Line at infinity - Wikipedia
Axiom of Parallels: Given a line and a point outside it there is exactly one line through lies in the plane of the given line and point so that the two lines do not meet. axiom is that even (apparently) parallel lines appear to meet at the horizon. Projection rays are parallel; Image plane is fronto-parallel. (orthogonal to Where do parallel lines meet? Parallel lines meet at the horizon (“vanishing line”) . In elementary Euclidian geometry, parallel lines do not "meet at infinity". .. There is a "line at infinity" (in art, this is called the "horizon line".
This is sort of untidy, and there are two ways to fix that. One is simply to decide to draw stuff on the sky. That leads to spherical geometry, an ancient and venerated area of study that is the foundation of astronomy.
But it's not what today's lecture is about. When we look in any direction, our VR helmet computes the infinite line through our head in the direction we're looking, and figures out where that infinite line intersects the plane below us.
Whatever is at the plane there is what we see. So when we look at the sky, what we see is the plane behind our head! Only when we look in a perfectly horizontal direction does this procedure not work, but we posit that there's nevertheless some points to look at there, which form a "line at infintity" as before.
Line at infinity
Note that since we're looking at the sky plane from below, things in the sky will be the mirror image of the same thing if the turn around and look down at them instead. In projective geometry any two different lines have exactly one point in common. If they are non-parallel lines in the original plane they will cross once on the ground, and we will also see that crossing in the sky -- but these are just two images of the same point in the projective plane.
Two parallel lines will cross exactly once at the line at infinity -- again we see two images of that crossing when we turn around, but they are by definition the same point. And any line in the plane will cross the line at infinity once. The main thing that makes this cool is that the line at infintity now has no special properties.
If we tilt our head and forget which way was up and down, there is no way we can deduce from the geometrical properties of what we see which direction the "true" horizon lies.
Projective geometry is about properties of figures that are invariant of rotating our view of the world.
geometry - The intersection of two parallel lines - Mathematics Stack Exchange
It is also about properties of figures that don't change as we walk around on the plane, or move our head closer to or farther from the plane.
Such a room or device as you can build miniature versions is called a camera obscura. The image as would have been projected in the camera obscura in case the camera obscura was placed on the middle of the road the observed image is turned upside down in this figure From the image of the road with trees in Fig.
Note that not all parallel lines in 3D will meet in a point in the projection.
Two lines parallel to the horizon will not meet in a point. Euclidean geometry is not capable of dealing with points and lines at infinity. In projective geometry the points at infinity are treated as all other points in space. The most important invariance in projective space indeed its defining property is that colinearity is preserved. Points on a line will stay points on a line although maybe a different line. For an introduction to homogeneous coordinates we refer to section Homogeneous Coordinates.
By definition a projective transform preserves colinearity of points. It can be shown that any 2D projective transform is a linear operator in homogeneous 3D space i. In this chapter on the pinhole camera we will write homogeneous vectors with a tilde: