---
title: Pre- and Co-Image
categories: exercises
---
+ 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
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