<p> Circle rkfixed </p>

3bfd19f6-db6c-412c-8742-320d7c788a54 32 4 5 decimal

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

Example 1: A system "as is"

4 appVersion 0.98.6179.21440 ≡

r 0.25 :

NOT effective ε 109 -^:

"FindRoot" needs to
explicit the two 
specific algorithms eq_1 x 4^2 x 2^* y * + 3 x * y 2^* - y 3^+ : eq_2 x 2^y 2^+ r - : 21 sys

x y q x 2^y 2^+ r - : x y f x 4^2 x 2^* y * + 3 x * y 2^* - y 3^+ : x y axes x y / - : 31 sys

guess wisely ... init 0.5 0.125 - 21 mat 0.25 - 0.5 - 21 mat 0.25 0.45 21 mat 0.3 0.42 21 mat 14 mat :

roots roots eval :

roots j 123414 mat : ct 1 : j j sol j el eq_1 0 ≡ eq_2 0 ≡ 21 mat init 1 j el ε FindRoot : ct ct 1 + : 21 line for all sol 1 el transpose sol 2 el transpose sol 3 el transpose sol 4 el transpose stack : 41 line :

f q roots axes 4 1 sys

roots 0.4913 0.0928 - 0.2155 - 0.4512 - 0.2379 0.4398 0.2993 0.4005 42 mat

f q roots axes 41 sys

Example 2: A general explicit system

α 0.842 :

β 1.75 :

NOT effective ε 109 -^:

r 0.25 :

x y f1 1 x 2^y α - (y α + (* + (2^2 x * y α - (* (2^+ / β - :

make ellipse (2*y)^2 x y circle x 2^y 2^+ r - :

f1 circle 2 1 sys

r 0.25 :

NOT effective ε 109 -^:

x y axes x y / - : 11 sys

"FindRoot" needs to
explicit the two 
specific algorithms eq_1 1 x 2^y α - (y α + (* + (2^2 x * y α - (* (2^+ / β - : eq_2 x 2^y 2^+ r - : 21 sys

guess wisely ... init 0.25 0.5 - 21 mat 0.25 - 0.5 - 21 mat 0.5 - 0.25 21 mat 0.5 0.25 21 mat 14 mat :

roots j 123414 mat : ct 1 : j j sol j el eq_1 0 ≡ eq_2 0 ≡ 21 mat init 1 j el ε FindRoot : ct ct 1 + : 21 line for all sol 1 el transpose sol 2 el transpose sol 3 el transpose sol 4 el transpose stack : 41 line :

roots roots eval : 0.2339 0.4419 - 0.2339 - 0.4419 - 0.4618 - 0.1917 0.4618 0.1917 42 mat

plot f1 circle roots axes 41 sys :

plot

roots 0.2339 0.4419 - 0.2339 - 0.4419 - 0.4618 - 0.1917 0.4618 0.1917 42 mat

Example 3: A parametric system

inactive
oblate/prolate 
orientation s 25 : α 0.625 : β 0.5 - : 31 sys

x y axes x y / - :

Draghilev Parametric Ellipse ... else use. x y Ellipse s α x * (2^β y * α x * * - β y * (2^+ s - (* :

a 816 π * sqrt * :

f s 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ a / * - (* 0.01 - s 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ a / * - (* 0.05 - s 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ a / * - (* 0.1 - 31 sys :

x y f0 f 1 el eval : x y f1 f 2 el eval : x y f2 f 3 el eval : 31 sys

t0 1 time :

f0 f1 f2 axes Ellipse 5 1 sys

97 s to plot ►

1 time t0 - 106 s *

scale pair wise s s :

Below is SYMBOLIC

x y Electron s 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ 816 π * sqrt * / * - (* 0.1 - : x y Ellipse s α x * (2^β y * α * x * - 0.5 - (y * (2^+ s - (* : 21 sys

At this point of the project ... You scale pair wise eq_1, eq_2 You may have to adapt 'init' Try ...eq_2(x,y):=x^2+y-3

"FindRoot" needs to
explicit the two 
specific algorithms eq_1 f 3 el : eq_2 s α x * (2^β y * α * x * - 0.5 - (y * (2^+ s - (* : 21 sys

guess wisely ... set Options ... init 8 7.5 - 21 mat 5 - 4 - 21 mat 8 - 7.5 21 mat 5421 mat 14 mat :

ε 109 -^:

collect j 123414 mat : ct 1 : j j sol j el eq_1 0 ≡ eq_2 0 ≡ 21 mat init 1 j el ε FindRoot : ct ct 1 + : 21 line for all sol 1 el transpose sol 2 el transpose sol 3 el transpose sol 4 el transpose stack : 41 line :

collect collect eval :

