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I recently started reading the book Multiple View Geometry by Hartley and Zisserman. In the first chapter I came across the following concepts..

  1. Projective geometry is an extension of Euclidean geometry with two lines always meeting at a point.
  2. In Perspective geometry parallelism does not exist.

Then he goes on to explain how the points at infinity in the world like the points at the horizon appear as a line in the image of the world taken by a camera. Then he says a line which I cannot relate is the following..

The geometry of the projective plane and a distinguished line is known as Affine Geometry and any projective transformation that maps the distinguished line in one space to the distinguished line of the other space is known as an Affine transform.

I have the following questions...

  1. Is the camera plane the projective space of the real world?
  2. Is the line which is the image of the horizon the distinguished line?
  3. Whenever we do an Affine transform do we need to look out for a distinguished line?
  4. Why does just a distinction of the geometry a line in the perspective plane make the geometry an Affine geometry?
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Loosely speaking when one is looking at geometries from an axiomatic point of view projective geometries are ones where every pair of lines meet at a point and affine geometries are ones where given a point P not on a line l there is a unique parallel to l through P. Affine geometries with additional structure lead to the Euclidean plane. If one has a projective plane, and one singles out some particular line m and deletes this line (sometimes thought of as the line at "infinity") the lines that used to meet at a fixed point Q on the line you threw away (line m) now can be thought of as parallel. So all the lines that met at Q are now "parallel" in the new geometry. Lines that used to meet at points not on m still intersect.

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As far as I understand Affine transformation preserves parallelism and ratios of lengths. Consider a transformation which maps distinguished line from one space to the distinguished line of other space. Now consider two parallel lines in a world plane which meet at a point Q on the distinguished line in one space. There will definitely be a corresponding point say Q' on the distinguished line in the other space where the same two parallel lines meet. So the property of parallelism remains relevant on removal of the distinguished line. Hence it could be seen as Affine transformation.

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