Quaternion

d3d44454-af28-4580-b169-b86c4cf809a6 50 3 5 decimal

&[DATE] &[TIME] - &[FILENAME]

The three principal 3D rotation systems

a 6 :

b 3 :

n 20 :

u v Torus a b v cos * + (u cos * a b v cos * + (u sin * b v sin * 31 mat :

TORUS u v Torus 0 0.5 π * 02 π * n n CreateMesh :

Slice u v Torus 0.5 π * 1 π * 02 π * n n CreateMesh :

angle Ω1 1000 angle cos angle sin - 0 angle sin angle cos 33 mat :

Euler Angles Rotation: Ψ Θ Φ Rzyx sΨ Ψ sin : cΨ Ψ cos : sΘ Θ sin : cΘ Θ cos : sΦ Φ sin : cΦ Φ cos : cΘ cΨ * cΦ sΨ * - sΦ sΘ * cΨ * + sΦ sΨ * cΦ sΘ * cΨ * + cΘ sΨ * cΦ cΨ * sΦ sΘ * sΨ * + sΦ cΨ * - cΦ sΘ * sΨ * + sΘ - sΦ cΘ * cΦ cΘ * 33 mat 71 line :

Матрица преобразования для поворота вокруг оси y angle Ω2 angle cos 0 angle sin 010 angle sin - 0 angle cos 33 mat :

Матрица преобразования для поворота вокруг оси z angle Ω3 angle cos angle sin - 0 angle sin angle cos 000133 mat :

Combined rotation around [x, y, z] & 't' x y z t Parametric x 2^1 x 2^- (t cos * + 1 t cos - (x * y * z t sin * - 1 t cos - (x * z * y t sin * + x y * 1 t cos - (* z t sin * + y 2^1 y 2^- (t cos * + y z * 1 t cos - (* x t sin * - x z * 1 t cos - (* y t sin * - y z * 1 t cos - (* x t sin * + z 2^1 z 2^- (t cos * + 33 mat :

Ψ 0 :

Θ 0 :

Φ 30 :

This Smath document:

Exemplifies the 3 most common systems of rotations.

1. Euler [Roll/Pitch/Yaw] ... [Roulis/Azimuth/Horizon]

2. Parametric around axes x,y,z,(t)

3. Quaternion from a, b ... this system is rich of

pre-defined rotations from the assigned "q" in [4].

4. Either system supports user zoom

5. Eiter system will rotate 3D matrix of the buckyball style.

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E Ψ Ω1 Θ Ω2 * Φ Ω3 * eval :

x 1 : y 0 : z 0 : 13 mat

t 30 - (° * :

P x y z t Parametric :

Solving Quaternion [Smath 6179]

Observe:

1. init [10e-15] => col(modulo,1)

2. init [1] => col(modulo,2)

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2. Solving constrains/init α 0.75 : β 0.25 - :_init 1015 -^: 14 mat

Quaternion

1. Instruct/code the solver a b GivenFind a b FindRoot :

Solutions from the GivenFind solver are in lexicographic order wrt "init".

2. System to solve v1 p q + r - 0 ≡ p 2^q 2^+ r 2^+ 1 ≡ α p * β r * + 0 ≡ 31 mat :_v2 p init ≡ q init ≡ r init ≡ 31 mat : 13 mat

solutions p q r 31 mat ≡

Quaternion v1 v2 GivenFind : 0.267 0.535 0.802 31 mat

p q r 31 mat 0.267 0.535 0.802 31 mat

3. Define the Quaternion "modulo"
... and the Quaternion vector modulo 2 β 2^* 2 β * α * + 2 α 2^* + sqrt :

V β - modulo / β modulo / β α + (modulo / β α + (- modulo / α modulo / α - modulo / 32 mat : 0.2673 0.2673 - 0.5345 0.5345 - 0.8018 0.8018 - 32 mat

