Quaternion

384bc680-39af-44ad-9e75-cf6a1684ba33 5 4 5 decimal

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

Part 1: Quaternions rotation

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 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

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

Quaternion Companion . . . SS 6179

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. Each modulo vector is permutable by 24

5. Renders nicely: Torus, Horn, Meduse ... horribly Breather.

]]> 2.5 HORN * q quat *

'Unit norme' modulo2 
we feed in 'permute' q 0.258 - 0.516 - 0.775 - 0.258 41 mat

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 2 :

q p 1 j el :

Quaternion rotation => α,β

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

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

r 6 :

591015161920 # # 91 mat

u v Sph r u cos * v sin * r u sin * v sin * r v cos * 31 mat :

j 15 :

q p 1 j el :

SPHERE u v Sph 02 π * 0 π 2 / 126 CreateMesh : sphere u v Sph 02 π * π 2 / π 126 CreateMesh : 21 sys

Quaternion rotation => α,β

SPHERE q quat * sphere q quat * 2 1 sys

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

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

Earth

r 1 :

u v Sph r u cos * v sin * r u sin * v sin * r v cos * 31 mat :

earth_N u v Sph 02 π * 0 π 2 / 126 CreateMesh : earth_S u v Sph 02 π * π 2 / π 126 CreateMesh : 21 sys

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

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 :

FRAME 24 :

α π 2 / - :

φ π 2 / :

θ 1 :

β 0 π * :

γ 0.733 0.667 0.133 0.133 - 0.333 0.933 - 0.667 - 0.667 0.333 33 mat :

γ ° 42 38.22 7.62 7.62 - 19.08 53.46 - 38.22 - 38.22 19.08 33 mat

earth_N t f t B earth_N θ t * Ω3 * γ * eval : t B 2 col t B 1 col augment 21 line :

Pole 0 1.25 0 1.25 - 22 mat :

earth_S t f t B earth_S θ t * Ω3 * γ * eval : t B 2 col t B 1 col augment 21 line :

disabled step1 1 FRAME range :

Earth around poles axis

earth_N t f earth_S t f 2 1 sys

User/orient. γ 1101 - 101 - 0033 mat :

zoom 1.25 :

HORN Orient B HORN zoom * γ * eval : B 1 col B 2 col augment 21 line :

Lovely views ! 11 - 01101 - 0033 mat 11 - 011010033 mat 1101 - 101 - 0033 mat 01010010033 mat # 15 mat

"User" orientations, the 4 quadrants ready to display.

HORN Orient

Part 2: Parametric rotation [X,Y,Z,t°]

Prepare Smath CreateMesh u v f u v 2 u 2^v 2^+ (0.125 / - exp * 31 mat :

Gauss u v f 1 - 11 - 12020 CreateMesh :

[x,y,z] ... ±[0 .. 1] x 1 : y 1 : z 0.5 : 13 mat

t 45 ° * : zoom 6 : 12 mat

Combined rotation around [x, y, z] & 't°' Q 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 :

NOT commutative G Gauss Q * :

Other axes rotations x π 12 / : y π 6 / : z 1 : 13 mat

zoom G *

Q 1 0.061 - 0.854 0.646 1 0.561 - 0.561 - 0.854 0.78 33 mat

Q ° 57.296 3.476 - 48.905 37.039 57.296 32.123 - 32.123 - 48.905 44.71 33 mat

Part 3: Euler rotation

angle Ω1 1000 angle cos angle sin - 0 angle sin angle cos 33 mat : angle Ω2 angle cos 0 angle sin 010 angle sin - 0 angle cos 33 mat : angle Ω3 angle cos angle sin - 0 angle sin angle cos 000133 mat : 31 sys

angle Ω1 Roll [Roulis, rotation around X] ≡ angle Ω2 Pitch [Azimuth, up/down aroud Y] ≡ angle Ω3 Yaw [Horizon, rotation around Z] ≡ 31 line

angle 45 ° * :

zoom 5 :

γ angle Ω1 angle Ω2 * angle Ω3 * :

