rotation of 3D XYZ points by an ijk unit vector

I have a XYZ co-ordinate points that I would like to rotate around the origin from one vector of a defined plane to another. For example I have a unit surface vector of a plane of [0.9997 -0.0240 -0.0053] and I would like to rotation my points so that this planes normal is parallel to the X axis [1 0 0].

How can I take my [X Y Z] co-ordinates and rotate them in 3 dimensions from vector [0.9997 -0.0240 -0.0053] to [1 0 0]

Answer 1

[Bruno Luong]

% source and target unit vectors
u =  [0.9997; -0.0240; -0.0053] ; u = u/norm(u);
v =  [1; 0; 0]; v = v/norm(v);

% Compute 3 x 3 rotation matrix R so that R*u is v
% see here foe ref of angle calculation
% https://www.mathworks.com/matlabcentral/answers/101590-how-can-i-determine-the-angle-between-two-vectors-in-matlab?s_tid=srchtitle
M = makehgtform("axisrotate",cross(u,v),2*atan(norm(u-v)/norm(u+v))); 
R = M(1:3,1:3);

XYZ = [u, randn(3,6)], % (3 x n) your n data point coordinates

XYZ = 3×7

0.9997 0.6224 0.2960 1.1053 0.4341 0.6850 0.5836 -0.0240 -1.6776 -1.3716 -0.9371 1.1838 -0.0971 -0.5146 -0.0053 -1.1380 -1.2065 -0.3211 0.1301 -0.7294 -0.0976

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    XYZ_Rotates = R*XYZ % observe the first vector u after rotation becomes v

XYZ_Rotates = 3×7

1.0000 0.6685 0.3352 1.1291 0.4049 0.6910 0.5963 -0.0000 -1.6621 -1.3641 -0.9103 1.1939 -0.0806 -0.5004 -0.0000 -1.1346 -1.2049 -0.3151 0.1324 -0.7257 -0.0944

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[Bruno Luong]

Note that the choice here of axis rotation vector r := cross(u,v) is not unique; but it's the one that implies a smallest rotation angle.

Any unit vector that has the same distance to u and v can be setected as axis of rotation.

For example normalized (u+v)/2. The angle here is pi, the largest possible choice.

% source and target unit vectors
u =  [0.9997; -0.0240; -0.0053] ; u = u/norm(u);
v =  [1; 0; 0]; v = v/norm(v);

% Compute 3 x 3 rotation matrix R so that R*u is v
M = makehgtform("axisrotate",(u+v)/2,pi); 
R = M(1:3,1:3);

XYZ = [u, randn(3,6)], % (3 x n) your n data point coordinates

XYZ = 3×7

0.9997 0.3643 -1.2357 0.2286 -0.0913 0.7236 -0.7353 -0.0240 0.3797 -1.1702 -0.3897 -1.1961 -0.6616 -0.5746 -0.0053 0.7244 0.2336 -0.6793 0.9005 1.2260 -1.4620

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XYZ_Rotates = R*XYZ % observe the first vector u after rotation becomes v

XYZ_Rotates = 3×7

1.0000 0.3513 -1.2084 0.2415 -0.0674 0.7327 -0.7135 -0.0000 -0.3883 1.1996 0.3841 1.1980 0.6441 0.5920 0.0000 -0.7263 -0.2271 0.6781 -0.9001 -1.2298 1.4658

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