The computer graphics.

So we are right now in the middle of the section on transformations.

So on Tuesday I started describing affine transformations and I explained how we can

use homogeneous coordinates to have a nice way to describe affine transformations using

matrices, 4x4 matrices.

And before we will expand our class of transformations that we're looking at to projective transformations.

Today I would like to have a closer look at a particularly important transformation that

we will use in computer graphics very, very often and that's rotations, 3D rotations.

And so before we look at that I would once again have a look at coordinate system transformations

because we will use that for the rotations all the time.

And I already told you an affine transformation is nothing but a simple transformation to

another coordinate system.

And if we now want to describe such transformations what we can do is that we look at how the

unit vectors or the unit coordinate vectors are transformed and we will see that in these

matrices these vectors appear again and by this we can derive simple constraints on matrices

that have certain properties of our transformations.

So let's look for an example.

So when we speak about coordinate transformations what we always have in mind is that we have

two coordinate systems.

We have a global coordinate system that has its origin, yeah, in the origin.

There's some origin in the world and we have basis vectors x and y.

So these are, our origin is always 0, 0 at coordinate 0, 0.

This is the x-axis and this is the y-axis.

And now we have another local coordinate system.

This has its origin somewhere and it has two new basis vectors u and v. So that would be

an example here.

That's our global coordinate system and now we have a new coordinate system with origin

e and coordinate vectors u and v.

And now a particular point here, p, can be expressed in this coordinate system.

So if we say this point has coordinates 5, 1, this is implicitly meant in this coordinate

system.

But in this coordinate system it has different coordinates.

And so you see in this coordinate system we have to go one step along u and minus two

steps along v.

So the coordinates of that point p in coordinate system u, v here is 1 minus 2 in this new

coordinate system.

And OK, now all affine transformations can also be represented or also always mean such

transformation of coordinates.

And that means we can always compute such a transformation or compute such a matrix

that does this transformation here.

And because we also learned that if we use homogeneous coordinates and describe transformations

using these matrices, the inverse transformation can be described by the inverse matrix.

So that means if we find a matrix that maps new coordinates to old coordinates, the inverse

does it the other way around.

So if we find a transformation from one coordinate system to the other, we also immediately have

the matrix for the other way.

And now that's quite simple.

Because if we look at that example, again, that's our global coordinate system.

Now we have a particular point p in that coordinate, a particular point.

And we have it both in this coordinate system and in this coordinate system.