γ ° 28.65 28.65 - 40.51 48.91 8.39 28.65 - 8.39 48.91 28.65 33 mat

zoom Gauss * γ *

γ # Ω1 # Ω2 * # Ω3 * ≡

r 1 :

u v Sph r u cos * v sin * r u sin * v sin * r v cos * 31 mat :

c 00 + 200 red 15 mat : Figure 44 Sphere 14 black 15 mat : 21 line

earth u v Sph 02 π * 0 π 2412 CreateMesh :

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

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 :

FRAME 24 :

α π 2 / - :

φ π 2 / :

θ 1 :

β 0 π * :

γ α Ω1 β Ω2 * φ Ω3 * eval :

Pole 0 1.25 0 1.25 - 22 mat :

earth t f t B earth θ t * Ω3 * γ * eval : t B 2 col t B 1 col augment 21 line :

disabled step1 1 FRAME range :

Earth around poles axis

earth t f Pole 2 1 sys

User/orient. γ 1101 - 101 - 0033 mat :

HORN 1 1 0 1 - 1 0 1 - 0 0 3 3 mat * 2 *

zoom 1.25 :

HORN Orient New HORN zoom * γ * eval : New 1 col New 2 col augment 21 line :

Lovely views ! 11 - 01101 - 0033 mat 11 - 011010033 mat 1101 - 101 - 0033 mat 01010010033 mat # 15 mat

"User" orientations, the 4 quadrants ready to display.

HORN Orient 0 0 + 100 red 1 5 mat 2 1 sys

γ ° 57.3 57.3 0 57.3 - 57.3 0 57.3 - 0033 mat

Breather in 4 quadrants orientation

t0 1 time :

ρ 1 b 2^- :

w ρ sqrt :

ρ 0.7333

b 0.4 :

c 00 + 200 red 15 mat : Figure 40 Breather 14 red 15 mat : 21 line

u v d b w b u * cosh (2^* b w v * sin * (2^+ (* :

u v f u - 2 ρ * b u * cosh * b u * sinh * u v d / + 2 w * b u * cosh * w v cos * w v * cos * - v sin w v * sin * - (* u v d / 2 w * b u * cosh * w v sin * w v * cos * - v cos w v * sin * - (* u v d / 31 mat :

x,y mesh = even, practical limit 20 x 20 ... 30 x 30 Br u v f 13.2 - 13.2 37.4 - 37.4 3030 CreateMesh :

User/orient. γ 1101 - 1000033 mat :

45°<= 11 - 011000033 mat

45°=> 1101 - 1000033 mat

Vertical 01010000033 mat

Horizontal 10001000033 mat

zoom 0.5 :

Br Orient Ωmega Br zoom * γ * eval : Ωmega 1 col Ωmega 2 col augment 21 line :

1 time t0 - min 1.7

Br Orient x c Figure 4 1 sys Br 1 - 1 0 1 - 1 - 0 0 0 0 3 3 mat * Br 1 - 1 - 0 1 1 - 0 0 0 0 3 3 mat * Br 0 1 0 1 0 0 0 0 0 3 3 mat * Br 1 0 0 0 1 0 0 0 0 3 3 mat *

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

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 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

Meduse

u v Clear 1

u v Med u sin 0.5 v * cos * 0.5 u * sin 13 / v * cos * u sin v cos * 31 mat :

x,y mesh = even, practical limit 30 x 30 ... 40 x 40 Meduse u v Med 5 - 05 - 04040 CreateMesh :

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

zoom 10 :

permuted rotaion 1...24 j 8 :

q p 1 j el :

Quaternion rotation => α,β

zoom Meduse * q quat * 1 1 sys

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

Quaternion

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 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

Hyperboloid

a 2 :

c 2 :

n 20 :

zoom 1 :

One-sheet Hyperboloid x 2^a 2^/ y 2^a 2^/ + z 2^c 2^/ - 1 ≡

u v H a 1 u 2^+ sqrt * v cos * a 1 u 2^+ sqrt * v sin * c u * 31 mat :

Hyperboloid_1 u v H a - 002 π * 1020 CreateMesh : Hyperboloid_2 u v H 0 a 02 π * 1020 CreateMesh : 21 sys

