<p> Spool to file in there ... [6 sec] </p>

257d6580-6809-449c-a437-a5accf6296c6 103 15 12 decimal

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

Application of the circulation "fmesh"

3 appVersion 0.98.6179 ≡ Timing 2nd run 2 min ≡ 21 sys

x y Φ3 x 2^y 2^+ (2^1 x 3^3 x * y 2^* - (* - :

jct 0.5625 - :

t0 1 time :

x f x y Φ3 0 ≡ y solve maple :

x c 1 x 2^- sqrt ± :

jct f 1 el jct f 3 el - 0.000000000000000

x f1 x 32 x * + 916 x * + sqrt - (* - sqrt 2 sqrt / : x f2 x 32 x * + 916 x * + sqrt - (* - sqrt 2 sqrt / - : x f3 x 32 x * + 916 x * + sqrt + (* - sqrt 2 sqrt / : x f4 x 32 x * + 916 x * + sqrt + (* - sqrt 2 sqrt / - : 41 sys

x f1 x f2 x f3 x f4 4 1 sys

Trifolium constants X 0.5625 - ≡ Y 0.726184377413896 ± ≡ datum 1012 mat ≡ ArcLength 6.682 ≡ Area 0.785 ≡ Angle 120 ° * ≡ 61 mat

Trifolium constant jct f1 0.726184377413896

Algo style "circulation" discretizer ... 2ways  => # <= x f x0 x1 mesh fmesh U 0 : count 1 : i 1 mesh range x i el x0 i mesh / x1 x0 - (* + : U count el x i el f eval : count count 1 + : 31 line for x U augment 41 line :

N 100 :

Discretize/collect from fmesh(,,,,) b1 x f1 jct 1 N fmesh eval : b2 x f2 1 jct N fmesh eval : b3 x f4 jct 0 N fmesh eval : b4 x f3 0 jct N fmesh eval : Trifolium b1 b2 b3 b4 stack : 51 sys

b1 b2 b3 b4 4 1 sys Trifolium 1 0.5 1 0.5 - 2 2 mat jct jct f1 + 25 red 1 5 mat x c 4 1 sys

x y Φ3 x 2^y 2^+ (2^a x 3^3 x * y 2^* - (* - :

Trifolium 0.75 Trifolium * 0.5 Trifolium * 1.5 Trifolium * 4 1 sys

a 1 :

arc 6 a * 189 / t sin (2^* - sqrt t 0 0.5 π * int * : Area π 4 / a 2^* : 21 mat 6.682 0.785 21 mat

https://www.mathcurve.com/courbes2d/trifoliumregulier/trifoliumregulier.shtml

The wind mill rotation adapted Ber7

θ rot θ cos θ sin - θ sin θ cos 22 mat : n 60 : t Eol Trifolium 2 π * n / t * rot * : step1 0 n range : Wind Mill rotation system 51 sys

Eol t Eol 0.75 t Eol * 0.5 t Eol * 1.25 t Eol * 00.15 red 15 mat plot argument 1 x 2^- sqrt ± 71 sys :

Smath "Wind Mill" project

Eol

1 time t0 - 38 s *

Otherwise Trifolium based on symmetry

t0 1 time :

x y Φ3 x 2^y 2^+ (2^1 x 3^3 x * y 2^* - (* - : x C 1 x 2^- sqrt ± : 21 sys

x map x y Φ3 0 ≡ y solve maple :

junction 0.5625 - 0.5625 - map 1 el + 15 black 15 mat :

x plot x 32 x * + 916 x * + sqrt - (* - sqrt 2 sqrt / x 32 x * + 916 x * + sqrt - (* - sqrt 2 sqrt / - x 32 x * + 916 x * + sqrt + (* - sqrt 2 sqrt / x 32 x * + 916 x * + sqrt + (* - sqrt 2 sqrt / - 41 sys :

x plot x C junction 3 1 sys

Maple  symbolic x q1 x 32 x * + 916 x * + sqrt - (* - sqrt 2 sqrt / : x q2 x 32 x * + 916 x * + sqrt - (* - sqrt 2 sqrt / - : 21 sys

lo 0.56245 - : hi 0.99 : 12 mat

Hi 0.001 - :

find low limit u lo hi lo hi lo - 99 / + (range :

find low limit v lo Hi lo Hi lo - 51 / + (range :

T j 1 u length range rho j el dum u j el y Φ3 y roots at : 11 line for rho 21 line :

u T augment u reverse T - reverse augment v v q2 vectorize augment v reverse v q1 vectorize reverse augment 4 1 sys

1 time t0 - 33 s *

Next collect Trifolium

NUMERIC Ω u T augment u reverse T - reverse augment v v q2 vectorize augment v reverse v q1 vectorize reverse augment 41 sys :

Most important . . .

