Let me explain to SuperSentient what he (and EVERY OTHER FE'er) is failing to understand...
WHAT YOU'RE SAYING:
Take the train-track example. You see two lines (which in the real world are parallel) and you find the intersection line between them - then claim that this distance is closer than infinity.
You claim this because you can make a picture of some train tracks - draw a line down each rail - and observe that they intersect somewhere a just a little beyond what the camera lens shows.
You SEE with your eyes that the two lines obviously meet - not very far (it seems) beyond the limits of the camera's lens or our visual acuity. You never actually DO see the tracks meeting - but you presume that they must because you can draw the two lines on top of the photo - and it looks like they meet somewhere just fractionally beyond the resolution of the camera.
So you conclude that parallel lines meet at some distance from the eye like maybe 10 miles or 100 miles or something...and base all of FET's optical properties on this, seemingly reasonable, claim.
BUT THERE IS A PROBLEM:What you are MISSING (and it's kinda subtle - but VERY important) is shown in the photo below. (You can show it with the horizontal railroad ties on train tracks too - but they are rather closely spaced and that makes it hard to get a strong visual impression.)
Now, let's talk about the building on the left. You can draw a line at roof level and another one at ground level and see that they intersect...but what you're claiming is that this intersection is happening at some distance from us - 10 miles, 100 miles - whatever. I'm claiming that...yes, the lines obviously intersect - but the DISTANCE at which they intersect is infinite.
Now I hear you complaining.
But let's look at the vertical columns and the windows of that building - they are regularly spaced out there in the real world...maybe a few meters apart.
But look CAREFULLY at them in the picture. Do you see that the separation between the columns
in the photograph is smaller in the distance than it is near to the camera?
On my computer screen, the first two columns are about 3 centimeters apart. (Depends on the size of your computer screen)
The next two columns are only around 2 centimeters apart, and the next pair are maybe 1.2 cm apart...the distance between them gets smaller and smaller.
The last two columns are less than 2 millimeters apart - you can hardly see a gap between them.
So here is what's happening. The closer together the roof line and the pavement line get - the more and more compressed the distance INTO the scene the picture becomes. The horizontal spacing between columns get smaller and smaller.
What's actually happening is that as the VERTICAL distance is being compressed by "perspective" - so is the distance INTO THE SCENE. So when the roof/pavement lines would be VERY close to touching, they'd be representing something a billion miles into the scene - and at the precise point where the "perspective lines" touch - we are INFINITELY far into the scene.
Another way to think of this is that just as the left/right and up/down spacing of things shrinks with perspective, so does the near/far distance.
X, Y and Z are *ALL* shrinking as we go further into the distance.
So when the X or Y distance hits zero - so the Z spacings of our columns ALSO hits zero - and you get an infinite number of columns packed together into that last screen pixel as we approach the vanishing point.
An THAT is why parallel lines meet at INFINITE Z and not 10 miles or 100 miles as FE'ers seem to believe.
I can quite understand why this fooled you - and I have to say that it hurts my brain even thinking about it. But regardless - this is what truly happens.
USING MATH:This is MUCH clearer if you do it with math. Perspective is used all the time in photo-realistic 3D graphics - which is what I do for a living.
x' = k.x / z
y' = k.y / z
(x,y,z) is a point in the real/virtual world (in a coordinate system where the "camera" is at (0,0,0) and z is distance away from the camera).
(x', y') is the point on the screen where that point ends up (in a coordinate system where the center of the screen is (0,0)).
k is a constant that relates to the 'lens' of the virtual camera and the size/resolution of the screen.
These two equations are built into every 3D computer game - every simulation, every CGI movie. It's so fundamental that it's even built into the hardware of 3D graphics cards in your PC.
We do this because it's the only formula that produces realistic pictures.
So if one railroad rail is 1 meter to the right of the camera (x=+1) - then at what value of 'z' does it arrive at the vanishing point?
x' = 0
x = +1
What is 'z'?
0 = k . 1 / z
z = k / 0
...hmmm - that's a problem because you can't divide by zero without getting an infinity for 'z'.
And that's the mathematical reason why parallel lines meet at infinity under perspective.
DERIVATION:I can even derive those equations for you - from first principles - using a 'pinhole camera' analogy:
I drew this diagram for a thread about perspective and sunsets - but forget for a moment that this is about the sun...pretend that the blue line in the diagram a tree or something in the far distance. A pinhole camera is just a box with a pinhole punched in the front and a photographic plate at the back - it's the simplest possible camera - and it produces upside-down photographs.
The law of similar triangles says that:
Himage / Dimage = Hsubject / Dsubject
Himage = Hsubject x Dimage / Dsubject
The height of the image is the height of the subject (the tree) multiplied by the distance from the pinhole from the film and divided by the distance to the subject (the tree).
This is actually the exact same perspective equation that I used before:
y' = k.y / z
* k = Dimage
* y' = Himage
* y = Hsubject
* z = Dsubject
QED.
The only way to discount this derivation of the math for perspective is to deny that light travels in straight lines - or to deny that the method of similar triangles is valid.
So the pinhole camera is proof of the equations - and the equations are proof of the laws of perspective.
The observation that perspective operates in Z as well as in X and Y is further proof that FET's concept of finite vanishing points is untrue.
I think this argument is completely watertight - and so far, nobody in FE land has been willing to even discuss it. Tom just says "it's just a diagram"...which is a rather fundamentalist anti-science, anti-math position - and if he were honest and consistent then he'd have to call "bullshit" on all of Rowbothams diagrams too!