3cddb93a-deb0-4480-ade5-b645fc11bf90 40 4 5 decimal

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

Circle/line intersection

3 appVersion 0.98.6179 ≡

C 6 : Circle ≡

L 0 : Line ≡

Φ vector equalities Φ x 2^y 2^+ C - 0 ≡ x y + L - 0 ≡ 21 mat :

init x 1.5 ≡ y 1.5 - ≡ 21 mat :

TOL 1013 -^:

Does not support assignment Φ init TOL FindRoot 1.732051 1.732051 - 21 mat

Assign the solutions sol x y 21 mat : 1.732051 1.732051 - 21 mat

C x 2 ^ - sqrt ± L x - sol 1 el sol 2 el . 14 black sol 2 el sol 1 el . 14 black 2 5 mat 3 1 sys

assign the solutons [x,y] ... for sanity x y find Φ init TOL FindRoot : 1.7321 1.7321 - 21 mat

sanity x 2^y 2^+ x y + 21 mat 6.000000000000020 0.000000000000000 21 mat

x y Clear 1

Solving the ellipse/line intersection [Smath Designers]

============================================== solving 1rst intersection

determined condition
determined condition f 2 x 2^* 3 y 2^* + 59 - 0 ≡ x - 4 y * + 8 - 0 ≡ 21 mat :

Guess roughly x.0 x 1.25 ≡ y 0.75 ≡ 21 mat :

Accuracy TOL 1015 -^:

most robust for this example sol_1 f x.0 TOL LevenbergMarquardt :

sol_1 4321 mat

Ellipse in context x y E 2 x 2^* 3 y 2^* + 59 - :

u λine u 8 + 4 / :

x y axes x y / - :

============================================== solving 2nd intersection

x y Clear 1

determined condition
determined condition Φ 2 x 2^* 3 y 2^* + 59 - 0 ≡ x - 4 y * + 8 - 0 ≡ 21 mat :

Guess wisely y.0 x 7 - ≡ y 0.75 ≡ 21 mat :

Accuracy TOL 1015 -^:

most robust for this example sol_2 Φ y.0 TOL LevenbergMarquardt :

sol_2 5.371429 - 0.657143 21 mat

E u λine sol_1 1 el sol_1 2 el 1 2 mat axes sol_2 1 el sol_2 2 el 1 2 mat 5 1 sys

E u λine sol_1 1 el sol_1 2 el 12 mat axes sol_2 1 el sol_2 2 el 12 mat 51 sys

r 0.5 - : x y q x 2^y 2^+ r + : x y axes x y / - : 31 sys

solving  "one" pair u 0.5 :

x circle x y q y solve :

pair wise X u circle : 0.499998 - 0.499992 21 mat

xy u X 2 el augment :

q xy axes u 0.75 u 0.75 - 2 2 mat 4 1 sys

r 0.25 - : x y c x 2^y 2^+ r + : 21 sys

Collect pairwise

solving  "ω" pairs ω 0.31 0.45 0.315 range :

x circle collect x y c 0 ≡ y solve : collect 21 line :

pairs ω ω circle vectorize augment :

pairs 0.31 0.3923 - 0.3923 21 mat 0.315 0.3883 - 0.3883 21 mat 0.32 0.3842 - 0.3842 21 mat 32 mat

rejects  range that
create '0' circle. 0.3 0.45 0.31 range

Collect augmented

solving /augment "ω" pairs ω 0.25 0.45 0.251 range :

x y Clear r 0.25 - : x y c x 2^y 2^+ r + : 31 sys

x circle collect x y c y roots : collect 21 line :

pair wise pairs ω circle vectorize : 0.433013 0.432434 0.431852 0.431267 0.430679 51 mat

Collect ω ω circle vectorize ω circle vectorize - augment :

Collect 0.25 0.433 0.433 - 0.251 0.4324 0.4324 - 0.252 0.4319 0.4319 - 0.253 0.4313 0.4313 - 0.254 0.4307 0.4307 - 0.255 0.4301 0.4301 - 0.256 0.4295 0.4295 - 73 mat

cercle a 1 : b 1 : xo 1 : X 1 : Y 1 : 15 mat

ellipse a 1 : b 1.5 - : xo 1 : X 1 : Y 1 : 15 mat

strophoid a 1 : b 1.5 : xo 1 : X 0.75 : Y 1 : 15 mat

strophoid x f 1 b / 1 x xo - (2^a 4^/ - sqrt x xo - (/ * a 4^* x X - (* Y + :

x q x :

true
extraterrestrial
true sol x f x q - x 0.75 1.75 solve : 0.807 1.6746 21 mat

x f x f - 2 + a 5 a 10 - 2 2 mat 1 x q 0.807 0.807 f . 14 black 1 5 mat 1.6746 1.6746 f . 14 black 1 5 mat 7 1 sys

x f x f - 2 + a 5 a 10 - 22 mat 1 x q 0.807 0.807 f . 14 black 15 mat 1.6746 1.6746 f . 14 black 15 mat 71 sys

