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When trying to find the vanishing point of an image, why is it that the vector from the center of the camera to the vanishing point is parallel to the parallel lines of the image?

What's the mathematical reasoning behind this? Diagram of vanishing point

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  • $\begingroup$ What are "original lines"? $\endgroup$ – Andrew Wilson Oct 29 '17 at 4:08
  • $\begingroup$ Sorry for not being clear, I fixed that now. I hope its clear now $\endgroup$ – Zack1234 Oct 29 '17 at 4:35
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    $\begingroup$ This diagram simply shows how in perspective projection the parallel lines are not parallel in the resulting image (image plane). They would only be parallel in orthogonal projection. The author of the diagram could project any lines he wanted to on the image plane. They don't have to be parallel with the line going through the vanishing point. $\endgroup$ – Andrew Wilson Oct 29 '17 at 4:47
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    $\begingroup$ @AndrewWilson, absolutely. And the OP phrase to the parallel lines of the image is invalid, because there is no such thing that "THE" parallel lines. It's only meant to mean that some cubic looking, well aligned geometry, like architecture and streets, will tend to display lots of parallel lines that "engenders" the presence of a virtual vanishing point. Do anybody speaks of vanishing points in image of trees and ponds ? $\endgroup$ – v.oddou Oct 30 '17 at 2:23
  • $\begingroup$ This question seems nonsensical as it currently stands. Perhaps a photo showing which parallel lines you mean would clarify? $\endgroup$ – user1118321 Oct 30 '17 at 3:28
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FYI every set of parallel lines will have their own vanishing point. The reason why there is usually only a single vanishing point (the most obvious one) talked about in an image is due to aesthetic choice.

A perspective projection can be seen as a point projection of a 3D scene onto a 2D plane. The way to determine where a point in the 3D scene ends up on the 2D plane is to draw a line between the point and the camera (the sight line) and see where it intersects the projection target plane.

To find the vanishing point of a set of parallel lines you can take a the limit of traveling on that line infinitely far. At the limit the sight line will be parallel to the line.

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