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set_random_state

? set_random_state;

Calculate

set_random_state

? set_random_state;

Calculate

set_random_state

Run Example
(%i1)build_info();

Maxima version: 5.21.1
Maxima build date: 7:25 9/18/2010
Host type: x86_64-unknown-linux-gnu
Lisp implementation type: GNU Common Lisp (GCL)
Lisp implementation version: GCL 2.6.7

(%o1) 
(%i2) /* compute angles for intrinsic Tait-Bryan angles * on axes y-z'-x'', conventionally named pitch, yaw, and roll * (following terminology on Wikipedia page for "Euler angles") * via method expounded by G. Slabaugh: * http://www.staff.city.ac.uk/~sbbh653/publications/euler.pdf * * Argument R is a 3x3 rotation matrix. * Return value is a list of two triples, [[pitch1, yaw1, roll1], [pitch2, yaw2, roll2]], * because there are always two solutions. */intrinsic_yzx (R) :=  if equal (R[2, 1], 1) or equal (R[2, 1], -1)    then intrinsic_yzx_yaw_pi2 (R)    else block ([pitch1, pitch2, yaw1, yaw2, roll1, roll2],      yaw1 : asin (R[2, 1]),       yaw2 : %pi - yaw1,      pitch1 : atan2 (-R[3, 1]/cos(yaw1), R[1, 1]/cos(yaw1)),      pitch2 : atan2 (-R[3, 1]/cos(yaw2), R[1, 1]/cos(yaw2)),      roll1 : atan2 (-R[2, 3]/cos(yaw1), R[2, 2]/cos(yaw1)),      roll2 : atan2 (-R[2, 3]/cos(yaw2), R[2, 2]/cos(yaw2)),      [[pitch1, yaw1, roll1], [pitch2, yaw2, roll2]]);
(%o2) intrinsic_yzx(R) := if equal(R    , 1) or equal(R    , - 1)
                                    2, 1               2, 1
 then intrinsic_yzx_yaw_pi2(R) else block([pitch1, pitch2, yaw1, yaw2, roll1, 
roll2], yaw1 : asin(R    ), yaw2 : %pi - yaw1, 
                     2, 1
                - R         R                         - R         R
                   3, 1      1, 1                        3, 1      1, 1
pitch1 : atan2(---------, ---------), pitch2 : atan2(---------, ---------), 
               cos(yaw1)  cos(yaw1)                  cos(yaw2)  cos(yaw2)
               - R         R                        - R         R
                  2, 3      2, 2                       2, 3      2, 2
roll1 : atan2(---------, ---------), roll2 : atan2(---------, ---------), 
              cos(yaw1)  cos(yaw1)                 cos(yaw2)  cos(yaw2)
[[pitch1, yaw1, roll1], [pitch2, yaw2, roll2]])
(%i3) intrinsic_yzx_yaw_pi2 (R) :=  block ([pitch1, pitch2, yaw1, yaw2, roll1, roll2],    yaw1 : %pi/2,    roll1 : 0, /* could be anything */    pitch1 : -roll1 + atan2 (R[3, 2], R[3, 3]),    yaw2 : -%pi/2,    roll2 : 0, /* could be anything */    pitch2 : roll2 + atan2 (R[1, 3], R[1, 2]),    [[pitch1, yaw1, roll1], [pitch2, yaw2, roll2]]);
(%o3) intrinsic_yzx_yaw_pi2(R) := block([pitch1, pitch2, yaw1, yaw2, roll1, 
               %pi
roll2], yaw1 : ---, roll1 : 0, pitch1 : - roll1 + atan2(R    , R    ), 
                2                                        3, 2   3, 3
       - %pi
yaw2 : -----, roll2 : 0, pitch2 : roll2 + atan2(R    , R    ), 
         2                                       1, 3   1, 2
[[pitch1, yaw1, roll1], [pitch2, yaw2, roll2]])
(%i4) set_random_state (make_random_state (1));
(%o4)                                done
(%i5) random_3d_unit () := (random_3d (), %%/norm(%%));
                                                       %%
(%o5)             random_3d_unit() := (random_3d(), --------)
                                                    norm(%%)
(%i6) random_3d () := makelist (random (10), i, 1, 3);
(%o6)            random_3d() := makelist(random(10), i, 1, 3)
(%i7) norm (v) := sqrt (lsum (x^2, x, v));
                                              2
(%o7)                   norm(v) := sqrt(lsum(x , x, v))
(%i8) u : random_3d_unit ();
                            5          9          4
(%o8)                  [---------, ---------, ---------]
                        sqrt(122)  sqrt(122)  sqrt(122)
(%i9) v : random_3d_unit ();
                            8         3         3
(%o9)                   [--------, --------, --------]
                         sqrt(82)  sqrt(82)  sqrt(82)
(%i10) R(a,b):= block ([v, s, c], v:cross(a,b), s:norm(v), c:dot(a,b), ident(length(a)) + ssc(v) + (ssc(v).ssc(v))*(1 - c)/s^2, ratsimp(%%));
(%o10) R(a, b) := block([v, s, c], v : cross(a, b), s : norm(v), 
                                           (ssc(v) . ssc(v)) (1 - c)
c : dot(a, b), ident(length(a)) + ssc(v) + -------------------------, 
                                                       2
                                                      s
ratsimp(%%))
(%i11) cross(a,b) := [a[2]*b[3]-b[2]*a[3], -(a[1]*b[3]-b[1]*a[3]), a[1]*b[2]-b[1]*a[2]];
(%o11) cross(a, b) := [a  b  - b  a , - (a  b  - b  a ), a  b  - b  a ]
                        2  3    2  3      1  3    1  3    1  2    1  2
(%i12) norm(v):= sqrt(v[1]^2 + v[2]^2 + v[3]^2);
                                          2    2    2
(%o12)                   norm(v) := sqrt(v  + v  + v )
                                          1    2    3
(%i13) dot(a,b):= a[1]*b[1]+a[2]*b[2]+a[3]*b[3];
(%o13)                dot(a, b) := a  b  + a  b  + a  b
                                    1  1    2  2    3  3
(%i14) ssc(v) := matrix ([0, -v[3], v[2]], [v[3], 0, -v[1]], [-v[2], v[1], 0]);
(%o14)   ssc(v) := matrix([0, - v , v ], [v , 0, - v ], [- v , v , 0])
                                 3   2     3        1       2   1
(%i15) Ruv : float (R (u, v));
         [  0.80240796270139   0.58412740488141   0.12221553199131  ]
         [                                                          ]
(%o15)   [ - 0.5556446634958   0.80598226750272  - 0.20408721762966 ]
         [                                                          ]
         [ - 0.21771648840189  0.09585280036434   0.97129386456431  ]
(%i16) intrinsic_yzx (Ruv);