Strophoid

a π - :

xx q 1 xx * 0.5 + :

u v Strophoid u u 2^v 2^+ (* a a cos u 2^v 2^- (* * + 2 a sin * u * v * + :

Strophoid xx q 2 1 sys

x f x y Strophoid 0 ≡ y solve maple :

x f x π - (x π + (* - sqrt x * x π - / x π - (x π + (* - sqrt x * x π - / - 21 mat

x π - ( x π + ( * - sqrt x * x π - / ± π - 2 π - 2 - 2 2 mat x 3 1 sys

x f x π - (x π + (* - sqrt x * x π - / x π - (x π + (* - sqrt x * x π - / - 21 mat :

x π - :

x f 0.000000000000000 0.000000000000000 21 mat

xx f 1 el xx q - 0 ≡ xx 1 - 0 solve xx f 2 el xx q - 0 ≡ xx 2 - 2 solve 21 mat 0.260378 - 1.353725 - 1.114103 21 mat 21 mat

Solving Ellipse otherwise

NOT needed on new document x y Clear 1

determined condition
determined condition system 2 x 2^* 3 y 2^* + 59 - 0 ≡ x - 4 y * + 8 - 0 ≡ 21 mat :

Accuracy TOL 109 -^:

Guess roughly init x 1.75 ≡ y 0.75 ≡ 21 mat :

sol_1 system init TOL FindRoot : 4.000000000000000 3.000000000000000 21 mat

clear from previous use x y Clear 1

Guess roughly init x 7 - ≡ y 0.75 ≡ 21 mat :

sol_2 system init TOL FindRoot : 5.371428571428890 - 0.657142857142777 21 mat

Ellipse in context x y E 2 x 2^* 3 y 2^* + 59 - :

xx λine xx 8 + 4 / :

x y axes x y / - :

E xx λine sol_1 1 el sol_1 2 el 1 2 mat axes sol_2 1 el sol_2 2 el 1 2 mat 5 1 sys

E xx λine sol_1 1 el sol_1 2 el 12 mat axes sol_2 1 el sol_2 2 el 12 mat 51 sys

simply on new document x λine x 8 + 4 / :

The discrete iterative solve bloc

NASA(h) is kPa vs h[m] ... we solve [m] actual altitude vs measured [kPa] h NASA 101.29 288.14 0.00649 h * - 288.08 / (5.256^* 1000 - h ≤ (h 11000 ≤ (& 22.65 1.73 0.000157 h * - exp * 11000 h ≤ (h 25000 ≤ (& 2.488 141.89 0.00299 h * + 216.6 / (11.388 -^* cases :

60000 NASA 25000 NASA 11000 NASA 0 NASA 41 mat 0.0279 2.5223 22.7074 101.4009 41 mat

L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

L 60000 NASA : H 101.4009 : N 200 : 13 mat u L H N xd : 21 sys

altitude i 1 u rows range altitude i el h NASA u i el - 0 ≡ h 0100000 solve : for u 4 round vectorize altitude 0 round vectorize augment 21 line :

t0 1 time :

NASA(h) solved [m_altitude] from measured kPa

altitude 101.325 0 + 40 red 1 5 mat 11000 NASA 11000 + 100 red 1 5 mat 25000 NASA 25000 + 175 red 1 5 mat 4 1 sys

1 time t0 - min 0.8

MACH on canvas

γ 1.4 : (Air isentropic ≡ k 0.88128485 : (Pitot constant ≡ p static pressure ≡ qc dynamic pressure ≡ x Pitot dynamic ratio qc/P ≡ x qc p / ≡ (supersonic Pitot ratio ≡ 61 line

Observe the canvas solutions:

as 'x' in Supersonic(x) runs on the plot canvas,

the 'roots' solves and plots each canvas pixel result ...

each result on the basis of 96 ppi [pixel per inch].

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Plot limit x c 10 x ≤ (x 5 ≤ (& cases :

t0 1 time :

x Subsonic 2 γ 1 - / (x 1 + (γ 1 - γ /^1 - (* sqrt 0 x ≤ (x k ≤ (& Supersonic cases : x Supersonic U x 1 + (117 U 2^* / - (2.5^* sqrt / k - 0 ≡ U 1 roots k x ≤ Subsonic cases : 21 line

x Subsonic x Supersonic x c * k 1 + 20 black 1 5 mat 2 0 Pitot-Mach 12 red 1 5 mat 4 1 sys

1 time t0 - 70 s *

supersonic

subsonic

Otherwise profitable Discrete "roots" at(2) . . .

