The unit normal vector is controlled by the angles
,
, and
. The first two are just like the same angles in regular 3D spherical coordinates:
, with
, is the angle to the positive
axis (or more precisely, the


hyperplane), and
, with
, is the angle to the positive
axis (or rather the

plane). The angle
, with
, is the angle formed by the normal and the positive
axis and is analogous to
in that way. If
, then
and
have no effect, just as when
, the angle
has no effect. When the normal is changed, it appears that the fourcube rotates. But it is not moving; it is your view point that is moving. Of course, the two kinds of motion look the same.
Animate the distance
and view the series of sections orthogonal to the normal from one end to the other. The coordinates of the vertices of the fourcube are
, so the vertices are two units from the origin. The displacement
is the displacement of the hyperplane, and thus the section consists of all points of the fourcube, the projections of whose coordinate vectors onto the normal are equal to
times the normal.
Each cell (bounding 3D cube) of the fourcube is drawn with the same color as the face of its section by the hyperplane. By playing with the opacities, you can see the relationship between the cubecube's cells and the intersection.
The projection displayed is from a point 3 to infinity units away from the origin along the line through the origin parallel to the normal (infinity gives a parallel projection). The fourcube and section are projected onto the hyperplane through the origin and orthogonal to the normal. A 3D coordinate system for this hyperplane is attached to the normal and rotates around with it as the normal is changed. This coordinate system will "precess": as the normal moves around (you need to change at least two angles) and moves back (along a different trajectory) the section will appear to have changed position. But it has not. It is the coordinate system used to project the section onto the screen that has changed.
Some important angles:
,
,
,
.
Snapshot 1: cell first (normal parallel to center of a cubecell): cube
Snapshot 2: face first (normal parallel to center of a 2D face): rectangular box with a square face
Snapshot 3: edge first (normal parallel to midpoint of edge): equilateraltriangular prism
Snapshot 4: vertex first (normal parallel to midpoint of vertex): regular tetrahedron
Snapshot 5: hexagonal prism
Snapshot 6: regular octahedron