We stopped last week discussing projection models and we basically have seen two different

types of projection models.

We have seen the perspective and the parallel projection.

Let me switch the color from pink to a more suitable color.

I don't see any color here now.

Okay.

Sorry.

I have to restart.

Seems that it's not my day today.

Good.

Where did we stop?

Now once again, we discussed the two different types of projection models, perspective and

parallel projection.

And what we need to know is basically if we have parallel projection, we basically forget

about the z-coordinate.

We do an orthogonal projection.

And if I have here my image plane and we illustrate the image plane here by using a one-dimensional

line, I know that a point here is orthogonally projected onto this line here.

This point here is also orthogonally projected on this line.

This is orthographic projection.

And if this line here is defined, for instance, by this vector, this is the spanning vector,

we can basically define here so-called level sets.

This here, for instance, is a level set.

And all points on this line here share one property.

They fall on one and the same projection point.

That's why we call this a level set.

So we select a certain level and all points on this straight line here end up to be the

same in the projection space.

And here we get another level set.

So here we have all the points here.

That's also orthogonal.

They all map to the same point in the image plane.

This is also a level set.

And we know how to express here the orthogonal projection.

That's something we have discussed yesterday.

This can be described by a projection matrix.

And if we project from 3D to 2D using homogeneous coordinates, the matrix looks like this.

It's 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1.

That's the orthogonal projection or parallel projection or orthographic projection.

Good.

What's also interesting basically for all of you who will attend the lecture in summer

semester, if this here is the vector that spans the image plane and this here is the

vector that has to be projected, can you tell me what is the length of this vector?

That means the orthographic projection.

It's the dot product or the inner product.

So if this is the vector V and this is the vector U, it's V transposed U times U. Perfect.

So very simple, a very simple observation, but it will be a very powerful thing.

This easy observation we can lift to a very complex algorithm in summer semester for doing

3D reconstruction.

It's just an outlook.