rat: replaced -1.5556446634958 by -7758/4987 = -1.55564467615801

rat: replaced 0.4443553365042 by 4987/11223 = 0.44435534170899
(%o16) [[0.26495004690786, - 0.58913814199809, 0.24800273610235], 
             [- 2.876642606681936, %pi + 0.58913814199809, - 2.89358991748744]]
(%i17) 
Run Example
Rx(t) := matrix ([1, 0, 0], [0, cos(t), -sin(t)], [0, sin(t), cos(t)]);
(%o1) Rx(t) := matrix([1, 0, 0], [0, cos(t), - sin(t)], [0, sin(t), cos(t)])
(%i2) Ry(t) := matrix ([cos(t), 0, sin(t)], [0, 1, 0], [-sin(t), 0, cos(t)]);
(%o2) Ry(t) := matrix([cos(t), 0, sin(t)], [0, 1, 0], [- sin(t), 0, cos(t)])
(%i3) Rz(t) := matrix ([cos(t), -sin(t), 0], [sin(t), cos(t), 0], [0, 0, 1]);
(%o3) Rz(t) := matrix([cos(t), - sin(t), 0], [sin(t), cos(t), 0], [0, 0, 1])
(%i4) Rxyz(a, b, c) := Rx(a) . Ry(b) . Rz(c);
(%o4)              Rxyz(a, b, c) := Rx(a) . (Ry(b) . Rz(c))
(%i5) R(a,b):= block ([v, s, c, I],  v:cross(a,b),  s:norm(v),  c:dot(a,b),  I : ident (length (a)),  if equal(s, 0) /* a and b are parallel or antiparallel */    then (if c >
 0 then I else -I)    else I + ssc(v) + (ssc(v).ssc(v))*(1 - c)/s^2);
(%o5) R(a, b) := block([v, s, c, I], v : cross(a, b), s : norm(v), 
c : dot(a, b), I : ident(length(a)), if equal(s, 0)
                                                   (ssc(v) . ssc(v)) (1 - c)
 then (if c > 0 then I else - I) else I + ssc(v) + -------------------------)
                                                               2
                                                              s
(%i6) cross(a,b) := [a[2]*b[3]-b[2]*a[3], -(a[1]*b[3]-b[1]*a[3]), a[1]*b[2]-b[1]*a[2]];
(%o6)  cross(a, b) := [a  b  - b  a , - (a  b  - b  a ), a  b  - b  a ]
                        2  3    2  3      1  3    1  3    1  2    1  2
(%i7) norm(v):= sqrt(v[1]^2 + v[2]^2 + v[3]^2);
                                          2    2    2
(%o7)                    norm(v) := sqrt(v  + v  + v )
                                          1    2    3
(%i8) dot(a,b):= a[1]*b[1]+a[2]*b[2]+a[3]*b[3];
(%o8)                 dot(a, b) := a  b  + a  b  + a  b
                                    1  1    2  2    3  3
(%i9) ssc(v) := matrix ([0, -v[3], v[2]],                  [v[3], 0, -v[1]],                  [-v[2], v[1], 0]);
(%o9)    ssc(v) := matrix([0, - v , v ], [v , 0, - v ], [- v , v , 0])
                                 3   2     3        1       2   1
(%i10) intrinsic_xyz (R) :=  if equal (R[1, 3], +1) or equal (R[1, 3], -1)    then intrinsic_xyz_pitch_pi2 (R)    else block ([roll1, roll2, pitch1, pitch2, yaw1, yaw2],      pitch1 : asin (R[1, 3]),      pitch2 : %pi - pitch1,      roll1 : atan2 (-R[2, 3]/cos(pitch1), R[3, 3]/cos(pitch1)),      roll2 : atan2 (-R[2, 3]/cos(pitch2), R[3, 3]/cos(pitch2)),      yaw1 : atan2 (-R[1,2]/cos(pitch1), R[1, 1]/cos(pitch1)),      yaw2 : atan2 (-R[1,2]/cos(pitch2), R[1, 1]/cos(pitch2)),      [[roll1, pitch1, yaw1], [roll2, pitch2, yaw2]]);
(%o10) intrinsic_xyz(R) := if equal(R    , + 1) or equal(R    , - 1)
                                     1, 3                 1, 3
