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Page 4 of my computer vision textbook, Multiple View Geometry in Computer Vision (Second Edition), by Hartley and Zisserman, states the following:

Affine geometry. We will take the point of view that the projective space is initially homogeneous, with no particular coordinate frame being preferred. In such a space, there is no concept of parallelism of lines, since parallel lines (or planes in the three-dimensional case) are ones that meet at infinity. However, in projective space, there is no concept of which points are at infinity – all points are created equal. We say that parallelism is not a concept of projective geometry. It is simply meaningless to talk about it.

The textbook is freely available here.

Although I have read the textbook's description of homogeneous coordinates and projective space, I still don't understand why there cannot be a concept of parallelism in a homogeneous projective space?

I would greatly appreciate it if people could please take the time to clarify this.

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The sentence that begins, “However, in projective space, there is no concept...” is the key. You might be used to constructing, say, the projective plane $\mathbb{RP}^2$ in a “bottom-up” manner by starting with the Euclidean plane $\mathbb R^2$ and then adding “points at infinity” and the line that they all lie on. Hartley and Zisserman are instead taking a “top-down” approach here: starting with the projective plane, complete with all of those “extra” points, as a given. They point that they’re trying to make is that from this point of view there’s nothing special about those particular points—we can pick any line in the projective plane to be our “line at infinity” with a different set of lines that are then considered to be parallel for each choice.

A different model of the projective plane might help here. Instead of starting with the Euclidean plane and adding points, start with the unit sphere in $\mathbb R^3$ and identify antipodes. (This is equivalent to the model in which lines in $\mathbb R^3\setminus\{0\}$ are considered to be points in $\mathbb{RP}^2$.) Lines in the projective plane then correspond to great circles on this sphere, but with this construction there’s nothing special about any particular great circle. The definition of parallelism depends on knowing which of these lines is the “line at infinity,” so until you’ve chosen a particular line to be that, it doesn’t make any sense to talk about lines being parallel. Making that choice imposes an affine geometry on the space—you now know when lines are parallel—as the authors discuss in more detail later.

You can see this in action when you apply a projective transformation to the Euclidean plane. If the transformation maps the line at infinity to some other line, then parallel lines in the source don’t generally remain parallel in the image: they instead converge at a finite point (their “vanishing point”) that lies on the image of the line at infinity. (Identifying this image will become important in later chapters.) Parallelism is not a projective-geometric propery—it is not preserved by projective transformations. In a sense, considering points with homogeneous coordinates of the form $x:y:0$ as the points at infinity is an artifact of the coordinate system that you’ve chosen; the projective map in this paragraph can be viewed passively as a change of basis rather than actively as a “warping” of the plane. We could just as well have said that points with homogeneous coordinates of the form $0:y:w$ are at infinity instead, and some sources do just that.

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  • $\begingroup$ Thanks for the answer. So if we take the bottom-up approach by starting with Euclidean space and then adding the points at infinity to get projective space, then we do have parallelism, but if we take the top-down approach of starting with homogeneous projective space, then parallelism isn’t defined, since we have not defined the points at infinity? $\endgroup$ – The Pointer Apr 2 at 18:22
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    $\begingroup$ @ThePointer Pretty much. You inherit sets of parallel lines (and other things that give you a Euclidean geometry) from the Euclidean space and add a point at infinity for each such set. The projective plane with a Euclidean geometry imposed on it is sometimes called the “extended Euclidean plane.” H&Z later discuss the choices that need to be made in the top-down approach to end up with a Euclidean geometry. $\endgroup$ – amd Apr 2 at 18:31
  • $\begingroup$ Great! Thanks again. By the way, any chance you could come back and post an answer to this question? math.stackexchange.com/questions/3038984/… I have set a bounty for it, and it seems like you knew what they were referring to? $\endgroup$ – The Pointer Apr 2 at 18:33
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    $\begingroup$ I’ll take a look when I get a chance. $\endgroup$ – amd Apr 2 at 18:39
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    $\begingroup$ A good edge case to look at when figuring out what is and isn’t a projective-geometric property is the Fano plane, which is the minimal construct that satisfies all of the axioms of a projective plane. $\endgroup$ – amd Apr 2 at 18:40

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