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1. Circle/line .... Normal, normal-parallel

C 6 : Circle ≡

L 1.5 : (Line ≡

3 appVersion 0.98.6179 ≡

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 2.311249 0.811249 - 21 mat

Assign the solutions
[x,y] coordinates of
the point of normalcy sol x y 21 mat : 2.311249 0.811249 - 21 mat

normal(x) separation parameter θ 0.725 :

L x Normal L x + 2 sol 1 el * - :

x normal L x + θ sol 1 el * - :

I C x 2^- sqrt - L x Normal - 0 ≡ x 03 solve : 0.8112 2.3112 21 mat

r C sqrt : 2.4495

X I 1 el : Y I 2 el : 12 mat 0.8112 2.3112 12 mat

N C x 2^- sqrt x normal - 0 ≡ x 03 solve : 1.8177

Normal is tangent to circle for L = 0 only ... style 3 below

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 0 L + 20 red 1 5 mat L x Normal x normal X Y - o 7 black 1 5 mat N N normal o 7 black 1 5 mat L x + 9 1 sys

assign the solutons [x,y] ... for sanity x y find Φ init TOL FindRoot : 2.3112 0.8112 - 21 mat

sanity x 2^y 2^+ x y + 21 mat 6.000000000000 1.500000000000 21 mat

C x 2^- sqrt ± L x - sol 1 el sol 2 el . 14 black sol 2 el sol 1 el . 14 black 25 mat 0 L + 20 red 15 mat L x Normal x normal X Y - o 7 black 15 mat N N normal o 7 black 15 mat L x + 91 sys

N N normal 21 mat 1.8177 1.642 21 mat

x y Clear 1

2. Circle/line intersection create sector(s)

# exp exp

C 6 : Circle ≡

L 1 : (Line ≡

Normal for 'Γ' this value Γ C 4 nthroot : 1.5651

C 4 nthroot

α Γ :

Φ vector equalities Φ x 2^y 2^+ C - 0 ≡ Γ α^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 1.468552 1.960448 - 21 mat

Assign the solutions
[x,y] coordinates of
the point of normalcy sol x y 21 mat : 1.468552 1.960448 - 21 mat

Y X 21 mat x y find Φ init TOL FindRoot : (: 1.468552 1.960448 - 21 mat

r C sqrt : 2.4495

m X X - 21 mat Y Y - 21 mat Slope : I X X - 21 mat Y Y - 21 mat Intercept : 21 mat 0.7491 - 021 mat

x Line I m x * + :

Normal tangent circle shift Γ X * Line - Y - abs : 3.767

Circle & lines user project ...

C x 2 ^ - sqrt ± x Line sol 2 el sol 1 el . 14 black 1 5 mat sol 2 el - sol 1 el - . 14 black 1 5 mat 0 L + 20 red 1 5 mat Γ α ^ x * Line - π - L x + 7 1 sys

assign the solutons [x,y] ... for sanity x y find Φ init TOL FindRoot : 1.4686 1.9604 - 21 mat

'C'
'L' x 2^y 2^+ Γ α^x * y + 21 mat 6.000000000000 1.000000000000 21 mat

x y Clear 1

3. Circle/line intersection Normals-Tangents

C 6 : Circle ≡

L 0 : (Line ≡

Γ 2 :

Φ vector equalities Φ x 2^y 2^+ C - 0 ≡ Γ 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 1.095445 2.19089 - 21 mat

Assign the solutions
[x,y] coordinates of
the point of normalcy sol x y 21 mat : 1.095445 2.19089 - 21 mat

Y X 21 mat x y find Φ init TOL FindRoot : (: 1.095445 2.19089 - 21 mat

r C sqrt : 2.4495

m X X - 21 mat Y Y - 21 mat Slope : I X X - 21 mat Y Y - 21 mat Intercept : 21 mat 0.5 - 021 mat

∡ x i y * + arg :

x Line I m x * + :

Normal tangent circle shift Γ 2^X * Line - Y - abs : 5.4772

Normal(s) tangent circle ...

