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Suppose I have considered example of perspective projection.I have one railway track with two lines meets at point say,$X$ at infinity which is theoretical approach.And this X is called vanishing point.But practically this $X$ doesn't exists.

My first question is how we can say it doesn't exist, but when we see real image of railway track they intersects at $X?$

My second question is if it exists then during projection how this $X$ is projects in projection plane? Is it projecting like normal point projection $(x, y,z)$ to in view plane $(x_p, y_p, z_p)?$

N. B- I don't want that answer which is in details manner. I want just intuition which is brief and easy to understand.

S. M.
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  • In projective geometry it does exist. Check out this question and answers .. https://math.stackexchange.com/q/3795673/1257 – brainjam Oct 31 '21 at 16:49
  • @brainjam no this not my questions answerer. I have asked perspective projection. – S. M. Oct 31 '21 at 16:54
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    Isn't this image clear enough? https://en.wikipedia.org/wiki/Vanishing_point#/media/File:Vanishing_point.svg – Intelligenti pauca Oct 31 '21 at 22:51
  • @pauca you mean in 3D space they don't intersects, but in projection plane they intersects? – S. M. Oct 31 '21 at 23:45
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    False premise. In a real image of railway tracks the rails do not meet at the vanishing point, because no real railroad track goes perfectly straight without end. – David K Nov 01 '21 at 02:53
  • @Niyon Yes, that's quite clear from that image. They intersect at the point where a line from the point of view, parallel to the tracks, intersects the projection plane. – Intelligenti pauca Nov 01 '21 at 09:54
  • @Pauca could you insert your comments in answer, I will accept it. – S. M. Nov 01 '21 at 10:04
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    As already said by others, mathematicians have developed extended versions of the 3D euclidean space where the crossing point of parallel lines really exist. Me and others who know nothing (except the existence, maybe) of such geometry still can say "parallel lines meet in the vanishing point in the infinity". But that's lousy speak (=nonsense) because the 3D space that we think doesn't have such point. I have shaped up. I have ceased to talk about vanishing point in infinity. I'll say that the perspective images of parallel lines meet in their vanishing point on the 2D imaging plane. –  Nov 02 '21 at 10:04

1 Answers1

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X doesn't exist in reality, since the railroad tracks are parallel, but it exists in the projection plane.

In the following link, you have the explanation :

https://en.wikipedia.org/wiki/Vanishing_point

Projections of two sets of parallel lines lying in some plane πA appear to converge, i.e. the vanishing point associated with that pair, on a horizon line, or vanishing line H formed by the intersection of the image plane with the plane parallel to πA and passing through the pinhole.

Mateo.

Mateo_13
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  • It's not my question answer. Please elaborate more. – S. M. Oct 31 '21 at 16:57
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