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x y u Clear 1

r 0.25 - : x y c x 2^y 2^+ r + : 21 sys

lo 0.0001 : hi 0.4999 : N 99 : 13 mat

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

t0 1 time :

C j 1 u length range rho j el ρ u j el y c y roots at : 11 line for rho 21 line :

u C augment u C - augment u - C augment u - C - augment 4 1 sys

1 time t0 - 10.1 s *

x y u Clear 1

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 :

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

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 : N 99 : 13 mat

Hi 0.001 - :

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

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

t0 1 time :

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

1 time t0 - 27 s *

u T augment u T - augment v v q1 vectorize augment v v q2 vectorize augment 41 sys

Trifolium u T augment u T - augment v v q1 vectorize augment v v q2 vectorize augment stack :

Trifolium

Otherwise profitable Discrete-Difference "roots" at(3) . . .

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r 0.125 - :

x y u Clear 1

x y Γ x 2^y 2^+ r + :

x y Ω 1 x 4^(* 2 x 2^* y * + 3 x * y 2^* - 1 y 3^(* + :

Ω Γ 0.35355 0.04526 - 1 2 mat 3 1 sys

Intersection limit

t0 1 time :

lo 0.35 : hi 0.36 : N 1000 : 13 mat

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

last value "on error" ... NO real root. data j 1 u length range rho j el at3 u j el y Ω y roots u j el y Γ y roots at : break try for rho 21 line :

1 time t0 - 0.9 s *

data data :

"real root" exists in the positive region left of 0.35355. On the other side, the differential region is negative ... thus => "NO real root".

k 1 data rows range : junction u k el data augment : junction junction : 31 sys

u 0.35355 : Yjunction u y Γ u y Ω ≡ y roots : 21 sys

1 time t0 - min 0.8

Yjunction 0.04526 -

junction junction rows row 0.35355 0.0525 12 mat

junction 0.35355 0.5 0.35355 0.05 - 2 2 mat 0.35 0.0525 0.36 0.0525 2 2 mat 3 1 sys

real root

NO real root at(3)

x y Clear 1

a 816 π * sqrt * : 351.67021

x y gn 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ a / * - 0.0025 - :

About the Electron function . . .

A closed lobe that has no symbolic expression

of the ± branches. It complicates more familiar

expansion of f(x,y) but delights Smath.

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x y rd 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ a / * - 0.005 + :

x y axes x y / - :

solving 6 values u 7.5 :

====================================================

axes rd gn u 25 u 25 - 2 2 mat 14.2455293 5 14.2455293 5 - 2 2 mat 5 1 sys

x green user x y gn y solve : x user 12 mat 21 line :

axes rd gn u 25 u 25 - 22 mat 14.24553 5 14.24553 5 - 22 mat 51 sys

pair wise u green 7.5 18.063615 - 6.294379 - 6.294379 18.063615 41 mat 12 mat

x red user x y rd y solve : x user 12 mat 21 line :

pair wise u red 7.5 3.716206 - 3.716206 21 mat 12 mat

x f 5953 / 11 x 2 / - (2^- (* :

Solving a function, i.e: inverse

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1 f 3 f 2 f 1.75 f 41 mat 148.75 148.75 198.333 195.234 41 mat

x f 0 ≡ x 35 solve 4

x c 10 x ≤ (x 4 ≤ (& cases :

algo style "iterative inverser" on range [L, H, N] .... f L H N Inverse dx H L - N / : i 1 N range ω i el L dx - (dx i * + : for j 1 ω rows range sol j el x f ω j el - 0 ≡ x roots : for ω sol augment 41 line :

Inv f 0 198.333 100 Inverse :

x freak Inv 1 col Inv 2 col x linterp :

ξ 0.75 :

ξ f ξ f freak 21 mat 120.859 0.75 21 mat

ξ 148.75 :

=============================================

x f x c * 0 ξ ξ freak ξ ξ freak 0 3 2 mat 2 256 2 64 - 2 2 mat 3 1 sys

50 freak 148.75 freak 195.234 freak 198.333 freak 41 mat 0.27 1 1.753 1.883 41 mat

ξ freak 1

2. Maple classical/direct analytical inverser

x a b q 1 x b / - exp (a^- :

y x a b q ≡ x solve maple Pile vide.

y a b sol 1 y - ln b * a / - :

a 2 : b 1 : 12 mat

x middle b x * :

x a b q x a b sol x middle 3 1 sys

An inverse function:

1. may exist simply, this example: q(x,a,b)

2. may exist via "horrible maths" !!!.

3. may not exist at all ... inverse ∫(x)

4. inverse ∫(x) is evaluated via the iterative "solve bloc".

