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Pre- and co-image

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title: Pre- and Co-Image (Session 2022)
categories: session
title: Pre- and Co-Image
categories: exercises

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
+ 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

1.  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?  
2.  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).
3.  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?

# Debrief