--- 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