The Effects of Perspective
There is no way we can throw away something as important as a dimension without there being some consequences and there are some quite significant consequences, but we tend not to think about them.
If I look at this scene of parallel railway lines, they clearly appear to converge and we expect that. Looking at this scene we know that they are parallel lines and this comes from our everyday experience with what a three-dimensional world looks like, but it’s kind of surprising that things that are parallel in the world appear to us to be converging and we don’t give it any thought at all. It is unsurprising to us.
Here is another example, looking up at a bunch of trees and they appear to be converging even though these trees are all parallel to one another, emerging vertically from the ground plain.
Here is another example of the ferris wheel and we know that the ferris wheel is round. Now in this particular image, because I have looked at it obliquely, it clearly appears to be elliptical.
So what’s going on here? Well, mathematically the perspective projection maps lines to lines, but it doesn't preserve angles. That mean a parallel lines in the world will not necessarily appear to be parallel in an image. There are cases when there will be, but in general, parallel lines do not remain parallel after the projection.
The other shape mapping perceptive projection does, is it maps conics to conics. So a conic, short for conic section is the shape that you get by slicing through a solid cone and though shapes can be a circle, an ellipse, a parabola or a hyperbola.
What are the consequences of representing a three-dimensional scene using only two-dimensions? The appearance of parallel lines converging and circular objects being elliptical should be surprising but we take this for granted.
High school mathematics
This content assumes an understanding of high school-level mathematics, e.g. trigonometry, algebra, calculus, physics (optics) and some knowledge/experience of programming (any language).