# Pre- and Co-Image (Session 2022)

The format today will differ somewhat from the norm. Instead of having one debrief the very end, we will try to do two exercises with a shorted debrief after each one.

Feel free to look at the debrief notes as hints to solving the exercises; just take a couple of minutes first to try to make sense of the question and make a sketch.

**Briefing**Pre- and co-image Lecture- Exercise 3.9 are from Ma 2004 page 62ff.
- Exercise 3.10 are from Ma 2004 page 62ff.

# First Exercise (3.9)

Exercise 3.9 are from Ma 2004 page 62ff.

## Debrief Notes

### Part 1

You should first find the pre-image of the image of \(L\).

- What kind of object is the pre-image?
- How did we describe such an object previously?

- What is the relationship between this pre-image and a point \(x\in L\)?
- What is the relationship between the pre-image and and the vector \(\ell\)?

### Part 2

- If you read the points \(x^1\) and \(x^2\) as vectors in 3D, what do they look like?
- Can you describe the pre-image in terms of \(x^1\) and \(x^2\)?
- maybe as a span?

- What then is the relationship between \(\ell\) and \(x^1,x^2\in L\)?

How do you find a vector which is orthogonal on two known vectors in 3D?

### Part 3

- Note that \(x\) is an image point.
- \(\ell^1\) and \(\ell^2\) are vectors in 3D, and co-images of two image lines
- If you view \(x\) as a 3D vector instead of a point, what does it look like?
- What would be the relationship between this vector \(x\) and \(\ell^1\) and \(\ell^2\)?
- How do we find vector \(x\) with the right relationship with \(\ell^1\) and \(\ell^2\)?
- How do we make sure that the vector \(x\) is an image point \(x\)?

# Second Exercise (3.10)

Exercise 3.10 are from Ma 2004 page 62ff.

## Debrief Notes

- Here, it is necessary to look at the pre-images of the two lines.
- What does the pre-images look like?
- What is the intersection of the pre-images? Could it be empty?
- What is the intersection between the image plane and the pre-images?

- Here, you need to look at the co-images.
- What can you say about co-images of parallel lines?
- What can you say about the relationship between the co-images and the images? Is there are relationship between one line and the co-image of the other line?
- Now return to Part 3 of the previous exercise (3.9).

- Because the two lines are parallel, they lie in the same plane (not necessarily through the origin). Consider the orientation of this plane.
- Suppose first that it intersects the image plane close to the centre (image origin). Where is the vanishing point?
- Suppose you turn the plane. Where does the vanishing point go?
- At the extremity, the plane is parallel to the image plane. Where is the vanishing point now?