Good morning, welcome for computer graphics.

Today we will hopefully finalize the step to go from 2D to 3D.

So on Tuesday, no on Monday I already started with explaining how we can project 3D objects

to 2D images mimicking standard cameras.

And we saw that the correct way to generate images like a camera does or like the human

eye does is to take a point projection.

So we have our 3D object, we have a camera positioned somewhere in space and then we

generate rays in some sense from the object points to the eye and where these ray intersect

an image plane this is the position where we draw our image point.

So it's a point projection.

And the shape that contains all points that are projected onto a particular image is called

the view frustum and all points within that view frustum are then projected into our image

and this generates these perspective drawings or perspective images that we know.

Distant objects get smaller in the image and all these effects, parallel lines don't remain

parallel and we also learned that this is a new larger class of transformations which

we call protective transformations.

And the nice thing now is that we can describe this mapping using our framework with homogeneous

coordinates.

And here we derive that this perspective, if we have a very special setup or a simplified

setup where we already transformed the scene such that it's in the origin or that the camera

is in the origin and that we look, if we look along the set axis then this perspective projection

simply means dividing x and y coordinates by set and we can do that division with homogeneous

coordinates if we generate a matrix like this one, yeah, that looks very simple and the

important thing is that bottom line because that makes that after, if we take homogeneous

coordinates and we apply that matrix, the set goes to the homogeneous coordinate and

by de-homogenization, yeah, we divide by this set and this gives us exactly this perspective

division, yeah, this division that we want.

So we can, if we have homogeneous coordinates, we can describe this perspective mapping with

that very simple matrix.

And this brought us to, yeah, how now in general we define our matrices and usually just to

make things easier to understand or to handle it better, we separate it into two steps,

viewing and projection.

And the first one, viewing, is just a rigid transformation and it somehow, it positions

the camera in our scene and says in which direction are we looking into.

So that's just a rigid transformation, translation and rotation and after that, yeah, we are

in this setup, now we are looking along, yeah, the set axis and I told you that because usually

we use a right-handed coordinate system, we usually change the setup so that we look along

the negative set axis, yeah, and in that setup now we do the perspective projection.

And to this end, yeah, now we can even still use orthogonal projection, yeah, this is shown

here, now we are after viewing, we are in that setup, we look along the negative set

axis and now x and y span the image plane and now doing an orthogonal projection is

extremely simple because now we can simply take the x and the y coordinate of our point,

yeah, and consider these as image coordinates, yeah, now parallel projection just means projecting

parallel to the set axis so that just means, yeah, throwing away set, yeah, then we get

a 2D image, in fact, we don't throw away set because we will need it for occlusion later

on and the alternative is to do that perspective projection like here, yeah, and now this is

a little bit more complicated but in fact it is exactly the matrix that we have seen

before, yeah, it is just that the bottom row is a little bit simpler and then we get this

perspective projection here, yeah, so just by very taking two different very simple matrices

we can generate both of these projections, so this is the one, the matrix for the orthogonal