C x 2 ^ - sqrt ± x Line sol 2 el sol 1 el . 14 black 1 5 mat sol 2 el - sol 1 el - . 14 black 1 5 mat 0 L + 20 red 1 5 mat Γ 2 ^ x * Line - shift - Γ 2 ^ x * Line - shift + L Γ x * + 8 1 sys

assign the solutons [x,y] ... for sanity x y find Φ init TOL FindRoot : 1.0954 2.1909 - 21 mat

'C'
'L' x 2^y 2^+ Γ x * y + 21 mat 6.000000000000 1.011550697554 1014 -^* - 21 mat

∡ ° 63.43 -

x i y * + arg ° 63.43 -

x i y * + arg rad 1.1071 -

4. Parametric circle-ellipse

a 1 : b 1 : c 1.5 : r c sqrt : 41 line

General parametric equation x y Circle x 2^a 2^/ y 2^b 2^/ + c - :

Circle 0 0 r 0 2 2 mat 2 1 sys

r 1.2247

a 1 : b 0.5 : X 00 a 022 mat : Y 0 b 0022 mat : 41 line

General parametric equation x y Circle x 2^a 2^/ y 2^b 2^/ + 1 - :

Circle Y X 3 1 sys

5. Circle & roots_of_Unity

x y Clear 1

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

x f 1 x 2^- sqrt ± :

n 12 :

roots x n^1 - 0 ≡ x solve maple :

j 5 : jj 11 : 12 mat

U roots Re roots Im augment :

X U j 1 el U jj 1 el 21 mat :

Y U j 2 el U jj 2 el 21 mat :

X Y 12 mat 0.866 0.5 - 21 mat 0.5 0.866 21 mat 12 mat

rootsU U . 14 black plot :

roots j el 0.866 0.5 i * +

In X Y Intercept : Sl X Y Slope : 21 mat 0.7321 0.2679 - 21 mat

∡ roots j el arg : ° 30

roots --- U 13 mat 11 - i i - 0.866 0.5 i * + 0.866 - 0.5 i * - 0.866 0.5 i * - 0.866 - 0.5 i * + 0.5 0.866 i * - 0.5 0.866 i * + 0.5 - 0.866 i * + 0.5 - 0.866 i * - 121 mat --- 101 - 00101 - 0.866 0.5 0.866 - 0.5 - 0.866 0.5 - 0.866 - 0.5 0.5 0.866 - 0.5 0.866 0.5 - 0.866 0.5 - 0.866 - 122 mat 13 mat

x f In Sl x * + rootsU 3 1 sys

n 12

5. Circle ... segment-sector

x y Clear 1

a 1 : b 1 : c 2 : r c sqrt : 41 line

General parametric equation x y Circle x 2^a 2^/ y 2^b 2^/ + c - :

x y c x 2^a 2^/ - (b 2^* sqrt :

x y axes x y / - :

3 sqrt 2 / r * -

X 0.5 r * :

Y X y : 1.2247

XY 00 X Y 22 mat :

∡ X i Y * + arg : ° 60

vx XY 1 col : vy XY 2 col : n XY rows : 31 line

r 1.4142

β vx transpose vy * 1 n / vx sum (* vy sum (* - vx transpose vx * 1 n / vx sum (2^* - / : α 1 n / vy sum (β vx sum (* - (* : 21 mat

Segment-sector are user define

trig rotation ± r
sector-segment x y Line_0 β x * y - : x y Line_1 β x * y + : 21 sys

β 1.7321

plot Circle Line_0 X Y 12 mat axes 00 r 022 mat Line_1 X Y X Y - 22 mat X Y X - Y 22 mat 81 sys :

plot

∡ ° 60

6. Circle-circle intersection ... the Goat constant

http://mathworld.wolfram.com/GoatProblem.html

Tieing a goat to a point on the interior of the fence with radius a=1 using a chain of length L=r, consider the length of chain that must be used in order to allow the goat to graze exactly one half the area of the field. The answer is obtained by using the equation for a circle-circle intersection ... which cannot be solved exactly, but which has approximate solution: L= 1.15872847

R 1 :

L Clear 1

f 212 / R 2^* 2 L 2 R * / asin * 2 L 2 R * / asin * sin - (* (* L 2^L 2 R * / acos * + π R 2^* 2 / ≡ :

init L 1 ≡ :

ε 109 -^:

L f init ε FindRoot : 1.158728473

R L Goat 212 / R 2^* 2 L 2 R * / asin * 2 L 2 R * / asin * sin - (* (* L 2^L 2 R * / acos * + :

x c 10 x ≤ (x 2 R * ≤ (& cases :

R x Goat x c * L π R 2 ^ * 2 / + 15 red 1 5 mat 2 1 sys

... a more mathematical competitive solution R L Goat π R 2^* 2 / - 0 ≡ L 1.1 1.2 solve 1.158728473

