rotate(n,v)

e5a382b0-4932-4acd-8441-74066b57a35d 84 3 5 decimal

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

Revisit Quaternion [Smath 6179]

rotate(n,v)

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 :

GivenFind Smath solver ... NOT used

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

2. Solving constrains/init α 0.5 : β 1.5 :_init 1015 -^: 14 mat

3. 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

init = 1
........ 0.588 0.785 - 0.196 - 31 mat

init=
10e-15 0.588 - 0.784 0.196 31 mat

Quaternion v1 v2 GivenFind : 0.588 - 0.784 0.196 31 mat

mesh 20 : n 0 : ([n => 0,1,2,3 ... rotate(n,v) ≡ 21 line

mesh & rotate

Breather vector functions

ρ 1 b 2^- :

b 0.4 :

w ρ sqrt :

ρ 0.84

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 :

M u v f 13.2 - 13.2 37.4 - 37.4 mesh mesh CreateMesh :

inclined α 0.5 : β 1 : p 0.2 : vect 1 : 41 line

vertical α 0.25 : β 0 : p 0 : vect 1 : 41 line

Quaternion stuff

1. Given the Quaternion vector system, it has two orthogonal vector solution, of which the symbolic expansion is 'V'.

1. Define the "modulo" solver modulo 2 β 2^* 2 β * α * + 2 α 2^* + sqrt :

2. The "modulo" vectors V β - modulo / β modulo / β α + (modulo / β α + (- modulo / α modulo / α - modulo / 32 mat : 0.5345 - 0.5345 0.8018 0.8018 - 0.2673 0.2673 - 32 mat

3. Create the Quaternion vector ω V vect col p stack : 0.5345 - 0.8018 0.2673 0.2 41 mat

q n ω rotate (: 0.5345 - 0.8018 0.2673 0.2 41 mat

4. Normalise to "UnitVector"
'q' that feeds 'quat converter' vector norme Euclidean norm ≡ q q q norme / : 21 line

Euclidean norm => Quaternion vector ω 1 el 2^ω 2 el 2^+ ω 3 el 2^+ ω 4 el 2^+ sqrt 1.02

q 0.5241 - 0.7862 0.2621 0.1961 41 mat

5. The "Quaternion" converter quat 12 q 3 el (2^q 4 el (2^+ (* - 2 q 2 el q 3 el * q 1 el q 4 el * - (* 2 q 1 el q 3 el * q 2 el q 4 el * + (* 2 q 1 el q 4 el * q 2 el q 3 el * + (* 12 q 2 el (2^q 4 el (2^+ (* - 2 q 1 el q 2 el * - q 3 el q 4 el * + (* 2 q 1 el q 3 el * - q 4 el q 2 el * + (* 2 q 1 el q 2 el * q 3 el q 4 el * + (* 12 q 2 el (2^q 3 el (2^+ (* - 33 mat :

γ quat :

If you want a nice view of fixed orientation <= disable ... enable =>

The 4 quadrants orientations

45°<= 1 - 101 - 1 - 000033 mat

45°=> 1 - 1 - 011 - 000033 mat

Vertical 01010000033 mat

Horizontal 10001000033 mat

γ 0.7857 0.6177 0.0337 0.2065 0.3132 - 0.927 0.5831 0.7214 - 0.3736 - 33 mat

γ 1 - 101 - 1 - 000033 mat :

zoom 0.45 :

M rows 2380

t0 1 time :

Reversible orientation:
augment(col(B,2),col(B,1)) M Orient B M zoom * γ * eval : B 1 col B 2 col augment 21 line :

timing ~ 6 min M zoom * γ * 1 col M zoom * γ * 2 col augment

1 time t0 - 0.004 s *

c 00 + 200 red 15 mat :

n 0

Quaternion rotation modulo[α, β] ... α=0.5, β=1, p=0.2,zoom=0.45 supports "rot(n=1,v)" ... approximate slopes [n=0, n=1]

M Orient c 2.475 x * 0.78 x * 4 1 sys

modulo 2 β 2^* 2 β * α * + 2 α 2^* + sqrt ≡

Possible orientations:

1. 2 from rotate(n,v) ... n=0, n=1

2. 2 from reversible

augment(col(B,1),col(B,2))

augment(col(B,2),col(B,1))

3. 4 from "Quadrants"

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Breather

β - modulo / β modulo / β α + (modulo / β α + (- modulo / α modulo / α - modulo / 32 mat

The 3D solid Breather will produce few Quaternion rotation that will not hurt the eyes. The combined Quaternion is very reactive and Breather explodes easily ... α= 0.5, β= 1, p= 0.2 are acceptable. Euler rotation of Breather is more manageable.

u v X u 0031 mat :

u v Y 0 v 031 mat :

Xaxis u v X 100 - 10000 mesh mesh CreateMesh : Yaxis u v Y 0050 - 50 mesh mesh CreateMesh : 21 line

zoom 2 :

3D default plots are oriented on the Xaxis

zoom M * Xaxis Yaxis 4 1 sys

Quaternion to Euler Angles Conversion [Wikipedia] https://en.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles

Note, however, that the arctan and arcsin functions implemented in computer languages only produce results between −π/2 and π/2, and for three rotations between −π/2 and π/2 one does not obtain all possible orientations. To generate all the orientations one needs to replace the arctan functions in computer code by atan2:

φ 2 ω 1 el ω 3 el * ω 3 el ω 4 el * + (* 12 ω 2 el 2^ω 3 el 2^+ (* - atan : θ 2 ω 1 el ω 3 el * ω 4 el ω 2 el * - (* asin : ψ 2 ω 1 el ω 4 el * ω 2 el ω 3 el * + (* 12 ω 3 el (2^ω 4 el (2^+ (* - atan : 31 line

φ θ ψ 31 mat 2.746 - 0.652 - 0.27 31 mat

2.71 - 0.623 - 0.257 31 mat

Some nice Breather from Mathcad 11 Quaternion, meshed 199 x 199 Would be great if Smath could render something like GNU plot Kuen surface example. As it looks from work sheet above, Smath does not access Maxima 3D plot ?

mesh 20 :

zoom 2 :

B u v f 015 37.4 - 37.4 mesh mesh CreateMesh : Bslice u v f 15 - 0 37.4 - 37.4 mesh mesh CreateMesh : 21 line

zoom B * zoom Bslice * 2 1 sys

Imitate contour plot ... this section is stand alone

Breather vector functions

ρ 1 b 2^- :

b 0.4 :

w ρ sqrt :

ρ 0.84

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 :

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

u v X u 0031 mat :

u v Y 0 v 031 mat :

Xaxis u v X 20 - 2000100100 CreateMesh : Yaxis u v Y 0020 - 20100100 CreateMesh : 21 line

mesh 40 :

zoom 2 :

B u v f 01500 mesh 1 CreateMesh : Bslice u v f 15 - 000 mesh 1 CreateMesh : b u v Φ 01500 mesh 1 CreateMesh : bslice u v Φ 15 - 000 mesh 1 CreateMesh : 41 line

Opposite Oppos 01 - 010000033 mat :

zoom B * zoom Bslice * Xaxis zoom b * Oppos * zoom bslice * Oppos * Yaxis 6 1 sys