Let's continue in the text.

Last week we started to consider different representations for rotations.

Rotations in 3D.

Rotations in 3D can be characterized by different ways.

We need a linear mapping that characterizes the isotropic rotation of the object.

It's not shrinked and not scaled.

It's just rotated and does not change in its shape and size.

The most commonly used way to represent rotations,

and that's also something that you will find out by heart and by your basic linear algebra education,

are the Euler angles and the 3x3 rotation matrices that characterize the rotation around the x, y, and z-axis.

Then you multiply things together and you get the rotation matrix.

The only thing that you need to know is the columns of the rotation matrix are the images,

the image vectors of the base vectors.

If you know that and if you know a little bit of trigonometry, then you are fine.

And you can find it out.

Then we started out to look at the Rodriguez formula to represent rotations.

The Rodriguez formula requires a rotation axis and a rotation angle around the axis.

What we did not consider is that each rotation or all the rotations in 3D

can be represented by a rotation matrix and an angle.

That's something we did not prove.

That's something people in mechanical engineering do.

And you can attend lectures on mechanics where these proofs are provided.

For us, it was just important to look what happens to a point or a vector in space

if I rotate this point around a given axis u.

Then we were looking at what is the point where we end up with

and how can we characterize this mapping mathematically.

And we have drawn here a few figures.

We have looked at the vector v.

We span a coordinate system given by the rotation vector u

and the point that is required to be rotated.

These two vectors span a plane.

And then we take the cross product to get an orthogonal vector to this plane.

And based on this induced 3D coordinate system,

we were able to see where the rotated vector ends up.

And this rotation was at the end of the day characterized by a rotation matrix.

And then we have seen here how the matrix is built up.

We have the basic three components.

Let me just go one slide back here.

Where we can say the rotation matrix is the dyadic product of the rotation axis unit vector u.

Plus, and then we have here this type of projection matrix,

the identity, three by three identity matrix plus the dyadic product

times the cosine of the rotation angle around theta.

And then we have here the skew matrix that

defines basically the cross product in matrix notation.

If you compute the cross product u cross v,

this is a vector that is linear in the components of v.

And that means that this mapping can be expressed in terms of a matrix.

And this matrix is this one here.

You remember that the skew matrix with the symmetric matrix were just the sine of the matrix.

The symmetric matrix were just the sine flipped and on the diagonal we had just zeros.