8.5888 9.0496 - 5.4088 - 4.7261 - 8.5888 - 9.0496 5.4088 4.7261 42 mat

Electron Ellipse axes collect 4 1 sys

collect 8.5888 9.0496 - 5.4088 - 4.7261 - 8.5888 - 9.0496 5.4088 4.7261 42 mat

Circle/Ellipse Companion

r 0.25 : a 0.5 : b 1 : 13 mat

y x circle a x * (2^b y * (2^+ r - :

circle axes 2 1 sys

Example 4: Solve otherwise a triplet system

a 1 :

b 1 :

as 5 :

sy o : size 7 : clr black : 13 mat

Y 1.9109 - 1.6284 1.9109 1.6284 1.9109 - 1.6284 - 1.9109 1.6284 - 42 mat :

x y Hyperbola x 2^a 2^/ y 2^b 2^/ - (1 - :

ortho-axes x y axes x y / - :

x y Astroide x 2^y 2^+ as 2^- (3^27 as 2^* x 2^* y 2^* + :

radius r 2.5106 :

<p> Circle rkfixed </p>

Circle definition t u D u 2 el u 1 el - 21 mat :

t0 0 : t1 2 π * : N 200 : 31 line

IC 0 r 21 mat :

Either 'rkfixed' or 'Rkadapt' s IC t0 t1 N t u D rkfixed :

x s 1 col : X s 2 col : Y s 3 col : 31 line

circle X Y augment : (inverse the path Y <- X ≡

Hyperbola Astroide axes circle 1.9109 - 1.6284 1.9109 1.6284 1.9109 - 1.6284 - 1.9109 1.6284 - 4 2 mat 5 1 sys

x H x y Hyperbola 0 ≡ y solve maple :

x A x y Astroide 0 ≡ y solve maple :

X x H 1 el x A 1 el - 0 ≡ x 2 - 2 solve : 1.9109 - 1.9109 21 mat

conserve the 4 "true" decimals X X 4 round vectorize : 1.9109 - 1.9109 21 mat

from there ...
'Y' symmetric X X 1 el H 4 round vectorize augment 1.9109 - 1.6284 1.9109 1.6284 - 22 mat

x H 1 el x H 2 el x A 1 el x A 2 el circle X 1 el X 1 el H 1 el sy size clr 15 mat X 2 el X 2 el H 1 el sy size clr 15 mat X 1 el X 1 el H 2 el sy size clr 15 mat X 2 el X 2 el H 2 el sy size clr 15 mat 10 + 40 green 15 mat 111 sys

x H 1 el x H 2 el x A 1 el x A 2 el circle X 1 el X 1 el H 1 el sy size clr 1 5 mat X 2 el X 2 el H 1 el sy size clr 1 5 mat X 1 el X 1 el H 2 el sy size clr 1 5 mat X 2 el X 2 el H 2 el sy size clr 1 5 mat 1 0 + 40 green 1 5 mat 11 1 sys

0 A 1 el 0 H 1 el abs 21 mat 5121 mat

approximate 'r' inscribed 
Circle/Astroide/Hyperbola X 1 el 2^X 1 el H 1 el 2^+ sqrt 2.5106

jmG: 20230416

x y a b c roots Clear 1

Solve an hypersurface isocurve

c 1 :

t0 1 time :

f 2 c * y * 3 x 2^* y 2^* y 4^- c 2^x 2^y 2^+ (* - 4 c 4^* + (* x cos * y sinh * 2 c * y * x 4^3 y 2^* x 2^* - c 2^x 2^y 2^+ (* - 4 c 4^* - (* x sin * y cosh * x y * x 2^y 2^+ (y 2^x 2^- 2 c 2^* + (* (* 2 c * sin * 31 mat :

x y f f 1 el f 2 el + f 3 el + :

level_0 0 : level_1 1 : level_2 2 : 13 mat

y x f0 x y f level_0 + : y x f1 x y f level_1 + : y x f2 x y f level_2 + : 31 sys

r 1 : a 1 : b 1 : 13 mat y x Circle a x * (2^b y * (2^+ r - : 21 sys

symbolic x y f1 eval :

"FindRoot" needs 
the two algorithms
in explicit form. eq_1 symbolic : eq_2 a x * (2^b y * (2^+ r - : 21 sys

roots accuracy ε 1012 -^:

guess wisely ... init 0.25 - 1 - 21 mat 1 - 0.5 - 21 mat 0.25 121 mat 1 0.5 21 mat 14 mat :

roots j 123414 mat : ct 1 : j j sol j el eq_1 0 ≡ eq_2 0 ≡ 21 mat init 1 j el ε FindRoot : ct ct 1 + : 21 line for all sol 1 el transpose sol 2 el transpose sol 3 el transpose sol 4 el transpose stack : 41 line :

roots roots eval : 0.183161 - 0.983083 - 0.862777 - 0.505585 - 0.183161 0.983083 0.862777 0.505585 42 mat

◄ roots piled down ▲

You are at liberty to select oblate/prolate ellipse ... Check initials !

f0 f1 f2 Circle roots 5 1 sys

very long expansion symbolic

1 time t0 - min 1.3