Hyperboloid_1 zoom * Hyperboloid_2 zoom * 2 1 sys 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 0.516 0.258 0.775 0.258 41 mat 17 mat

zoom 2 :

permuted rotaion 1...24 j 5 :

q p 1 j el :

Quaternion rotation => α,β

Hyperboloid_1 zoom * q quat * Hyperboloid_2 zoom * q quat * 2 1 sys

Quaternion rotation => α,β

Hyperboloid_1 zoom * p 1 2 el quat * Hyperboloid_2 zoom * p 1 2 el quat * 2 1 sys

Rotate "Hedron"

Dirhombicosidodecahedron

r 10 :

n 6 : (Hedron Nunmber ≡

level 0 :

Sphere in space u v sph r u sin * v cos * r u sin * v sin * r u cos * 31 mat :

Circle in space u v circle r u cos * r u sin * level 31 mat :

S u v sph 0 π 01 π * n n CreateMesh : C u v circle 01 π * 01 π * n 1 CreateMesh : 21 sys

About Smath CreateMesh :

A much different creator than Mathcad that creates

a stack of 3 matrices. Smath outputs 3 augmented

vectors that seemingly represents an n-hedron.

For this demo, to see the Equator, we plot ½ sphere

and finish the view with a manual rotation.

Set n = 6 to view "Hedron".

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S rows C rows 21 mat 2103021 mat

Nrows 6 n 2^(* n - : 210

x 1 : y 0 : z 0 : 13 mat

t 45 - (° * :

Combined rotation around [x, y, z] & 't' Q 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 :

NOT commutative H S Q * : Equator C Q * : 21 sys

Q 1000 0.707 0.707 0 0.707 - 0.707 33 mat

sub S 21018031 - (range 13 range el :

sub Q * H 2 1 sys Equator H 2 1 sys

x 0 : y 0 : z 0 : 13 mat

t 45 - (° * :

Combined rotation around [x, y, z] & 't' Q 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 :

[1,0,0] [0,1,0] [0,0,1] [0,0,0] 1000 0.707 0.707 0 0.707 - 0.707 33 mat 0.707 0 0.707 - 010 0.707 0 0.707 33 mat 0.707 0.707 0 0.707 - 0.707 000133 mat 0.707 000 0.707 000 0.707 33 mat 24 mat

a 6 :

b 3 :

n 15 :

zoom 0.75 :

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

Parametric rotation system ...

TORUS u v Torus 0 0.5 π * 02 π * n n CreateMesh : torus u v Torus 0.5 π * 1 π * 02 π * n n CreateMesh : 21 sys

Combined rotation around [x, y, z] & 't' x y z t Prm 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 eval :

P 0.5 0.5 0.5 45 - (° * Prm :

P 0.78 0.427 0.28 - 0.28 - 0.78 0.427 0.427 0.28 - 0.78 33 mat

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

TORUS P * zoom * torus P * zoom * 2 1 sys

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

Pitch Y angle Ω2 angle cos 0 angle sin 010 angle sin - 0 angle cos 33 mat :

Yaw Z angle Ω3 angle cos angle sin - 0 angle sin angle cos 000133 mat :

a 15 ° * : b 30 ° * : c 45 - (° * : 13 mat

Euler Angles Rotation: Ψ Θ Φ ROT sΨ Ψ sin : cΨ Ψ cos : 12 mat sΘ Θ sin : cΘ Θ cos : 12 mat sΦ Φ sin : cΦ Φ cos : 12 mat 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 41 line :

Ep a b c 31 mat a c b 31 mat b a c 31 mat b c a 31 mat c a b 31 mat c b a 31 mat 16 mat :

permute 1...6 j 6 :

E Ep 1 j el :

NUMERIC MyRot E 1 el E 2 el E 3 el ROT :

