using any glass plate and being also able really to render any curve we want. This is the beauty of the bi-perspectograph