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- From: wijkstra@fwi.uva.nl (Marcel Wijkstra (AIO))
- Newsgroups: comp.graphics.visualization
- Subject: Rotation in nD space
- Message-ID: <1992Aug19.100559.13778@fwi.uva.nl>
- Date: 19 Aug 92 10:05:59 GMT
- Sender: news@fwi.uva.nl
- Organization: FWI, University of Amsterdam
- Lines: 65
- Nntp-Posting-Host: job.fwi.uva.nl
-
- In 3D, you can rotate an object by applying a 3x3 transformation
- matrix to all points. Rotation around the X axis over angle A
- requires the matrix:
- 1 0 0
- 0 C S where C = cos(A)
- 0 -S C and S = sin(A)
-
- Note: since I do not need translations, I do not need a 4x4 matrix and
- homogenous coordinates.
-
- I am trying to visualize objects in a higher space, e.g. a hypercube
- in 4D. To do so, I discard the fourth coordinate and treat it as a
- regular 3D object, i.e. use perspective projection to get it on the
- screen. A non-rotated hypercube thus results in a boring picture of
- a plain cube. Things get more interesting if the hypercube is rotated
- in 4D space. Then you would probably see something like two disformed
- cubes inside eachother, with their corners connected.
-
- The question is: how do I 'rotate' an object in 4D? Or even: what is
- rotation in 4D? I proved (I think) that true rotation of a 4D object
- around one of the four principal axes is impossible. In other words:
- there exists no matrix M:
-
- 1 0 0 0
- 0 a b c
- 0 d e f
- 0 g h i
-
- that rotates a point (xyzt) around the X axis over a certain angle A,
- such that applying MM is the same as rotating over 2A. Note: such an
- M could exist, if you use complex numbers, but I doubt if that is
- allowed.
-
- The only thing I could come up with, is to use three matrices
-
- 1 0 0 0 1 0 0 0 1 0 0 0
- 0 C S 0 , 0 S 0 C and 0 1 0 0
- 0 -S C 0 0 0 1 0 0 0 C S
- 0 0 0 1 0 C 0 -S 0 0 -S C
-
- for rotations around the XT, XZ and XY plane. Extended to an nD space,
- this approach results in (n^2 + n)/2 different rotation (n-2 D) spaces.
-
- My final questions are:
-
- (1) WHAT IS 'ROTATION' IN ND SPACE EXACLY (IF IT IS POSSIBLE AT ALL)?
- This question comes merely from the difficulties I have in abstract
- thinking, i.e. I haven't got the slightest idea what an object in
- nD space would 'look' like.
-
- (2) IS THE ABOVE APPROACH FOR ROTATIONS IN 4D SPACE CORRECT?
- If so, then rotation of a left shoe (being a 3D object, i.e. with
- t=0 for all points) can result in...a right shoe.
-
- (3) ARE THERE ANY OTHER COMMENTS ON THE STATEMENTS I MADE?
- Please, ALL comments/information/flames (yes, even flames) are very
- welcome.
-
- Thanks for reading,
- Marcel.
- --
- X Marcel Wijkstra AIO (wijkstra@fwi.uva.nl)
- |X| Faculty of Mathematics and Computer Science
- X University of Amsterdam The Netherlands
- ======Life stinks. Fortunately, I've got a cold.========
-