3 ms·

So, what's going on, is that one of the dimensions in the game world is getting switched out for a fourth dimension. The visual transformation that you see isn'

by lmkg 17y ago

So, what's going on, is that one of the dimensions in the game world is getting switched out for a fourth dimension. The visual transformation that you see isn't very clear about what's going on, and I guess it's basically not possible to be clear about this in a rectilinear coordinate system. I'll try to explain. At 0:12, the game world has 3 axis, aligned with the sides of the blocks. By convention, let's say the vertical axis is the Z-axis, that the one going up and to the right (on the screen) is the X-axis, and the one going up and to the left is the Y axis. There's also a "hidden" axis, which I'll call the W-axis, which can't be displayed (for obvious reasons). At 0:13, the player switches out the Y-axis for the W-axis. We can no longer see the extent of objects in the Y direction. However, we can now see the extent of objects in the W-axis direction. For example, in the XY-plane, the ground was all green, but in the XW-plane it looks stripped with green and brown. This means there are a series of XY-planes sitting "next to" each other in W-space, alternating in colors. The character was originally sitting on one of the green ones. Now that the Y-axis was switched for the X-axis, we see X-axis "slices" of these XY-planes, arranged along the W-axis. Note that the big rock structure is located on the W-axis only in the XYZ 3-D "slice" that the character originally occupied. The character then proceeds to move the wooden ring along the W-axis to a different XZ-slice, where the rock structure isn't in the way. At 0:18, he switches axes back to the XYZ plane, although this isn't strictly necessary. You'll notice he's now on an XYZ slice that, in W-space, has brown ground. The rock structure isn't in the way because it's on a different location on W-axis than everything on-screen (it's displayed as transparent, for reference). He moves the structure along the X-axis, past the rock obstruction. He then switches back to XWZ-space, moves the wooden thing back along the W-axis again, and switches back to the original coordinate system.

MikeCapone 17y ago
Thanks for taking the time to explain. I think I get it better now, but it's probably one of these things that I'll have to try for myself to really understand.
- zackattack 17y ago
  The Cartesian plane is in ℝ^2. Any point can be expressed as a combination of (x,y) where both x,y ∈ ℝ. That means that both x and y are members of the set of real numbers. A 3D graph is in ℝ^3. Any point can be expressed as (x,y,z), where x,y,z ∈ ℝ. A 4D graph is in ℝ^4. Any point can be expressed as a combination of (x,y,z,w), where x,y,z,w ∈ ℝ. Unfortunately, human vision is limited to three dimensions, so only four different perspectives are available (4 choose 3): (x,y,z), (x,y,w), (y,z,w), and (x,z,w). Moreover, the 3D must be collapsed into a 2D for presentation on the screen.

by lmkg 17y ago

MikeCapone 17y ago

Thanks for taking the time to explain. I think I get it better now, but it's probably one of these things that I'll have to try for myself to really understand.

zackattack 17y ago
The Cartesian plane is in ℝ^2. Any point can be expressed as a combination of (x,y) where both x,y ∈ ℝ. That means that both x and y are members of the set of real numbers. A 3D graph is in ℝ^3. Any point can be expressed as (x,y,z), where x,y,z ∈ ℝ. A 4D graph is in ℝ^4. Any point can be expressed as a combination of (x,y,z,w), where x,y,z,w ∈ ℝ. Unfortunately, human vision is limited to three dimensions, so only four different perspectives are available (4 choose 3): (x,y,z), (x,y,w), (y,z,w), and (x,z,w). Moreover, the 3D must be collapsed into a 2D for presentation on the screen.