Euler 6 permuted composite rotations

TORUS MyRot * zoom * torus MyRot * zoom * 2 1 sys

r 10 :

u v Sph r u cos * v sin * r u sin * v sin * r v cos * 31 mat :

x,y mesh = even, practical limit 30 x 30 ... 40 x 40 sph u v Sph 0 π 01 π * 2412 CreateMesh :

zoom 1 :

a 45 ° * : b 45 ° * : c 45 - (° * : 13 mat

Euler Angles Rotation: Ψ Θ Φ ROT sΨ Ψ sin : cΨ Ψ cos : 12 mat sΘ Θ sin : cΘ Θ cos : 12 mat sΦ Φ sin : cΦ Φ cos : 12 mat 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 41 line :

Ep a b c 31 mat a c b 31 mat b a c 31 mat b c a 31 mat c a b 31 mat c b a 31 mat 16 mat :

permute 1...6 j 6 :

E Ep 1 j el :

NUMERIC MyRot E 1 el E 2 el E 3 el ROT :

Euler 6 permuted composite rotations

sph MyRot * zoom * 1 1 sys

Part 4: Composite rotation/orientation

Composite Rot

Ψ θ Φ j Rot Ψ Θ Φ composite sΨ Ψ sin : cΨ Ψ cos : 12 mat sΘ Θ sin : cΘ Θ cos : 12 mat sΦ Φ sin : cΦ Φ cos : 12 mat 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 41 line : Ep Ψ Θ Φ 31 mat Ψ Φ Θ 31 mat Θ Ψ Φ 31 mat Θ Φ Ψ 31 mat Φ Ψ Θ 31 mat Φ Θ Ψ 31 mat 16 mat : E Ep 1 j el : E 1 el E 2 el E 3 el composite eval 41 line :