Ω & Eol NUMERIC

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NUMERIC Eol Ω 1 el Ω 2 el Ω 3 el Ω 4 el stack :

===================== Next rotate Trifolium ====================

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

θ π 5 / - :

x c 1 x 2^- sqrt ± :

Smath 2D vectors rotation system t Eol Eol θ π 30 / t * + rot * eval :

Animate rotation [60..0] Hub 00.20 red 15 mat : step1 060 range : 21 line

Smath "Wind Mill" project

t Eol x c Hub 3 1 sys Eol x c 2 1 sys

1 time t0 - 84 s *

t Eol x c Hub 31 sys

Piecewise Trifolium

t0 1 time :

x y Φ3 x 2^y 2^+ (2^1 x 3^3 x * y 2^* - (* - : x y Φ6 x 2^y 2^+ (4^1 x 3^3 x * y 2^* - (2^* - : x C 1 x 2^- sqrt : plot first the Sixfolium 41 line

x C Re x C Re - Φ3 3 1 sys

y F x y Φ3 0 ≡ x solve maple :

y F_Z_Z_Z 1 -_Z + (* 2 y 2^* + (* 3 y 2^* + (* y 4^+ RootOf

y F x y Φ3 0 ≡ x solve allvalues maple :

The Maple advanced expansion is in term of solutions "RootOf". The command "allvalues" restricts to explicit solutions wrt 't'. "simplify" is not needed either. "RootOf" has workaround Smath.

x c 1 x 2^- sqrt ± :

junction 0.5625 - 0.5625 - f 1 el + 15 black 15 mat :

x f x y Φ3 0 ≡ y solve maple :

plot x f 1 el x f 2 el x f 3 el x f 4 el 41 sys :

plot x c junction 3 1 sys

Solving for the junction ... value in the Trifolium domain x f 1 el x f 3 el - abs 0 ≡ x 0.563 - 0.562 - solve 0.5625 - f 1 el 21 mat 0.562500000000 - 0.726184377414 21 mat

junction 0.5625 - 0.5625 - f 1 el + 15 black 15 mat :

x f x 32 x * + 916 x * + sqrt - (* - sqrt 2 sqrt / x 32 x * + 916 x * + sqrt - (* - sqrt 2 sqrt / - x 32 x * + 916 x * + sqrt + (* - sqrt 2 sqrt / x 32 x * + 916 x * + sqrt + (* - sqrt 2 sqrt / - 41 mat :

x f 1 el x f 2 el x f 3 el x f 4 el junction 5 1 sys

Mesh U, V  within the bounds L, H L H N xd U 0 : dx H L - N / : i 1 N range U i el L dx i * dx - (+ : for U 41 line :

N 90 :

u 0 0.5625 - N xd : v 10 N xd : 21 line

The 3 lobes complete through two branches each for a total of 6 circulations

b1 i 1 u rows range U1 i el u i el f 1 el eval : for u U1 Re augment 21 line :

b3 i 1 v rows range U3 i el v i el f 1 el eval : for v U3 augment 21 line :

b5 i 1 u rows range U5 i el u i el f 2 el eval : for u U5 Re augment 21 line :

b2 i 1 u rows range V2 i el u i el f 3 el eval : for u u reverse : V2 V2 Re reverse : 21 line u V2 augment 31 line :

b4 i 1 v rows range V4 i el v i el f 2 el eval : for v v reverse : V4 V4 reverse : 21 line v V4 augment 31 line :

b6 i 1 u rows range V6 i el u i el f 4 el eval : for u u reverse : V6 V6 Re reverse : 21 line u V6 augment 31 line :

Unexplained shadow in context

b1 b2 b3 b4 b5 b6 6 1 sys

specific construction for Eol Trifolium ... ... can't explain ?

v v reverse : V V reverse : 21 line

Enable/disable observe Eol

<p> Spool to file in there ... [6 sec] </p>

CWD C:\SmathFile CurrentDirectory :

b2 CWD b2 wfile 1

b2 CWD b2 rfile :

b4 CWD b4 wfile 1

b4 CWD b4 rfile :

b6 CWD b6 wfile 1

b6 CWD b6 rfile :

b1 CWD b1 wfile 1

b1 CWD b1 rfile :

b3 CWD b3 wfile 1

b3 CWD b3 rfile :

b5 CWD b5 wfile 1

b5 CWD b5 rfile :

f1 b1 b2 stack : f2 b3 b4 stack : f3 b5 b6 stack : Eol f1 f2 f3 stack : 41 sys

Rotation system
plot arguments . t Eol x c Hub 31 sys

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

θ π 5 / - :

Smath 2D vectros rotation system t Eol Eol θ π 30 / t * + rot * eval :

Hub 00.15 red 15 mat : step1 060 range : 21 line

Smath "Wind Mill" project

t Eol x c Hub 3 1 sys f1 f2 f3 3 1 sys Eol 3 sqrt x * - 2 1 sys

1 time t0 - 43 s *