Explicit the Goat project ... Smath style

r 1 :

known accuracy from any solver R 1.158728473 :

x f r x 2^- sqrt ± :

x g R x 1 - (2^- sqrt ± :

Goat r x 2^- sqrt R x 1 - (2^- sqrt - 0 ≡ x 01 solve : 0.4206

x f x g 1 0 + 200 black 1 5 mat Goat 2 Goat 2 - 2 2 mat 4 1 sys

Goat f 0.907229604 0.907229604 - 21 sys

Goat g 0.907229604 0.907229604 - 21 sys

x f x g 10 + 200 black 15 mat Goat 2 Goat 2 - 22 mat 41 sys

0 g 0.398407421 0.398407421 - 21 sys

7. Circle ... segment area

1 Random earliest Smath 6179 ≡ 1 random latest Smath 6179 ≡ 21 line

1. Prepare the project

2. Construct the project

Density of random N 500 : (Density of random ≡ r 1 : (Unit circle ≡ 21 line

data spec pts i 1 data rows range p i el a b c spec : for Unest 'p' v p 1 el : j 2 p rows range v v p j el stack : for data v augment 51 line :

h 0.5 :

level 0 h ≤ (h 2 ≤ (& (≡

xmax 1 : xmin 1 - : 21 line

ymax 1 - h + (: ymin 1 - : 21 line

space_x xmax xmin - (: space_y ymax ymin - (: 21 line

x y1 x 1 + (2 r * x 1 + (- (* sqrt : x y2 x y1 - : 21 line

A function that tests an arbitrary point 
for its membership in the critical region. x y Test y x y1 + 0 ≥ (y x y1 - 0 ≤ (* :

t0 1 time :

A routine that finds an array of N random 
pairs in the critical region. Pts the routine is a compressor it maintains density 'N' within the variable region i 1 N range x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 1 x y Test - x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 21 line while P i 1 el x : P i 2 el y : 51 line for P 51 line :

a b c spec . 6 #A0A0A0 13 mat :

pts Pts a b c spec pts :

u f 0 u * 1 - h + (+ (:

1 time t0 - 7 s *

Mesh the datum level U x 11 - h + (2^- sqrt : u x - x 0.95 x * - range : i 1 u rows range datum i el u i el f : idx i el u i el : 21 line for idx datum augment 41 line :

3. View the project

plot x y1 x y2 pts λ 41 sys :

a b c spec . 8 black 13 mat :

4. Technical end use

λ U a b c spec pts :

The project runs on the UnitCircle ...

Partially filled

plot

level 0 h ≤ (h 2 ≤ (& (≡

... it must be converted wrt XTR [level transmitter 0..100%] and the C(x) function [0 .. 1]

Technical 'h'

h 0.5 h * :

x C 1 π / 12 x * - acos 12 x * - (4 x * 2 x * (2^- sqrt * - (* :

x cπ 2 0.5 x * 0.5 + C * 1 - :

h C 0.196

cπ x 1 - 1 0.995 - range : i 1 x rows range c i el x i el cπ : for x c augment 31 line :

References: the C(x) formula is the one used to tabuulate 'C' in: 1. Chemical Engineer's Handbook [Perry, Chilton, Kirkpatrik] 2. TEMA [Tubular Exchanger Manufacturers Association]

Explain ...

Figure:: Partially filled

x y1 x y2 pts λ cπ 5 1 sys

To compute the area of the segment, "solving" is not an option. The C(x) is pretty old, accurate to 5 decimals. It is a factor to 'π' in formulas involving the measure of 'h', for instance: the sole possible measurement in partially filled pipe, horizontal tank, spherical tank as well as partially filled elliptical bassin ... In the "Figure", we have superposed this curve over the partially filled circle. For experimental 'h', the area of the segment will be: C(h)*π*(D^2)/4. Observe that in the figure, the graph of C(x) has been shifted left/down to ease the pictorial understanding of the project.

Recast the C(x) function x Cπ 1 - x ≤ (x 1 ≤ (& x C invalid if :

C(x) graph and Cπ coefficient

x Cπ 0 h h Cπ + 20 red 1 5 mat 3 1 sys

h h Cπ 21 mat 0.25 0.196 21 mat

Plot arguments Figure x y1 x y2 pts λ cπ 51 sys ≡

jmGiraud: 2015_12_29