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Thread: Help needed to visualize parallel lines intersecting with homogeneous coordinates

  1. #1
    s3a
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    Help needed to visualize parallel lines intersecting with homogeneous coordinates

    Could someone please relate the notion of parallel lines intersecting to this diagram (at time 4:48 of the following video)?:
    https://www.youtube.com/watch?v=q3tu...outu.be&t=4m8s

    I do understand the algebraic explanation provided here (in the part that's below the text "Proof: Two parallel lines can intersect.").:
    Homogeneous Coordinates

    I also get that it has to do with perspective, such as railroad tracks, which are parallel in real life, visibly intersecting at a vanishing point.

    My only problem is that I'm struggling with visualizing the drawing(s) for homogeneous / projective coordinates that visually describe this phenomenon. Given that the last coordinates needs to approach zero for the phenomenon to occur, I visualize the plane that used to be at z = 1 almost overlapping with the one at z = 0, and it's like the two points cross the z = 0 plane, and though I see how they intersect at (0,0), I don't see how they are parallel.

    Could someone please help me demystify this for myself?

    Any input would be GREATLY appreciated!

    P.S.
    If this is in the wrong forum, let me know.
    Last edited by s3a; Feb 20th 2018 at 08:29 AM.
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  2. #2
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    Re: Help needed to visualize parallel lines intersecting with homogeneous coordinat

    Let's use a specific projective geometry, such as the Riemann Sphere. Every complex line translates to a circle on the Riemann Sphere. For instance, $\{x+0i\in \mathbb{C}|x \in \mathbb{R}\}$ and $\{x+i\in \mathbb{C} | x \in \mathbb{R} \}$ are parallel lines in the complex plane. But, in the complex projective plane, they intersect at infinity. They look like circles. Both include the point at infinity.
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  3. #3
    s3a
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    Re: Help needed to visualize parallel lines intersecting with homogeneous coordinat

    Keeping what you said in mind, I drew what I attached to this post as MyDrawingForTryingToVisualizeParallelLinesIntersec tingAtInfinityInHomogeneousProjectiveCoordinates.p df.

    (The dotted lines part is for the lines intersecting at infinity, in a circular manner, as you said.)

    I'm still unclear about the infinity/infinities part (shown in the drawing with the infinitely-close planes). If I imagine the plane as a piece of paper and the lines as being drawn on that paper with a marker, I could fold the paper such that the end of the lines meet (in addition to meeting at the origin), but how does that make the lines parallel? Could you please clarify that for me? Is my drawing completely off?
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  4. #4
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    Re: Help needed to visualize parallel lines intersecting with homogeneous coordinat

    I'm not sure I understand your drawing completely. You are adding both positive and negative infinity (I think). My example was with the Riemann Sphere. That is the one-point compactification of the complex plane. Basically, it adds one point ($\infty$). It can be represented by a sphere. One "pole" of the sphere is zero and the other is infinity. The real line is a circle on that sphere that passes through zero and infinity. The second line I mentioned that becomes a circle is just a shift of the same line. I added $i$ to it. So, it is $i$ plus the real line.

    To help you visualize, do a google search for the Riemann Sphere. There are several images that can show you what I mean.
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  5. #5
    s3a
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    Re: Help needed to visualize parallel lines intersecting with homogeneous coordinat

    I understand wrapping the complex plane around a sphere and having that be the Riemann sphere. I also understand that all four directions' infinities meet up at the “one-point compactification” labelled as “∞”. Also, when you say every complex line translates to a circle on the Riemann sphere, I believe you mean like this ( https://upload.wikimedia.org/wikiped...ctive_line.svg ), with each segment going from 0 to ∞ being half of the line, right? (I did not understand these things when I made my previous post.)

    Also, I see why two parallel lines intersect in your Riemann sphere example (so, thanks for that), but how is the Riemann sphere related to the three-dimensional* representation of parallel lines in homogeneous coordinates? I can see wrapping the Euclidean R^2 plane around a sphere, but why does one need a homogeneous coordinate system to do that? Can’t one just do that with a regular R^2 Euclidean space? (Sorry if that's a stupid question, but I believe that that’s my biggest confusion that remains.) (Also, please let me know if I’m using any terminology incorrectly.)

    *: I mean three-dimensional as in (x,y,w). That is, the Euclidean plane is in two dimensions, and the third dimension is the distance between the Euclidean and projective planes. That is not to be confused with three dimensions in the Euclidean plane (with w, the distance between the Euclidean and projective planes, as a fourth dimension). (Maybe this is obvious to you, but I just want to be clear.)

    P.S.
    Sorry for the huge delay in my response; this is not directly related to what I am doing in school, and I have very little free time.
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