 then intrinsic_xyz_pitch_pi2(R) else block([roll1, roll2, pitch1, pitch2, 
yaw1, yaw2], pitch1 : asin(R    ), pitch2 : %pi - pitch1, 
                            1, 3
                - R           R
                   2, 3        3, 3
roll1 : atan2(-----------, -----------), 
              cos(pitch1)  cos(pitch1)
                - R           R
                   2, 3        3, 3
roll2 : atan2(-----------, -----------), 
              cos(pitch2)  cos(pitch2)
               - R           R
                  1, 2        1, 1
yaw1 : atan2(-----------, -----------), 
             cos(pitch1)  cos(pitch1)
               - R           R
                  1, 2        1, 1
yaw2 : atan2(-----------, -----------), 
             cos(pitch2)  cos(pitch2)
[[roll1, pitch1, yaw1], [roll2, pitch2, yaw2]])
(%i11) intrinsic_xyz_pitch_pi2 (R) :=  block ([roll1, roll2, pitch1, pitch2, yaw1, yaw2],    pitch1 : +%pi/2,    yaw1 : 0, /* could be anything */    roll1 : -yaw1 + atan2 (R[2, 1], R[2, 2]),    pitch2 : -%pi/2,    yaw2 : 0, /* could be anything */    roll2 : +yaw2 + atan2 (R[3, 2], R[3, 1]),    [[roll1, pitch1, yaw1], [roll2, pitch2, yaw2]]);
(%o11) intrinsic_xyz_pitch_pi2(R) := block([roll1, roll2, pitch1, pitch2, 
                      + %pi
yaw1, yaw2], pitch1 : -----, yaw1 : 0, roll1 : - yaw1 + atan2(R    , R    ), 
                        2                                      2, 1   2, 2
         - %pi
pitch2 : -----, yaw2 : 0, roll2 : + yaw2 + atan2(R    , R    ), 
           2                                      3, 2   3, 1
[[roll1, pitch1, yaw1], [roll2, pitch2, yaw2]])
(%i12) intrinsic_xyz_minimum_norm (R) :=  block ([p, q],    [p, q] : intrinsic_xyz (R),    if norm(p) <
 norm(q) then p else q);
(%o12) intrinsic_xyz_minimum_norm(R) := 
   block([p, q], [p, q] : intrinsic_xyz(R), if norm(p) < norm(q) then p else q)
(%i13) set_random_state (make_random_state (8));
(%o13)                               done
(%i14) random_3d_unit () := (random_3d (), %%/norm(%%));
                                                       %%
(%o14)            random_3d_unit() := (random_3d(), --------)
                                                    norm(%%)
(%i15) random_3d () := makelist (random (10), i, 1, 3);
(%o15)           random_3d() := makelist(random(10), i, 1, 3)
(%i16) a0:float (random_3d_unit ());
(%o16)      [0.34874291623146, 0.92998110995055, 0.11624763874382]
(%i17) norm(a0);
(%o17)                                1.0
(%i18) Rx1:Rx(0.1);
                 [ 1          0                   0          ]
                 [                                           ]
(%o18)           [ 0  0.99500416527803   - 0.099833416646828 ]
                 [                                           ]
                 [ 0  0.099833416646828   0.99500416527803   ]
(%i19) a1:flatten (args (Rx1.a0));
(%o19)      [0.34874291623146, 0.91372967907776, 0.20851007637721]
(%i20) norm(a1);
(%o20)                                1.0
(%i21) R(a0,a1);

