Context-1 Consider a picture of rails being taken from a camera(and our eyes is at position as shown in the figure ) and consider the picture formed in image plane to be as shown in figure.
Questions:
I know that image plane will be perpendicular to our eyes but will it be always the case why?
How do we prove that distance of vanishing point from bottom of the image is same as height of our eye that is $X$= $H$ ?
Context-2: In this video from $8:10-8:50$ they explain that by telling that the image plane and the plane containing eye point and the line $L_1$ is perpendicular to line $L_1$ as the rails scenery is observed going to infinity but why So ? I think its doesnt need to be as such one can observe in the video figure itself it seems a bit tilted and not exactly $90°$ .
- Further more even the rail lines which are not exactly at $90°$ to the image plane , still their projective images(the two lines at some same acute angle kinda) in the image plane they still meet at a point which is collinear with the intial vanishing point and parallel to the bottom plane why thats is also (i.e.both meeting at horizon why ??)
The image plane is usually chosen to be a flat plane perpendicular to the gaze direction. This is because the image is usually later reproduced as a flat photo or on the flat screen of a television, and we generally look at those head-on. There are cases where you have an image plane that is not perpendicular, such as when you want a Trompe-l'œil effect in a painting on a floor or ceiling which looks great when viewed from a particular vantage point. If you are creating an image for use in a 3d virtual tour, then you would use a cylindrical or spherical image surface rather than a plane.
Your second question is a little confused. The image that is eventually reproduced on a photo or screen is generally a relatively small rectangle, and is not an infinite plane. So it is only a small section of the plane shown in your drawing, like a window in a wall, and won't go all the way to the ground. The bottom part that you can see through the window / in the picture will then be a point on the ground that is further away than where the foot of that wall is, so it doesn't really make sense to measure from the bottom of the image.
What you can say is that if you are looking horizontally in the distance, your gaze parallel to the ground and train tracks, then you are looking at the horizon. The horizon is in the centre of your view, because if you lift your gaze up by any amount you see the sky and if you drop your gaze by any amount you see the ground. Presumably the horizon is also approximately in the centre of your image window, but you might of course crop the resulting photo and essentially block off part of the window, so the horizon might not be exactly in the middle.
To prove that the vanishing point of the rails is a point on the horizon in your image window, you can consider two planes - the plane containing your eye and the left rail, and the plane containing your eye and the right rail. These flat planes intersect in a line parallel to the rails and through your eye, namely the line of your gaze to the horizon. These two planes and the ground enclose an infinitely long triangular tube, with the rails and your gaze as its edges.
Now look at how this triangular tube intersects the image plane. It forms a triangle, where the two sloping sides are the images of the rails, and the apex is the point on the horizon that you are looking at. So in the image the rails meet in a point on the horizon.
Note that if you are not looking at the horizon but a bit up or down from that, then the image plane follows that and also tilts up or down. Nevertheless, the two planes containing your eye and each rail still form the same triangular tube. If this tube still intersects the image window in the now tilted image plane, that intersection is still a triangle. The image of the rails will still converge to a vanishing point on the horizon, even though this is no longer the centre of your view but in your peripheral vision.