4. Create the Quaternion vector q V 2 col V 11 el stack :

q 0.2673 - 0.5345 - 0.8018 - 0.2673 41 mat

Euclidean norm => Quaternion vector q 1 el 2^q 2 el 2^+ q 3 el 2^+ q 4 el 2^+ sqrt 1.035

q norme 1.035

# norme Euclidean norm ≡

5. Define the "UnitV", apply to 'q' q q q norme / :

q 0.2582 - 0.5164 - 0.7746 - 0.2582 41 mat

q <= that goes in 'quat' converter

6. The rotation 'quat' converter quat 12 q 3 el (2^* - 2 q 4 el (2^* - 2 q 2 el * q 3 el * 2 q 1 el * q 4 el * - 2 q 1 el * q 3 el * 2 q 2 el * q 4 el * + 2 q 1 el * q 4 el * 2 q 2 el * q 3 el * + 12 q 2 el (2^* - 2 q 4 el (2^* - 2 q 1 el * q 2 el * - 2 q 3 el * q 4 el * + 2 q 1 el * q 3 el * - 2 q 4 el * q 2 el * + 2 q 1 el * q 2 el * 2 q 3 el * q 4 el * + 12 q 2 el (2^* - 2 q 3 el (2^* - 33 mat :

E 0.154 0.988 0 0.988 - 0.154 000133 mat

P 1000 0.866 0.5 0 0.5 - 0.866 33 mat

quat 0.333 - 0.933 0.133 0.667 0.333 0.667 - 0.667 - 0.133 - 0.733 - 33 mat

zoom 0.75 :

Euler => Ψ,Θ,Φ

TORUS E * Slice E * 2 1 sys

Parametric => x,y,z,(t)

TORUS P * Slice P * 2 1 sys

Quaternion => α,β

zoom TORUS * quat * zoom Slice * quat * 2 1 sys

Rotate the Buckyball C60

buckyball

buckyball 0.343 0 0.939 0.106 0.327 0.939 0.278 - 0.202 0.939 0.278 - 0.202 - 0.939 0.106 0.327 - 0.939 0.343 0 0.939 0.687 0 0.727 0.793 0.327 - 0.515 0.555 0.653 - 0.515 0.212 0.653 - 0.727 0.106 0.327 - 0.939 0.278 - 0.202 - 0.939 0.555 - 0.404 - 0.727 0.833 - 0.202 - 0.515 0.833 - 0.202 0.515 0.555 - 0.404 0.727 0.278 - 0.202 0.939 0.106 0.327 0.939 0.212 0.653 0.727 0.555 0.653 0.515 0.793 0.327 0.515 0.687 0 0.727 0.793 0.327 - 0.515 0.964 0.202 - 0.172 0.899 0.404 - 0.172 - 0.661 0.73 - 0.172 - 0.49 0.855 - 0.172 0.555 0.653 - 0.515 0.212 0.653 - 0.727 0.066 - 0.855 - 0.515 0.449 - 0.73 - 0.515 0.555 - 0.404 - 0.727 0.833 - 0.202 - 0.515 0.899 - 0.404 - 0.172 0.964 - 0.202 - 0.172 - 0.964 - 0.202 0.172 - 0.899 - 0.404 0.172 0.833 - 0.202 0.515 0.555 - 0.404 0.727 0.449 - 0.73 0.515 0.066 - 0.855 0.515 0.212 0.653 0.727 0.555 0.653 0.515 0.49 0.855 0.172 0.661 0.73 0.172 - 0.899 0.404 0.172 - 0.964 0.202 0.172 0.793 0.327 0.515 0.687 0 0.727 0.793 0.327 - 0.515 0.964 0.202 - 0.172 0.964 0.202 0.172 0.899 0.404 0.172 - 0.833 0.202 0.515 - 0.555 0.404 0.727 - 0.449 0.73 0.515 - 0.661 0.73 0.172 - 0.49 0.855 0.172 0.106 0.979 0.172 0.066 - 0.855 0.515 0.106 0.979 0.172 0.106 - 0.979 0.172 - 0.49 - 0.855 0.172 - 0.661 - 0.73 0.172 0.449 - 0.73 0.515 0.661 - 0.73 0.172 0.899 - 0.404 0.172 0.964 - 0.202 0.172 - 0.793 - 0.327 0.515 - 0.687 - 0 0.727 - 0.793 - 0.327 - 0.515 - 0.964 - 0.202 - 0.172 - 0.899 - 0.404 - 0.172 0.661 - 0.73 - 0.172 0.449 - 0.73 - 0.515 0.066 - 0.855 - 0.515 0.106 0.979 - 0.172 0.106 - 0.979 - 0.172 - 0.49 - 0.855 - 0.172 - 0.661 - 0.73 - 0.172 0.49 - 0.855 - 0.172 - 0.555 - 0.653 - 0.515 - 0.793 - 0.327 - 0.515 - 0.687 - 0 0.727 - 0.343 - 0 0.939 - 0.106 - 0.327 - 0.939 - 0.212 - 0.653 - 0.727 - 0.555 - 0.653 - 0.515 - 0.212 - 0.653 - 0.727 - 0.066 0.855 - 0.515 - 0.106 - 0.979 - 0.172 - 0.106 0.979 - 0.172 0.49 0.855 - 0.172 0.661 0.73 - 0.172 - 0.449 0.73 - 0.515 - 0.066 0.855 - 0.515 - 0.449 0.73 - 0.515 - 0.555 0.404 - 0.727 - 0.278 0.202 - 0.939 - 0.106 - 0.327 - 0.939 - 0.278 0.202 - 0.939 - 0.555 0.404 - 0.727 - 0.833 0.202 - 0.515 - 0.899 0.404 - 0.172 - 0.833 0.202 - 0.515 - 0.833 0.202 0.515 - 0.555 0.404 0.727 - 0.278 0.202 0.939 - 0.278 0.202 - 0.939 - 0.278 0.202 0.939 - 0.106 - 0.327 0.939 - 0.212 - 0.653 0.727 - 0.066 0.855 0.515 - 0.449 0.73 0.515 - 0.066 0.855 0.515 - 0.106 - 0.979 0.172 - 0.49 - 0.855 0.172 - 0.555 - 0.653 0.515 - 0.793 - 0.327 0.515 - 0.555 - 0.653 0.515 - 0.212 - 0.653 0.727 - 0.106 - 0.327 0.939 - 0.343 - 0 0.939 - 1233 mat :