rat: replaced 0.093579964804628 by 7533/80498 = 0.09357996471962
       [   0.99946512237808    0.004253693727404  - 0.032424916953477 ]
       [                                                              ]
(%o21) [ - 0.0070814490636194  0.99613054842092   - 0.087600134599152 ]
       [                                                              ]
       [  0.031926826164306    0.087782894645276   0.99562786279753   ]
(%i22) R(a0,a1).a0 - a1;

rat: replaced 0.093579964804628 by 7533/80498 = 0.09357996471962
                         [           0.0            ]
                         [                          ]
(%o22)                   [ - 1.1102230246251565E-16 ]
                         [                          ]
                         [  2.7755575615628914E-17  ]
(%i23) R(a1,a0).a1 - a0;

rat: replaced 0.093579964804628 by 7533/80498 = 0.09357996471962
                         [           0.0            ]
                         [                          ]
(%o23)                   [           0.0            ]
                         [                          ]
                         [ - 1.3877787807814457E-17 ]
(%i24) transpose(R(a0,a1)) - R(a1,a0);

rat: replaced 0.093579964804628 by 7533/80498 = 0.09357996471962

rat: replaced 0.093579964804628 by 7533/80498 = 0.09357996471962
                               [ 0.0  0.0  0.0 ]
                               [               ]
(%o24)                         [ 0.0  0.0  0.0 ]
                               [               ]
                               [ 0.0  0.0  0.0 ]
(%i25) intrinsic_xyz_minimum_norm (Rx1);
(%o25)                            [0.1, 0, 0]
(%i26) intrinsic_xyz_minimum_norm (R(a0, a1));

