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Hi folks.
Firstly, has anyone looked into it? (4D geometry in POV-Ray that is.)
Part of the problem is that the fourth dimension is a little bit
mind-numbing to think about!
Anyway, I'd like to use POV-Ray to draw animations of a hypercube
rotating in 4 dimensions, orthographically projected into 3 dimensions.
If I'm not mistaken, a hypercube can be defined as the set of all points
that satisfy
-k < (x, y, z, w) < +k
It can also be defined as the intersection of 8 hyperplanes - and this
lents itself for to rotation.
Suppose we have a hyperplane
Ax + By + Cz + Dw - E = 0
Then the "normal" would be <A, B, C, D> and the distance from the origin
would be E. (In 2D, a normal defines a line. In 3D, a normal defines a
plane. So presumably in 4D a normal defines a hyperplane...)
If (say) <A, B, C, D> = <0, 1, 0, 0>, then an orthographic projection
into 3D gives us an ordinary plane with normal <0, 1, 0>.
However...
As far as I can tell, if you rotate that hyperplane off the axis even
slightly (in such a way that D <> 0), then the 3D projection of that
hyperplane fills the entire 3D space. That would mean that if I do any
double rotation by a small angle, *all* 8 projections would fill all of
3D space, resulting in... well... nothing.
Question: is the intersection of the 3D projections of the 8 hyperplanes
equal to the 3D projection of the intersection of the 8 hyperplanes?
(Question: Does that even make sense?!)
Thoughts / suggestions?
Andrew @ home.
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