# Should/do parallel lines curve when rendered with perspective?

Simple perspective calculations used in rendering 3D points onto a 2D screen take the form of dividing the camera-relative coordinates by distance from camera and multiplying by a field-of-view parameter, as described here: http://en.wikipedia.org/wiki/Perspective_transform#Perspective_projection

So if I have a set of long horizontal parallel lines, like looking across a huge chess board, shouldn't the lines take on a slight curve on the 2D rendered output, since the lines are closer to the camera in the middle than at the ends?

I don't see this happening in real 3D rendering so I wondered if I have a mistake in my math, or if it's simply the case that the effect is so small it's not noticed?

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I think you should make the first paragraph in the question more explicit. As it stands I can't make any sense of it, since it seems to be a prescription for scaling three-dimensional coordinates, which leads again to three-dimensional coordinates, whereas the result of calculations for rendering three-dimensional objects in two dimensions would have to be two-dimensional coordinates? –  joriki Apr 5 '12 at 11:29
Sorry I'm probably using specialized terminology from the field of 3D computer graphics. See en.wikipedia.org/wiki/… I've edited slightly. –  Mr. Boy Apr 5 '12 at 11:40
Your image of a chess board, where the lines are wider apart in the middle: that is not perspective projection. Perhaps it is a "fish-eye" image. en.wikipedia.org/wiki/Fisheye_lens –  GEdgar Apr 5 '12 at 12:14

Your interpretation of the math is slightly off. In perspective projection, you divide the camera-relative $x$- and $y$-coordinates by the camera-relative $z$-coordinate, not by the full distance $\sqrt{x^2+y^2+z^2}$. This has the nice property that all straight lines in 3D project to straight lines in the image plane. If you are looking directly at a chessboard, for example, everything has the same $z$-coordinate, and so all the lines still appear as straight and parallel.