4D to 3D Projection Animations

by John Dick

(5) Wheel Animation with added WY and XY Rotations

This animation shows added 180 degree WY and XY rotations of the viewpoint in 4D (WXYZ). The thumbnail shows the configuration halfway through the WY rotation where the W dimension is shown and the Y direction is "away".

This Animation

If you're having trouble following this one--during the WY rotation the top (orange) track is rotating "towards" you, eventually ending on the bottom.

Placeholder text! While it seems apparent that we are observing a simple XY rotation in 3-space, the rotation preserves both Z (as we would expect) and W. Thus it is actually not a rotation about the Z axis, but about the WZ plane.

Discussion (continued)

Placeholder text! A simple way to allow my 2D person to view the picture, though, might for me to draw only dots on the page, so that he/she could see around the dots to those "inside" and could, in a natural way, infer the lines that those dots implied, and then also the 2D forms that those lines contained. You can see where I'm going with this -- a line drawing is our equivalent to the 2D person's dotted drawing, and in a natural way, we infer not only the surfaces defined by these lines, but also the volumes implied by those surfaces.

Placeholder text! The viewer might very well only see the picture's frame, or leaving off the frame, only the edges of every shape that we've drawn, the interiors being blocked from view. It's now clear that a complete view could only be possible if everything were semi-transparent, allowing her/him to perceive shading, shadows, and all the aspects of that drawing that allow us 3D folk to view into that drawing to infer what is happening away from the viewer, in the 3rd dimension.