<p> Quaternion </p>

629c53d0-9624-4f38-933e-19914625bece 27 3 5 decimal

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User Quaternion parameters ... 
8 sets of 3D display from [α,β] α 0.75 : β 0.25 - : 12 mat vectormodulo 2 : (select modulo[1], modulo[2] ≡ n 1 : (rotate(n,v) ... n:0,1,2,3 ≡ 31 line

Solving Quaternion [Smath 6179]

<p> Quaternion </p>

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

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

init 1 :

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

Observe:

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

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

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solutions p q r 31 mat ≡

Quaternion v1 v2 GivenFind : 0.267 - 0.534 - 0.802 - 31 mat

p q r 31 mat 0.267 - 0.534 - 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 ω V vectormodulo col V 11 el stack :

ω 0.2673 - 0.5345 - 0.8018 - 0.2673 41 mat

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

ω norme 1.035

# norme Euclidean norm ≡

n v rotate N v length : p n N mod : p p p 0 ≥ p N + p 0 < NOT accounted cases : Ω v : p 0 ≡ i 1 p range Ω i N + p - el v i el : for i p 1 + (N range Ω i p - el v i el : for 21 line cases Ω 51 line :

q n ω rotate : 0.535 - 0.802 - 0.267 0.267 - 41 mat

5. Define the "UnitV", apply to 'q' q q q norme / : 0.5164 - 0.7746 - 0.2582 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 :

r1 6 :

r2 3 :

m 15 :

u v Torus r1 r2 v cos * + (u cos * r1 r2 v cos * + (u sin * r2 v sin * 31 mat :

TORUS u v Torus 0 0.5 π * 02 π * m m CreateMesh : Slice u v Torus 0.5 π * 1 π * 02 π * m m CreateMesh : 21 line

n 1

zoom 0.75 :

Quaternion rotation => α,β

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

Quaternion Companion:

1. The solver GivenFind is given as reference but not used

2. ... instead: we use the modulo symbolic.

3. modulo outputs two vectors of solutions that you select

from "vector modulo".

4. The function rotate(n,v) circulates the raw quaternion 'q'

n => 0, 1, 2 ,3

5. After circulation , 'q' is normalised, then 'quat' converter applied.

6. TORUS & Slice are rotated and displayed.

7. This document outputs 2*4 = 8 displayed rotations from a single

set of α, β

8. Rotation is not commutative: quat*TORUS

9. Each modulo vector is permutable by 24 for a total 192 possible

displays ! Permuation is not available except from the web

to be installed manually.

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

a ω 1 el : b ω 2 el : c ω 3 el : d ω 4 el : 14 mat

a,b,c,d a,b,d,c a,c,b,d a,c,d,b a,d,b,c a,d,c,b b,a,c,d b,a,d,c b,c,a,d b,c,d,a b,d,a,c b,d,c,a c,a,b,d c,a,d,b c,b,a,d c,b,d,a c,d,a,b c,d,b,a d,a,b,c d,a,c,b d,b,a,c d,b,c,a d,c,a,b d,c,b,a

Permute  ... = 24 ... http://juliusdavies.ca/andre/permutations.html P a b c d 41 mat a b d c 41 mat a c b d 41 mat a c d b 41 mat a d b c 41 mat a d c b 41 mat b a c d 41 mat b a d c 41 mat b c a d 41 mat b c d a 41 mat b d a c 41 mat b d c a 41 mat c a b d 41 mat c a d b 41 mat c b a d 41 mat c b d a 41 mat c d a b 41 mat c d b a 41 mat d a b c 41 mat d a c b 41 mat d b a c 41 mat d b c a 41 mat d c a b 41 mat d c b a 41 mat 124 mat :

P 0.267 - 0.535 - 0.802 - 0.267 41 mat 0.267 - 0.535 - 0.267 0.802 - 41 mat 0.267 - 0.802 - 0.535 - 0.267 41 mat 0.267 - 0.802 - 0.267 0.535 - 41 mat 0.267 - 0.267 0.535 - 0.802 - 41 mat 0.267 - 0.267 0.802 - 0.535 - 41 mat 16 mat

u v Horn 2 v u sin * + (2 π * v * sin * v u cos * 2 v u sin * + (2 π * v * cos * 2 v * 2 - (+ 31 mat : HORN u v Horn 02 π * 012020 CreateMesh : 21 line

2.5 HORN * quat *