rat: replaced 0.093579964804628 by 7533/80498 = 0.09357996471962

rat: replaced -1.03242491695348 by -5190/5027 = -1.03242490551024

rat: replaced 0.96757508304652 by 4864/5027 = 0.96757509448976
(%o26)  [0.087758825274045, - 0.032430601435794, - 0.0042559444543473]
(%i27) intrinsic_xyz_minimum_norm (R(a1, a0));

rat: replaced 0.093579964804628 by 7533/80498 = 0.09357996471962

rat: replaced -0.9680731738357 by -5185/5356 = -0.968073188947

rat: replaced 1.031926826164306 by 5527/5356 = 1.031926811053025
(%o27)   [- 0.087940974857543, 0.031932252607692, 0.0070851202417696]
(%i28) Rz1:Rz(0.1);
                 [ 0.99500416527803   - 0.099833416646828  0 ]
                 [                                           ]
(%o28)           [ 0.099833416646828   0.99500416527803    0 ]
                 [                                           ]
                 [         0                   0           1 ]
(%i29) a1:flatten (args (Rz1.a0));
(%o29)      [0.25415746263813, 0.96015127488945, 0.11624763874382]
(%i30) norm(a1);
(%o30)                                1.0
(%i31) R(a0, a1);

rat: replaced 0.0991582498431 by 589/5940 = 0.099158249158249
        [ 0.99507784201325   - 0.098464987313879   - 0.01116846486926 ]
        [                                                             ]
(%o31)  [ 0.098503645529863    0.9951322745843     0.0029644389006664 ]
        [                                                             ]
        [ 0.010822206410214  - 0.0040499819686495   0.99993323651855  ]
(%i32) R(a0,a1).a0 - a1;

rat: replaced 0.0991582498431 by 589/5940 = 0.099158249158249
                          [ 5.5511151231257827E-17 ]
                          [                        ]
(%o32)                    [ 1.1102230246251565E-16 ]
                          [                        ]
                          [          0.0           ]
(%i33) R(a1,a0).a1 - a0;

rat: replaced 0.0991582498431 by 589/5940 = 0.099158249158249
                         [ - 5.5511151231257827E-17 ]
                         [                          ]
(%o33)                   [  1.1102230246251565E-16  ]
                         [                          ]
                         [  1.3877787807814457E-17  ]
(%i34) transpose(R(a0,a1)) - R(a1,a0);

rat: replaced 0.0991582498431 by 589/5940 = 0.099158249158249

rat: replaced 0.0991582498431 by 589/5940 = 0.099158249158249
                               [ 0.0  0.0  0.0 ]
                               [               ]
(%o34)                         [ 0.0  0.0  0.0 ]
                               [               ]
                               [ 0.0  0.0  0.0 ]
(%i35) intrinsic_xyz_minimum_norm (Rz1);
(%o35)                            [0, 0, 0.1]
(%i36) intrinsic_xyz_minimum_norm (R(a0, a1));

rat: replaced 0.0991582498431 by 589/5940 = 0.099158249158249

rat: replaced -1.01116846486926 by -10774/10655 = -1.01116846550915

rat: replaced 0.98883153513074 by 10536/10655 = 0.98883153449085
(%o36)   [- 0.0029646281447198, - 0.01116869706464, 0.098630965900368]
(%i37) intrinsic_xyz_minimum_norm (R(a1, a0));

rat: replaced 0.0991582498431 by 589/5940 = 0.099158249158249

rat: replaced -0.9891777935898 by -7038/7115 = -0.9891777933942

rat: replaced 1.010822206410215 by 7192/7115 = 1.010822206605762
(%o37)   [0.0040502302303021, 0.010822417671092, - 0.098669438486904]
(%i38) intrinsic_xyz (Rxyz (0.0, 0.0, 0.1));

rat: replaced -1.0 by -1/1 = -1.0

rat: replaced 1.0 by 1/1 = 1.0
(%o38) [[0.0, -0.0, 0.1], [3.141592653589793, %pi, - 3.041592653589793]]
(%i39) intrinsic_xyz (Rxyz (0.0, 0.1, 0.0));

