Have you ever wondered why perspective drawing using vanishing points works ?

What are these mysterious points that all lines point towards ?, Where are they ?

Well I can do better than just tell you where they are – I can show you what it looks like from there ! All vanishing points are actually in the same place. This is because they are, in fact, none other than our point at infinity.

If we take a simple orthogonal grid like this :

and invert it with respect to a sphere at the centre of the grid, we get something like this :

The video starts inside the central cube of the grid (which has now been turned completely inside out) and moves towards the origin of the inversion sphere. Once there the view rotates and we see that what looks at first strangely similar to a normal perspective view of a straight orthogonal grid, ( compare it with this video of the view inside a normal grid) but if you pause it a moment and take a closer look, you will see that everything is in fact reversed. In a normal perspective all the lines appear to converge as they get farther away. Here they are converging as they get closer to us. This is because we are at the vanishing point!

As the camera turns around a second time it also moves away from the origin and the distortion becomes more apparent.

P.S. I know Googlevideo have had a few server problems recently so here are the same videos on Youtube

[see chapter 4 there, it's the best animation I know of]

except in this case all the vertices appear straight from the central vantage point. But the similarity is striking – I am no geometer. Is it coincidence, or is inverted 3-space connected to 4-space in some way?

That is a very good point – inverted 3-space is indeed connected to 4-space.

If you do a reverse stereographic projection from flat 3-space to a 3-sphere in 4-space, then a 4-dimensional rotation*, then project back down to flat 3-space, it is equivalent to inversion in a 2-sphere plus a reflection.

If this is hard to follow,look at this great animation, which at 1.53 shows something similar in one dimension lower, so 2D space is projected to a 2-sphere and rotated in 3D.

I have actually made a script which performs these 4D projections and rotations, and I’ll post some stuff on it as soon as I can.

August 2, 2007 at 3:09 pm

Daniel. This is pretty impressive. I may have to study it a little closer. Nice work. Be interesting to see if you do anything more in this vein.

- Jonathan

October 14, 2008 at 2:09 pm

It looks very similar to the projections of 120-cell and 600-cell 4-Dimensional polyhedra into 3-space…

http://www.dimensions-math.org/Dim_regarder_E_E.htm

[see chapter 4 there, it's the best animation I know of]

except in this case all the vertices appear straight from the central vantage point. But the similarity is striking – I am no geometer. Is it coincidence, or is inverted 3-space connected to 4-space in some way?

October 14, 2008 at 3:12 pm

That is a very good point – inverted 3-space is indeed connected to 4-space.

If you do a reverse stereographic projection from flat 3-space to a 3-sphere in 4-space, then a 4-dimensional rotation*, then project back down to flat 3-space, it is equivalent to inversion in a 2-sphere plus a reflection.

If this is hard to follow,look at this great animation, which at 1.53 shows something similar in one dimension lower, so 2D space is projected to a 2-sphere and rotated in 3D.

I have actually made a script which performs these 4D projections and rotations, and I’ll post some stuff on it as soon as I can.

*actually, a particular type of 4D rotation,

you can find more technical stuff on this here