Annalema

Annalema 3.388 - 23 - 3.839 - 22.914 - 4.284 - 22.82 - 4.725 - 22.718 - 5.16 - 22.609 - 5.59 - 22.493 - 6.013 - 22.369 - 6.429 - 22.238 - 6.838 - 22.099 - 7.24 - 21.954 - 7.635 - 21.801 - 8.021 - 21.641 - 8.399 - 21.474 - 8.768 - 21.301 - 9.128 - 21.12 - 9.478 - 20.933 - 9.819 - 20.739 - 10.15 - 20.539 - 10.471 - 20.332 - 10.781 - 20.119 - 11.08 - 19.899 - 11.369 - 19.673 - 11.646 - 19.441 - 11.912 - 19.204 - 12.166 - 18.96 - 12.408 - 18.711 - 12.638 - 18.456 - 12.857 - 18.195 - 13.063 - 17.929 - 13.256 - 17.657 - 13.437 - 17.381 - 13.605 - 17.099 - 13.76 - 16.812 - 13.903 - 16.52 - 14.033 - 16.224 - 14.149 - 15.923 - 14.253 - 15.617 - 14.344 - 15.307 - 14.422 - 14.992 - 14.486 - 14.674 - 14.538 - 14.351 - 14.577 - 14.024 - 14.602 - 13.693 - 14.615 - 13.359 - 14.616 - 13.021 - 14.603 - 12.679 - 14.578 - 12.334 - 14.541 - 11.986 - 14.491 - 11.634 - 14.429 - 11.28 - 14.355 - 10.923 - 14.269 - 10.562 - 14.172 - 10.199 - 14.063 - 9.834 - 13.942 - 9.465 - 13.811 - 9.095 - 13.669 - 8.722 - 13.516 - 8.347 - 13.353 - 7.97 - 13.18 - 7.591 - 12.997 - 7.211 - 12.805 - 6.828 - 12.603 - 6.444 - 12.393 - 6.058 - 12.174 - 5.671 - 11.946 - 5.283 - 11.711 - 4.894 - 11.468 - 4.503 - 11.218 - 4.112 - 10.961 - 3.72 - 10.697 - 3.326 - 10.427 - 2.933 - 10.152 - 2.539 - 9.87 - 2.144 - 9.584 - 1.749 - 9.293 - 1.354 - 8.998 - 0.958 - 8.698 - 0.563 - 8.395 - 0.167 - 8.089 - 0.228 7.781 - 0.623 7.469 - 1.017 7.156 - 1.411 6.841 - 1.805 6.525 - 2.197 6.209 - 2.589 5.891 - 2.98 5.574 - 3.371 5.257 - 3.76 4.941 - 4.148 4.626 - 4.534 4.313 - 4.92 4.002 - 5.303 3.692 - 5.686 3.386 - 6.066 3.082 - 6.445 2.782 - 6.822 2.486 - 7.197 2.193 - 7.57 1.905 - 7.941 1.622 - 8.31 1.344 - 8.676 1.071 - 9.04 0.804 - 9.402 0.542 - 9.76 0.288 - 10.117 0.039 - 10.47 0.202 10.82 0.437 11.168 0.664 11.512 0.884 11.854 1.096 12.192 1.3 12.526 1.495 12.857 1.683 13.185 1.861 13.509 2.031 13.83 2.193 14.146 2.345 14.459 2.487 14.767 2.621 15.072 2.745 15.373 2.86 15.669 2.965 15.961 3.061 16.248 3.146 16.531 3.222 16.81 3.289 17.083 3.345 17.352 3.392 17.617 3.429 17.876 3.456 18.13 3.474 18.379 3.482 18.623 3.48 18.862 3.469 19.095 3.449 19.323 3.419 19.546 3.38 19.763 3.332 19.975 3.275 20.18 3.21 20.38 3.135 20.575 3.053 20.763 2.962 20.945 2.863 21.122 2.756 21.292 2.642 21.456 2.521 21.614 2.392 21.766 2.256 21.911 2.114 22.05 1.965 22.183 1.811 22.309 1.651 22.429 1.485 22.542 1.314 22.649 1.138 22.749 0.957 22.842 0.773 22.929 0.584 23.009 0.392 23.082 0.197 23.149 0.002 - 23.208 0.202 - 23.261 0.405 - 23.307 0.61 - 23.347 0.816 - 23.379 1.024 - 23.405 1.232 - 23.423 1.44 - 23.435 1.649 - 23.44 1.857 - 23.438 2.065 - 23.429 2.271 - 23.413 2.476 - 23.391 2.679 - 23.361 2.881 - 23.325 3.079 - 23.282 3.275 - 23.232 3.467 - 23.175 3.656 - 23.112 3.841 - 23.041 4.022 - 22.965 4.198 - 22.881 4.369 - 22.791 4.535 - 22.694 4.696 - 22.591 4.851 - 22.481 4.999 - 22.365 5.141 - 22.242 5.276 - 22.113 5.404 - 21.977 5.525 - 21.836 5.638 - 21.688 5.744 - 21.534 5.841 - 21.374 5.93 - 21.207 6.011 - 21.035 6.083 - 20.857 6.146 - 20.673 6.199 - 20.483 6.244 - 20.288 6.279 - 20.086 6.304 - 19.88 6.32 - 19.667 6.325 - 19.45 6.321 - 19.226 6.306 - 18.998 6.281 - 18.764 6.246 - 18.526 6.201 - 18.282 6.145 - 18.033 6.078 - 17.779 6.001 - 17.521 5.914 - 17.257 5.816 - 16.989 5.707 - 16.717 5.588 - 16.44 5.459 - 16.158 5.319 - 15.872 5.169 - 15.582 5.008 - 15.288 4.837 - 14.989 4.657 - 14.687 4.466 - 14.381 4.266 - 14.071 4.056 - 13.757 3.836 - 13.44 3.607 - 13.119 3.369 - 12.794 3.122 - 12.466 2.866 - 12.135 2.602 - 11.801 2.33 - 11.463 2.049 - 11.122 1.761 - 10.779 1.465 - 10.433 1.162 - 10.083 0.851 - 9.732 0.535 - 9.377 0.211 - 9.02 0.118 8.661 0.453 8.299 0.793 7.935 1.138 7.569 1.489 7.201 1.843 6.831 2.202 6.459 2.564 6.086 2.929 5.71 3.297 5.333 3.668 4.955 4.041 4.575 4.415 4.193 4.791 3.811 5.168 3.427 5.545 3.043 5.922 2.657 6.299 2.271 6.676 1.883 7.051 1.495 7.424 1.107 7.796 0.718 8.165 0.328 8.531 0.061 - 8.894 0.451 - 9.254 0.841 - 9.609 1.232 - 9.961 1.621 - 10.307 2.011 - 10.648 2.401 - 10.983 2.79 - 11.313 3.179 - 11.636 3.567 - 11.952 3.954 - 12.261 4.341 - 12.563 4.727 - 12.856 5.112 - 13.142 5.495 - 13.419 5.878 - 13.687 6.259 - 13.946 6.639 - 14.195 7.018 - 14.434 7.395 - 14.664 7.77 - 14.882 8.143 - 15.09 8.515 - 15.287 8.884 - 15.473 9.251 - 15.648 9.617 - 15.81 9.979 - 15.961 10.34 - 16.099 10.698 - 16.225 11.053 - 16.339 11.405 - 16.44 11.755 - 16.528 12.101 - 16.603 12.445 - 16.664 12.785 - 16.713 13.122 - 16.748 13.455 - 16.77 13.785 - 16.778 14.111 - 16.772 14.434 - 16.753 14.753 - 16.72 15.067 - 16.673 15.378 - 16.613 15.684 - 16.539 15.986 - 16.451 16.283 - 16.349 16.576 - 16.234 16.865 - 16.106 17.148 - 15.963 17.427 - 15.808 17.7 - 15.639 17.969 - 15.457 18.232 - 15.262 18.49 - 15.055 18.742 - 14.834 18.989 - 14.602 19.23 - 14.356 19.466 - 14.099 19.695 - 13.83 19.919 - 13.549 20.137 - 13.257 20.348 - 12.953 20.553 - 12.639 20.752 - 12.314 20.945 - 11.979 21.13 - 11.634 21.31 - 11.279 21.482 - 10.914 21.648 - 10.541 21.807 - 10.159 21.958 - 9.768 22.103 - 9.37 22.241 - 8.964 22.371 - 8.55 22.495 - 8.13 22.611 - 7.703 22.719 - 7.271 22.82 - 6.832 22.914 - 6.388 23 - 5.94 23.079 - 5.486 23.15 - 5.029 23.213 - 4.569 23.268 - 4.105 23.316 - 3.638 23.356 - 3.169 23.389 - 2.698 23.413 - 2.226 23.43 - 1.753 23.438 - 1.28 23.439 - 0.806 23.433 - 0.333 23.418 - 0.14 - 23.395 - 0.611 - 23.365 - 1.081 - 23.327 - 1.548 - 23.281 - 2.013 - 23.227 - 2.475 - 23.166 - 2.934 - 23.096 - 3.388 - 23.02 - 3662 mat :