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x a b q x a b sol - 0 ≡ x 0.5 1 solve 0.797

4. Golden-Ratio

x f x 4^2 x 3^* - 7 x 2^* - 2 - : a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : 21 line

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema y1 y2 ≥ max in range ≡ y1 y2 ≤ min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 61 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1 ≥ y2 ≡ "max in range" y1 ≤ y 2≡ "min in range"

X f 1.5 - 1 - y1 y2 ≤ 1012 -^Extrema 1 el f 0.5 - 0.5 y1 y2 ≥ 1012 -^Extrema 1 el f 2.5 3 y1 y2 ≤ 1012 -^Extrema 1 el 31 mat :

x f X 1 el X 1 el f + 15 red X 2 el X 2 el f + 15 red X 3 el X 3 el f + 15 red 3 5 mat 2 1 sys

X 1.265564 - 2.816018 108 -^* - 2.765564 31 mat

x f x 4^2 x 3^* - 7 x 2^* - 2 - : x g x : 21 line

solve root(s) X x f x 3 - 5 solve : 1.9225 - 3.8522 21 mat

solve intersection(s) C x f x g - 0 ≡ x 3 - 5 solve : 1.837 - 3.8963 21 mat

x f x g X 1 el X 1 el f . 15 red 1 5 mat X 2 el X 2 el f . 15 red 1 5 mat C 1 el C 1 el f o 10 green 1 5 mat C 2 el C 2 el f o 10 green 1 5 mat 6 1 sys

1. roots
2. intersections X C 21 mat 1.9225 - 3.8522 21 mat 1.837 - 3.8963 21 mat 21 mat

x y Clear 1

Solving numeric a user system ======================

b 1 : c π : 21 line

f a0 0 ≡ a1 c ≡ a1 a2 + a3 + 0 ≡ a1 2 a2 * + 3 a3 * + b ≡ 41 mat :

init a0 1 ≡ a1 1 ≡ a2 1 ≡ a3 1 ≡ 41 mat :

a0 a1 a2 a3 sol f init FindRoot :

solutions in lexicographic order of "init" solutions a0 a1 a2 a3 sol vectorize : 0 3.1416 7.2832 - 4.1416 41 mat

Solving quaternion ======================================

α 0.5 :

β 1.5 :

p q r Clear 1

The Quaternion system f p q + r - 1012 -^≡ p 2^q 2^+ r 2^+ 1 ≡ β p * α r * + 1012 -^≡ 31 mat :

init p 1012 -^≡ q 1012 -^≡ r 1012 -^≡ 31 mat :

Sanity check vs Mathcad 11 f init FindRoot 0.1961 - 0.7845 0.5883 31 mat

p q r 31 mat 0.1961 - 0.7845 0.5883 31 mat

Quaternion p α - 2 α 2^* 2 α * β * + 2 β 2^* + sqrt / α 2 α 2^* 2 α * β * + 2 β 2^* + sqrt / q 12 α 2^* 2 α * β * + 2 β 2^* + sqrt / α β + (* 1 - 2 α 2^* 2 α * β * + 2 β 2^* + sqrt / α β + (* r 12 α 2^* 2 α * β * + 2 β 2^* + sqrt / β * 1 - 2 α 2^* 2 α * β * + 2 β 2^* + sqrt / β * 33 mat :

Next: evaluate numeric

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

α 0.5 :

β 1.5 :

Quaternion 2 col 0.1961 - 0.7845 0.5883 31 mat

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

V 0.1961 - 0.1961 0.7845 0.7845 - 0.1961 0.1961 - 32 mat

Next: stack a 'p' value

p 0.6 :

q Quaternion 2 col p stack :

q 0.1961 - 0.7845 0.5883 0.6 41 mat

Nex: the unit vector

q 0.1961 - 0.7845 0.5883 0.6 41 mat :

q q q norme / : 0.1682 - 0.6727 0.5045 0.5145 41 mat

solutions are in lexicographic order

Solving a chemical balance =======================

k1 0.5 : k2 0.25 : k3 0.1 : 31 line

init γ1 1012 -^≡ γ2 1012 -^≡ 21 mat :

f γ1 γ2 + k1 k2 + k3 + ≡ γ1 γ2 * k1 k2 * k1 k3 * + ≡ 21 mat :

f init FindRoot 0.5006 0.3494 21 mat

γ1 γ2 21 mat 0.501 0.349 21 mat

Clear "initial" to renew the solver

γ1 γ2 Clear 1

k1 10.052 : k2 14.836 : k3 4.982 : 13 mat

init γ1 1012 -^≡ γ2 1012 -^≡ 21 mat :

f γ1 γ2 + k1 k2 + k3 + ≡ γ1 γ2 * k1 k2 * k1 k3 * + ≡ 21 mat :

f init FindRoot 19.818 10.052 21 mat

γ1 γ2 21 mat 19.818 10.052 21 mat