matlab - Change from one cartesian 3D co-ordinate system to another by translation and rotation -

there 2 reasons me ask question:

1. i want know if understanding on issue correct.
2. to clarify doubt have.

i want change co-ordinate system of set of points (old cartesian coordinates system new cartesian co-ordinate system). transformation involve translation rotation. plan do:

with respect image have set of points in xyz coordinate system (red). want change respect axes uvw (purple). in order so, have understood there 2 steps involved: translation , rotation.

when translate, change origin. (say, want uvw origin @ (5,6,7). then, points in data, x co-ordinates subtracted 5, y 6 , z 7. doing so. set of translated data.)

now have apply rotation transform (on translated data). rotation matrix shown in image. values ux, uy , uz co-ordinates of point on u axis has unit distance origin. similarly, values vx, vy , vz coordinates of point on v axis has unit distance origin. (i want know if right here.) wx, wy, wz calculated ((normalized u) x (normalised v))

(also, if serves purpose, let know using matlab.)

edit:

i have set of 42 points in 3d (42 x 3 matrix a) want first point considered origin of uvw plane. values of first point translation vector. correct?

next, calculate rotation vector: according requirement, 6th row of matrix a has u axis while 37th row has v axis. consequently, vector u (1st row minus 6th row) of matrix a. while vector v (1st row minus 37th row) of matrix a.

the first row of rotation matrix vector u/|u| (normalized). second row vector v/|v| (v normalized). third row (u x v) . right here?

given information, how can calculate value of wx, wy , wx. how can calculate 3rd row of rotation matrix r?

since have u , v, 2 basis vectors of orthonormal uvw system, w basis vector cross product of u , v. cross product gives out vector perpendicular operands; hence w = u × v. components of w fill in third row of rotation matrix.

is approach correct?

the order of transforms matter; changing order lead different results. when doing transformations of systems, scaling , rotation tackled first , translation dealt lastly. reason rotation respect origin. if new system isn't on old one's origin applying rotation rotate new system not around own origin around old system's origin. see rightside case of figure 3-4 on this page understand difference happen if it's not on origin; imagine pot uvw coordinate system.

think of both coordinate systems being super-imposed (laid 1 atop other). when rotate uvw system respect origin of xyz, end effect of rotating uvw w.r.t own origin. once rightly oriented, can apply translation it. however, if you'd translated, rotating lead translated rotation.

if you're using column-vector convention tr order i.e. rotation followed translation. if you're using row-vector convention rt order, again order rotation followed translation.