zoom 8 :

Buckyball under "parametric "

zoom buckyball * P *

Native Buckyball

zoom buckyball *

Buckyball under "quat rotation"

zoom buckyball * quat * Connect, Coord

Connect 2151611232112234316334542141545265167610667873077898428891093896109101210211121115111012111213121213141337131314151433141115141517153161716201615171617181717181918321818192019531916201920222042122212521202221222322222324235223232425244824212524252725526272630262527262728272728292847282829302943297302630293032313531543118323132333214333233343333343534363431353435563534363736403613373637383793837383938383940394139364039405740394142414541842414243422943424344434344454446444145444558454446474650462847464748472448474849484849504951494650495059504951525155512352515253521953525354533154535455545155545560553556575660564057565758574558575859585059585960595560566059601802 mat :

Coord 0 0.343 0 0.939 1 0.106 0.327 0.939 2 0.278 - 0.202 0.939 3 0.278 - 0.202 - 0.939 4 0.106 0.327 - 0.939 5 0.343 0 0.939 6 0.687 0 0.727 7 0.793 0.327 - 0.515 8 0.555 0.653 - 0.515 9 0.212 0.653 - 0.727 10 0.106 0.327 - 0.939 11 0.278 - 0.202 - 0.939 12 0.555 - 0.404 - 0.727 13 0.833 - 0.202 - 0.515 14 0.833 - 0.202 0.515 15 0.555 - 0.404 0.727 16 0.278 - 0.202 0.939 17 0.106 0.327 0.939 18 0.212 0.653 0.727 19 0.555 0.653 0.515 20 0.793 0.327 0.515 21 0.687 0 0.727 22 0.793 0.327 - 0.515 23 0.964 0.202 - 0.172 24 0.899 0.404 - 0.172 - 25 0.661 0.73 - 0.172 - 26 0.49 0.855 - 0.172 27 0.555 0.653 - 0.515 28 0.212 0.653 - 0.727 29 0.066 - 0.855 - 0.515 30 0.449 - 0.73 - 0.515 31 0.555 - 0.404 - 0.727 32 0.833 - 0.202 - 0.515 33 0.899 - 0.404 - 0.172 34 0.964 - 0.202 - 0.172 - 35 0.964 - 0.202 0.172 - 36 0.899 - 0.404 0.172 37 0.833 - 0.202 0.515 38 0.555 - 0.404 0.727 39 0.449 - 0.73 0.515 40 0.066 - 0.855 0.515 41 0.212 0.653 0.727 42 0.555 0.653 0.515 43 0.49 0.855 0.172 44 0.661 0.73 0.172 - 45 0.899 0.404 0.172 - 46 0.964 0.202 0.172 47 0.793 0.327 0.515 48 0.687 0 0.727 49 0.793 0.327 - 0.515 50 0.964 0.202 - 0.172 51 0.964 0.202 0.172 52 0.899 0.404 0.172 - 53 0.833 0.202 0.515 - 54 0.555 0.404 0.727 - 55 0.449 0.73 0.515 - 56 0.661 0.73 0.172 - 57 0.49 0.855 0.172 58 0.106 0.979 0.172 59 0.066 - 0.855 0.515 60 0.106 0.979 0.172 61 0.106 - 0.979 0.172 - 62 0.49 - 0.855 0.172 - 63 0.661 - 0.73 0.172 64 0.449 - 0.73 0.515 65 0.661 - 0.73 0.172 66 0.899 - 0.404 0.172 67 0.964 - 0.202 0.172 - 68 0.793 - 0.327 0.515 - 69 0.687 - 0 0.727 - 70 0.793 - 0.327 - 0.515 - 71 0.964 - 0.202 - 0.172 - 72 0.899 - 0.404 - 0.172 73 0.661 - 0.73 - 0.172 74 0.449 - 0.73 - 0.515 75 0.066 - 0.855 - 0.515 76 0.106 0.979 - 0.172 77 0.106 - 0.979 - 0.172 - 78 0.49 - 0.855 - 0.172 - 79 0.661 - 0.73 - 0.172 80 0.49 - 0.855 - 0.172 - 81 0.555 - 0.653 - 0.515 - 82 0.793 - 0.327 - 0.515 - 83 0.687 - 0 0.727 - 84 0.343 - 0 0.939 - 85 0.106 - 0.327 - 0.939 - 86 0.212 - 0.653 - 0.727 - 87 0.555 - 0.653 - 0.515 - 88 0.212 - 0.653 - 0.727 - 89 0.066 0.855 - 0.515 - 90 0.106 - 0.979 - 0.172 - 91 0.106 0.979 - 0.172 92 0.49 0.855 - 0.172 93 0.661 0.73 - 0.172 - 94 0.449 0.73 - 0.515 - 95 0.066 0.855 - 0.515 - 96 0.449 0.73 - 0.515 - 97 0.555 0.404 - 0.727 - 98 0.278 0.202 - 0.939 - 99 0.106 - 0.327 - 0.939 - 100 0.278 0.202 - 0.939 - 101 0.555 0.404 - 0.727 - 102 0.833 0.202 - 0.515 - 103 0.899 0.404 - 0.172 - 104 0.833 0.202 - 0.515 - 105 0.833 0.202 0.515 - 106 0.555 0.404 0.727 - 107 0.278 0.202 0.939 - 108 0.278 0.202 - 0.939 - 109 0.278 0.202 0.939 - 110 0.106 - 0.327 0.939 - 111 0.212 - 0.653 0.727 - 112 0.066 0.855 0.515 - 113 0.449 0.73 0.515 - 114 0.066 0.855 0.515 - 115 0.106 - 0.979 0.172 - 116 0.49 - 0.855 0.172 - 117 0.555 - 0.653 0.515 - 118 0.793 - 0.327 0.515 - 119 0.555 - 0.653 0.515 - 120 0.212 - 0.653 0.727 - 121 0.106 - 0.327 0.939 - 122 0.343 - 0 0.939 - 1234 mat :

k 122 :