rat: replaced -0.9001665833532 by -7565/8404 = -0.9001665873394

rat: replaced 1.099833416646828 by 9243/8404 = 1.099833412660638
(%o39) [[0.0, 0.1, 0.0], [3.141592653589793, %pi - 0.1, 3.141592653589793]]
(%i40) intrinsic_xyz (Rxyz (0.1, 0.0, 0.0));

rat: replaced -1.0 by -1/1 = -1.0

rat: replaced 1.0 by 1/1 = 1.0
(%o40) [[0.1, -0.0, 0.0], [- 3.041592653589793, %pi, 3.141592653589793]]
(%i41) intrinsic_xyz (Rxyz (0.1, 0.1, 0.0));

rat: replaced -0.9001665833532 by -7565/8404 = -0.9001665873394

rat: replaced 1.099833416646828 by 9243/8404 = 1.099833412660638
(%o41) [[0.1, 0.1, 0.0], [- 3.041592653589793, %pi - 0.1, 3.141592653589793]]
(%i42) intrinsic_xyz (Rxyz (0.1, 0.0, 0.1));

rat: replaced -1.0 by -1/1 = -1.0

rat: replaced 1.0 by 1/1 = 1.0
(%o42) [[0.1, -0.0, 0.1], [- 3.041592653589793, %pi, - 3.041592653589793]]
(%i43) intrinsic_xyz (Rxyz (0.0, 0.1, 0.1));

rat: replaced -0.9001665833532 by -7565/8404 = -0.9001665873394

rat: replaced 1.099833416646828 by 9243/8404 = 1.099833412660638
(%o43) [[0.0, 0.1, 0.1], [3.141592653589793, %pi - 0.1, - 3.041592653589793]]
(%i44) intrinsic_xyz (Rxyz (0.1, 0.1, 0.1));

rat: replaced -0.9001665833532 by -7565/8404 = -0.9001665873394

rat: replaced 1.099833416646828 by 9243/8404 = 1.099833412660638
(%o44) [[0.1, 0.1, 0.1], [- 3.041592653589793, %pi - 0.1, - 3.041592653589793]]
(%i45) Rz(c);
                            [ cos(c)  - sin(c)  0 ]
                            [                     ]
(%o45)                      [ sin(c)   cos(c)   0 ]
                            [                     ]
                            [   0        0      1 ]
(%i46) Rxyz(a, b, c);
                [            cos(b) cos(c)             ]
                [                                      ]
(%o46)  Col 1 = [ cos(a) sin(c) + sin(a) sin(b) cos(c) ]
                [                                      ]
                [ sin(a) sin(c) - cos(a) sin(b) cos(c) ]
           [           - cos(b) sin(c)            ]         [     sin(b)      ]
           [                                      ]         [                 ]
   Col 2 = [ cos(a) cos(c) - sin(a) sin(b) sin(c) ] Col 3 = [ - sin(a) cos(b) ]
           [                                      ]         [                 ]
           [ cos(a) sin(b) sin(c) + sin(a) cos(c) ]         [  cos(a) cos(b)  ]
(%i47) 
Run Example
a[i,j]:=random(15);
(%o1)                         a     := random(15)
                               i, j
(%i2) genmatrix(a,3,3);
                                [ 2   12  14 ]
                                [            ]
(%o2)                           [ 5   4   1  ]
                                [            ]
                                [ 14  5   3  ]
(%i3) set_random_state (true);
(%o3)                                done
(%i4)  a[i,j]:=random(15);
(%o4)                         a     := random(15)
                               i, j
(%i5) genmatrix(a,3,3);
                                [ 2   12  14 ]
                                [            ]
(%o5)                           [ 5   4   1  ]
                                [            ]
                                [ 14  5   3  ]
(%i6) for i:1 thru 6 do print(random(15));
2 
0 
5 
2 
8 
8 
(%o6)                                done
(%i7) 
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