OK.

So welcome for our computer graphics lecture.

And we are right now right in the middle

of speaking about transformations.

I think I told you that a large part of the rendering pipeline

is transforming coordinates from one coordinate system

to the other, and thus to map geometry, which is typically

a triangle mesh, to screen space.

And you also learned that we have a very nice machinery

to describe that.

And that is using homogeneous coordinates and matrices

to describe all these single steps of our transformations.

And in general, with such homogeneous matrices,

we are able to describe projective mappings, which

we will come to later on, maybe today, or I hope today.

But we are right now looking at a subclass

of these transformations, and that's affine mappings.

And affine mappings essentially are just

changes of coordinate systems.

So it's a linear mapping and a translation.

And we looked at certain classes of these affine mappings.

And right now, we are looking into rotations.

And rotations is a special kind of coordinate system

transformation, because it just transforms

to a new coordinate system, which again is orthonormal.

So objects are not deformed under these transformations.

And that's why it's a particular interesting and important type

of transformation.

And in fact, it is necessary to look

at this, because our current way to describe this,

and that is using a matrix, is not the best one

to describe these transformations.

It's perfect if you want to use that in your rendering pipeline

to map objects.

But for instance, if interpolation is required,

or user input is required to describe such a rotation,

then we need other representations.

And yeah, so for the case of matrices,

essentially if we assume a rotation around the origin,

then we have nine degrees of freedom.

That's essentially the linear part of the transformation.

But six of these entries are fixed due to some constraints.

And that's very unintuitive.

We are in a nine-dimensional space,

and we take a three-dimensional subset of this.

For user interfaces, that's not very good to use.

And an alternative or a more intuitive way

are Euler angles.

So probably the most easy or accessible example

is the typical way that is used in aeronautics,