Connect rows Coord rows 21 mat 18012321 mat

sanity Coord 1 k + (row : 122 0.343 - 0 0.939 - 14 mat

Coord Coord 2 col Coord 3 col Coord 4 col augment :

zoom 10 :

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

Matlab/Mathcad Buckyball C60

Coord zoom * 1 1 sys

a b c spec + 5 black 13 mat :

pts Connect a b c spec pts :

The adjacency matrix

pts

XY Coord 1 col Coord 3 col augment :

data r s k NestRow r k * data rows ≤ memo ≡ k s k range i 1 r range j 1 data cols range M i j el data i r k * + r - (j el : for for Q k el M : k k 1 + : 31 line for Q 31 line :

a b c spec . 14 black 13 mat :

Bucky XY a b c spec pts :

Nest XY 2131 NestRow :

n 1 :

Nest n el XY Bucky 3 1 sys

Nest n el 0.343 0.939 0.106 0.939 22 mat

2d rotation of a data set

L H N xd i 1 N range x i el L i H L - N / * + : for x 21 line :

θ rot θ cos θ sin - θ sin θ cos 22 mat :

θ π - 2 / :

θ 1.5708 -

t 02 π * 200 xd :

a 0.5 :

xx yy Axes xx yy / - :

G i 1 t rows range vx i el a t i el sin * : vy i el a t i el sin * t i el cos * : 21 line for vy vx augment 21 line :

Rot G θ rot * :

G Rot Axes 3 1 sys

The 2d incremental plots in Smath native 3D

zoom 10 :

zoom G * zoom Rot * 2 1 sys

Combined rotation around [x, y, z] & parameter 't' x y z t Parametric x 2^1 x 2^- (t cos * + 1 t cos - (x * y * z t sin * - 1 t cos - (x * z * y t sin * + x y * 1 t cos - (* z t sin * + y 2^1 y 2^- (t cos * + y z * 1 t cos - (* x t sin * - x z * 1 t cos - (* y t sin * - y z * 1 t cos - (* x t sin * + z 2^1 z 2^- (t cos * + 33 mat :

x 0 : y 0 : z 1 : 13 mat

t 90 - (° * :

L H N xd i 1 N range x i el L i H L - N / * + : for x 21 line :

P x y z t Parametric :

u 02 π * 100 xd :

a 0.5 :

zoom 10 :

G3 i 1 u rows range vx i el a u i el sin * eval : vy i el a u i el sin * u i el cos * eval : 21 line for vy vx 0 vx * augment 21 line :

zoom G3 * P * zoom G3 * x y z 0 - ( ° * Parametric * 2 1 sys

P 0101 - 0000133 mat

Rotate image 4 quadrants

M

M 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mat :

Explode mirror Matrix [5346] M mirror ct 1 : j M cols ct range mir 1 ct el M j col : ct ct 1 + : 21 line for mir Unest N v mir 1 el : j 2 mir cols range v v mir j el augment : for v 71 line :

Mirror M mirror :

M Min M Max 21 mat 8616721 mat

Inverse 91 M - :

M rotate_90 M mirror transpose :

M rotate90 M transpose :

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Mirror 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M 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M rotate_90 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M rotate90

Davide Carpi: 10 April, 20017 M mirror M 1 M rows range M cols 1 range el :

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N M mirror :

Rotate the W truss

W Truss W 008 0.5 22 mat 24016 0.5 22 mat 8 0.5 16 0.5 22 mat 006 2.5 22 mat 24018 2.5 22 mat 6 2.5 12522 mat 18 2.5 12522 mat 8 0.5 12522 mat 16 0.5 12522 mat 6 2.5 8 0.5 22 mat 18 2.5 16 0.5 22 mat 111 mat :

θ rot θ cos θ sin - θ sin θ cos 22 mat :

θ π - 2 / :

θ 1.5708 -

RotW i 1 W rows range member i el W i el θ rot * : for member 21 line :

xy RotW :

collect xy 1 el xy 2 el xy 3 el xy 4 el xy 5 el xy 6 el xy 7 el xy 8 el xy 9 el xy 10 el xy 11 el 111 sys :

collect