Quaternion

f860f2ca-0335-41b3-a2c8-347662d0a1fe 3 15 12 decimal

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

Quaternions explorer ... spooled to file ... auto calc

Quaternions system.
native modulo 6 permute
go to permute oder 'j' α 0.75 : β 0.25 - : 12 mat vectormodulo 2 : (select modulo[1,2] ≡ 21 sys

Quaternion

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 - 0.535 - 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 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 ≡

5. Define the "UnitV", apply to 'q' q ω ω norme / : 0.2582 - 0.5164 - 0.7746 - 0.2582 41 mat

q <= that goes in 'quat' converter

6. The rotation 'quat' converter q 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

u v Clear 1

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

CWD C:\SmathFile CurrentDirectory :

TORUS CWD TORUS wfile 1

TORUS CWD TORUS rfile :

Slice CWD Slice wfile 1

Slice CWD Slice rfile :

HORN CWD HORN wfile 1

HORN CWD HORN rfile :

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

q quat ° 19.1 - 53.48 7.64 38.2 19.1 38.2 - 38.2 - 7.64 - 42.02 - 33 mat

Native Quaternion ...

0.75 TORUS * q quat * 0.75 Slice * q quat * 2 1 sys 2.5 HORN * q quat * Permute

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

Permute  [a,b,c,d]... = 24 ... 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 0.258 - 0.516 - 0.775 - 0.258 41 mat 0.258 - 0.516 - 0.258 0.775 - 41 mat 0.258 - 0.775 - 0.516 - 0.258 41 mat 0.258 - 0.775 - 0.258 0.516 - 41 mat 0.258 - 0.258 0.516 - 0.775 - 41 mat 0.258 - 0.258 0.775 - 0.516 - 41 mat 16 mat

permuted rotaion 1...24 j 14 :

q p 1 j el :

Ω q quat :

Spool to file tori

r1 4 :

r2 1.2 :

m 30 :

u v Clear 1

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

torus u v Torus 02 π * 02 π * m m CreateMesh :

A0 torus 1 col torus 2 col torus 3 col 4 + augment : A1 torus 3 col torus 2 col 4 + torus 1 col 4 + augment : 21 sys

CWD C:\SmathFile CurrentDirectory :

Ω CWD Ωtorus wfile 1

A0 CWD A0torus wfile 1

Ω CWD Ωtorus rfile :

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

A1 CWD A1torus wfile 1

A0 CWD A0torus rfile :

θ π - 4 / :

A1 CWD A1torus rfile :

θ 0.7854 -

permuted rotaion 1...24 j 15 :

q p 1 j el :

Ω q quat :

# # # 131415189 # # # 111 mat

Quaternion rotation => α,β

1 TORUS * q quat * 1 Slice * q quat * 2 1 sys 3 HORN * q quat * 1.5 A0 * Ω * 1.5 A1 * Ω * 2 1 sys