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.

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

  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?