rotation - Rotate normal vector onto axis plane in 4D -


the problem need solve rotate 4-simplex given in 4d on hyperplane normal vector (1, 1, 1, 1) can draw in 3d. instance need know rotation regular 1 having vertices e_i (that coordinate vectors), , sub-simplices after division.

in order understand problem, let's go 1 dimension back. if have 3-simplex in 3d on hyperplane normal vector (1, 1, 1) here (http://upload.wikimedia.org/wikipedia/commons/thumb/3/38/2d-simplex.svg/150px-2d-simplex.svg.png), 1 can follow idea of nosredna question

rotate normal vector onto axis plane

it works fine in 3d, in 4d there no cross products, cannot extend answer question. on other hand using rotation matrices managed rotate simplex around x axes -45 degree, rotating around y axes around 35 degree (atan(sqrt(2)/2) using coordinate rotation matrices (http:// upload.wikimedia.org/math/2/8/5/2851c9dc2031127e6dacfb84b96446d8.png).

i tried calculate rotation matrix axes rotations in http://ken-soft.com/2009/01/08/graph4d-rotation4d-project-to-2d/ not find out should angles use. used r=rotxu*rotyu*rotzu angles pi/4, -atan(sqrt(2)/2, , -pi/6, looked good, somehow result wasn't ok.

sorry, not put images directly i'm newbie...

thank answer!

there no rotation axes in 4d, same reason there no cross product: group of 4d rotations 6-dimentional, , space rotating 4-dimentional. imagine, example, simultaneous rotation in xy plane , zt plane: has no non-zero stationary vectors, , therefore no axes.

the appropriate thing work usual transormation matrix, applicable in dimention n:

[ a11 ... a1n d1 ] ... [ an1 ... ann dn ] [ 0 ...   0   1  ]

here d1 ... dn represent translations, , nxn submatrix aij represent rotations, dilations, projections, , mirroring. limit rotations make matrix orthogonal: product own transpose should unit matrix. common practice n=2 , n=3, same n=4.

to find appropriate rotation matrix in case write down add requirement entire 4th row of orthogonal 4x4 matrix zero, , give bunch of solutions, each being acceptable answer question.


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