xyData n# Decimate collect every 'n' pairs range 1 xyData rows n# range : k 1 : i range uX k el xyData i 1 el : vY k el xyData i 2 el : k k 1 + : 31 line for uX vY augment 31 line 31 line :

Analema Annalema 3 Decimate :

Analema rows 183

Analemma in spherical coordinates N λon Analema 1 col : λat Analema 2 col 90 - : i 1 Analema rows range X i el λat deg * (i el sin λon deg * (i el cos * eval : Y i el λat deg * (i el sin λon deg * (i el sin * eval : Z i el λat deg * (i el cos eval : 31 line for X Y Z augment 21 line 31 line :

N N :

zoom 10 :

λongitude λatitude

Sphere u v f u sin v cos * u sin v sin * u cos 31 mat :

Anl u v f 02 π * 02 π * 2424 CreateMesh :

Ψ 0 ° * : Θ 23.5 - (° * : Φ 23.5 ° * : 13 mat

permute 1...6 j 3 :

MyRot Ψ θ Φ j Rot :

MyRot 0.917 0.366 0.159 - 0.399 - 0.841 0.366 - 0 0.399 0.917 33 mat

Anl MyRot * zoom * N MyRot * zoom * 2 1 sys

Composite . . .

1. 6 angles permutations [Ψ,Θ,Φ]

2. 6 plot ' j ' permutations.

3. [Ψ,Θ,Φ] has fixed orientations.

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