Why does perspective skew some lines and not others?

[ed2288]

Imagine looking at a set of x-y-z axes, from directly along the z-axis.

Say in this image, the road is the line z=0.

Why is it, that only lines on or parallel to the z-axis get skewed? All the surfaces that are in or parallel to the x-y plane remain undistorted, and their angles are at 90 degrees.

Just to highlight the point, imagine if this image was extended to include more building and streets at the side. The streets at the edges would get increasingly more skewed but again, only in the z-axis. What is it that is causing this increase in skewing?

 

[mfb]

That is not a perfect 3D sketch of cuboids (blocks). Those lines should converge somewhere outside the image, too, but you can neglect this for most images.

On the other hand, a real view is not planar, you need some projection technique to show it on a computer monitor. If you project it on a XY-plane close to us, shapes in XY-direction stay the same. Shapes with a Z-component get distorted.

 

[ed2288]

That is not a perfect 3D sketch of cuboids (blocks). Those lines should converge somewhere outside the image, too, but you can neglect this for most images.

In the above image, all the z-direction lines are converging at the vanishing point, as we would expect. Are you saying the x or y lines should be skewed too and converge too?

 

[mfb]

It depends on the way you draw an image.

Some more stuff:
http://en.wikipedia.org/wiki/Vanishing_point
http://en.wikipedia.org/wiki/Curvilinear_perspective

 

[I like Serena]

They are not actually lines - they are curves.

In the direction that you are looking, lines start out as parallel, but then curve toward a vanishing point.
This is usually simply drawn as straight lines, since the part where they are parallel are outside the range of the picture frame.

Perpendicular to the direction you are looking, lines also start out as parallel, and within the range of the image, they remain close enough to parallel to draw them as parallel lines.
If you extend your the range of your picture frame, they also curve towards a vanishing point.

 

[ed2288]

I'm still confused unfortunately. Let's take a very simple example. I know from life experience, that if I have a perfect square of green paper, lie it flat on a table, and put it a little distance in front of me, then look down on it - this is what my eyes will see:

http://www.englishtap.com/library/maths/images/trapezium.png

I suppose my question boils down to: how do I go about actually proving this?
How do I prove that the sides will slant inwards, the front will be longer and the back will be shorter?

 

[mfb]

You can model yourself as a point in space and the paper/monitor as 2D sheet somewhere in space. Project everything onto this sheet along rays which originate at you.
For objects parallel to the paper, this will conserve angles and relative lengths. For lines with a component perpendicular to the paper/monitor, this is not true.

 

[I like Serena]

It's more like this.

 

[ed2288]

Project everything onto this sheet along rays which originate at you.
For objects parallel to the paper, this will conserve angles and relative lengths.

Thanks for your help. So I've started to draw a ray diagram to try and help understand what you said. The green line is a side view of the square of paper on the table.

Am I going in the right direction? From this diagram, how can I now get what the eye is seeing?

 

[mfb]

If you want a picture of a town, it would be more useful to hold the paper vertically.

Something like this:

 

[Drakkith]

Ed, your drawing will not help because it does not correctly show how the light works. I can't draw, so I hope this explanation will help a little.

It really boils down to the fact that objects further away look smaller. Since a road is generally the same distance across at all points, the distance from one side to the other APPEARS to gradually get smaller as the distance increases, just like how a moving object appears to get smaller as it recedes from you.

If you really want to understand how the image formation works you must learn a bit about basic optics. Unfortunately I don't have a good link for you.

 

[ed2288]

Ok so I think I've now convinced myself that if I were to shine a light at an opaque square (slanted so it casts a shadow), then the shadow it would cast would be narrower at the top than it is at the bottom. Is this diagram that I've drawn physically accurate?

Is this the same principle that is occurring in perspective drawing? If so, how does my diagram relate to drawing an object with perspective? I've heard people talk about "transforming" 3d things onto a 2d surface but not too sure what this means.