Rodrigues’ rotation formula: Difference between revisions
Tidied task description / link slightly, add an alternate Raku entry
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;Task
;Reference
* [[wp:Rodrigues%27_rotation_formula|Wikipedia: Rodrigues' rotation formula]]
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<pre>2.232221073308229 1.3951381708176411 -8.370829024905852</pre>
Alternately, differing mostly in style:
<lang perl6>sub infix:<•> { sum @^v1 Z× @^v2 } # dot product
sub infix:<❌> (@v1, @v2) { # cross product
my \a = <1 2 0>; my \b = <2 0 1>;
@v1[a] »×« @v2[b] »-« @v1[b] »×« @v2[a]
}
sub norm (*@v) { sqrt @v • @v }
sub normal (*@v) { @v X/ @v.&norm }
sub angle-between (@v1, @v2) { acos( (@v1 • @v2) / (@v1.&norm × @v2.&norm) ) }
sub infix:<> is equiv(&infix:<×>) { $^a × $^b } # invisible times
sub postfix:<°> (\d) { d × τ / 360 } # degrees to radians
sub rodrigues-rotate( @point, @axis, $θ ) {
my \cos𝜃 = cos $θ;
my \sin𝜃 = sin $θ;
my \𝑡 = 1 - cos𝜃;
my (\𝑥, \𝑦, \𝑧) = |@axis;
map { @point • $_ }, [
[ cos𝜃 + 𝑥²𝑡, 𝑥𝑦𝑡 - 𝑧sin𝜃, 𝑥𝑧𝑡 + 𝑦sin𝜃 ],
[ 𝑥𝑦𝑡 + 𝑧sin𝜃, cos𝜃 + 𝑦²𝑡, 𝑦𝑧𝑡 - 𝑥sin𝜃 ],
[ 𝑧𝑥𝑡 - 𝑦sin𝜃, 𝑧𝑦𝑡 + 𝑥sin𝜃, cos𝜃 + 𝑧²𝑡 ]
]
}
sub point-vector (@point, @vector) {
rodrigues-rotate @point, normal(@point ❌ @vector), angle-between @point, @vector
}
put "Task example - Point and composite axis / angle:\n" ~
point-vector [5, -6, 4], [8, 5, -30];
put qq:to/END/;
Perhaps more useful, (when calculating a range of motion for a robot appendage,
for example), feeding a point, axis of rotation and rotation angle separately;
since theoretically, the point vector and axis of rotation should be constant:
END
for 0, 10, 20 ... 180 { # in degrees
printf "Rotated %3d°: %.13f, %.13f, %.13f\n", $_,
rodrigues-rotate [5, -6, 4], ([5, -6, 4] ❌ [8, 5, -30]).&normal, .°
}</lang>
{{out}}
<pre>Task example - Point and composite axis / angle:
2.232221073308228 1.3951381708176427 -8.370829024905852
Perhaps more useful, (when calculating a range of motion for a robot appendage,
for example), feeding a point, axis of rotation and rotation angle directly;
since theoretically, the point vector and axis of rotation should be constant:
Rotated 0°: 5.0000000000000, -6.0000000000000, 4.0000000000000
Rotated 10°: 5.7240554466526, -6.0975296976939, 2.6561853906284
Rotated 20°: 6.2741883650704, -6.0097890410223, 1.2316639322573
Rotated 30°: 6.6336832449081, -5.7394439854392, -0.2302810114435
Rotated 40°: 6.7916170161550, -5.2947088286573, -1.6852289831393
Rotated 50°: 6.7431909410900, -4.6890966233686, -3.0889721249495
Rotated 60°: 6.4898764214992, -3.9410085899762, -4.3988584118384
Rotated 70°: 6.0393702908772, -3.0731750048240, -5.5750876118134
Rotated 80°: 5.4053609500356, -2.1119645522518, -6.5819205958338
Rotated 90°: 4.6071124519719, -1.0865831254651, -7.3887652531624
Rotated 100°: 3.6688791733663, -0.0281864202486, -7.9711060171693
Rotated 110°: 2.6191688576205, 1.0310667150840, -8.3112487584187
Rotated 120°: 1.4898764214993, 2.0589914100238, -8.3988584118384
Rotated 130°: 0.3153148442246, 3.0243546928699, -8.2312730024418
Rotated 140°: -0.8688274150348, 3.8978244887705, -7.8135845280911
Rotated 150°: -2.0265707929363, 4.6528608599741, -7.1584842417190
Rotated 160°: -3.1227378427887, 5.2665224084086, -6.2858770340300
Rotated 170°: -4.1240220834695, 5.7201633384526, -5.2222766334692
Rotated 180°: -5.0000000000000, 6.0000000000000, -4.0000000000000</pre>
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