<p> Revisit Pythagorean 'π' Export New Page </p>

3ab26c75-0a0f-4f25-82ca-5005e6dbdb9f 25 4 5 decimal

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Part 1 ► Otherwise cis(θ)

u v N# var u v u v u - N# / + (range : x xo x1 N# var : u 0 xo N# var : 31 sys

xo 0.125 : x1 1 : N# 40 : r 1 : ρ 1.5 : 15 mat

cis r i 2 * π * x * exp vectorize * : ucis 0.375 i 2 * π * u * exp vectorize * : cis cis Re cis Im augment : ucis ucis Re ucis Im augment : 41 sys

x y axes x y / - :

Numeric radius 00 cis 11 el cis 12 el 0032 mat :

CIS ρ i 2 * π * x * exp * vectorize Conjugate : Ncis CIS Re CIS Im augment (: 21 sys

∠ cis 11 el i cis 12 el * + arg : ° 45

'cis' trig circulation 'Ncis' clockwise circulation

cis radius Ncis ucis axes 5 1 sys

∠ ° 45

Part 2 ... Generalized complex circle/ellipse

u v N var u v u v u - N / + (range : v xo x1 N var : 21 sys

S j 1 v rows range y j el i 2 * π * v j el * exp eval : for a y Re * xc + b y Im * yc + augment 21 line :

a 1 : b 0.5 : xc 0.5 : yc 0.5 : 14 mat

xo 0.125 : x1 0.875 : N 40 : 13 mat

Numeric radius xc yc S 11 el S 12 el xc yc 32 mat :

S radius xc yc + 20 green 1 5 mat 3 1 sys

radius 0.5 0.5 1.2071 0.8536 0.5 0.5 32 mat

λ S 11 el xc - (2^S 12 el yc - (2^+ sqrt :

λ 0.7906

Try xo [0.25] for 90° ∠ S 11 el xc - S 12 el yc - (i * + arg :

∠ ° 26.5651

Part 3 ... Roots of unity ... "unit circle"

x 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 circle 1 x 2^- sqrt ± :

n 12 :

root explorer j 8 :

roots x n^1 - 0 ≡ x solve maple :

[n ► 12]... Except for the 4 first roots NSEW, others don't repeat from solve to solve. For polygon, isolate/compose from a single set.

P roots Re roots Im augment :

rootsP P . 14 black plot :

explorer P j 1 el P j 2 el + 50 green 15 mat :

x circle rootsP explorer 3 1 sys

n 12

roots P 12 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 12 mat

p 10 0.866 0.5 0.5 0.866 01 0.5 - 0.866 0.866 - 0.5 1 - 0 0.866 - 0.5 - 0.5 - 0.866 - 01 - 0.5 0.866 - 0.866 0.5 - 10132 mat :

NSEW 10011 - 001 - 42 mat :

NSEW NSEW . 14 red plot :

Polygon of 'n 12' roots of unity

p NSEW 2 1 sys

Part 4 ... Fourier 'n' polygon ...

elliptic coeff.
'n' polygon
size polygon ε 1 : pad 5 : ø 1 : 31 sys

Fourier "ifft" from padded created vector [Numeric] v ifft N v length : n 0 N range I n 1 + el 1 N sqrt / v k 1 + el e i 2 π * n * N / (* k * -^* (k 0 N 1 - sum eval * : for I 31 line :

The population of the '0' padding 
sets the size of the coplanar
Circles  have  2*pad  points Y n 2 pad * 1 - (: i 1 n range Y i el 1 : for i 2 n 1 - range Y i el 0 : for Y 41 line :

Y transpose 10000000119 mat

sides 2*pad-1 ≡

U Y ifft mat2sys sys2mat : a U Re mat2sys sys2mat : b U Im mat2sys sys2mat : ab ø a 0.5 a 1 el (* - ε b * augment (* : 41 line

ab 0.5 a 1 el ( * 0 + 40 red 1 5 mat 2 1 sys

ab 0.3333 0 0.2553 0.2143 0.0579 0.3283 0.1667 - 0.2887 0.3132 - 0.114 0.3132 - 0.114 - 0.1667 - 0.2887 - 0.0579 0.3283 - 0.2553 0.2143 - 0.3333 0102 mat

x ab 1 col : y ab 2 col : 21 sys

Area Polygon wrt vertices coordinates A 12 / x i el y i 1 + el * x i 1 + el y i el * - (i 1 x rows 1 - sum * abs :

Perimeter  Polygon wrt vertices coordinates P x i 1 + el x i el - (2^y i 1 + el y i el - (2^+ sqrt i 1 x rows 1 - sum :

Center of gravity Xo 16 A * / x i el x i 1 + el + (x i el y i 1 + el * x i 1 + el y i el * - (* i 1 x rows 1 - sum * : Yo 16 A * / y i el y i 1 + el + (x i el y i 1 + el * x i 1 + el y i el * - (* i 1 x rows 1 - sum * : 21 mat

Area
Perimeter A P 21 mat 0.3 2.1 21 mat

Center of gravity Xo abs Yo abs 21 mat 31015 -^* 51016 -^* 21 mat

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

θ π - 3 / : ° 60 -

Polygon ab θ rot * :

Polygon

Perimeter/Area/XoYo are same for any rotation

<p> Revisit Pythagorean 'π' Export New Page </p>

Fourier 'n' polygon circles/ellipses

elliptic coeff.
'n' polygon
size polygon ε 1 : pad 100 : ø 14.1073 : 31 sys

Fourier "ifft" from padded created vector [Numeric] v ifft N v length : n 0 N range I n 1 + el 1 N sqrt / v k 1 + el e i 2 π * n * N / (* k * -^* (k 0 N 1 - sum eval * : for I 31 line :

The population of the '0' padding 
sets the size of the coplanar
Circles  have  2*pad  points Y n 2 pad * 1 - (: i 1 n range Y i el 1 : for i 2 n 1 - range Y i el 0 : for Y 41 line :

U Y ifft mat2sys sys2mat : a U Re mat2sys sys2mat : b U Im mat2sys sys2mat : ab ø a 0.5 a 1 el (* - ε b * augment (* : 41 line

ab 1 1 sys

ab 10 0.9995 0.0316 0.998 0.0631 0.9956 0.0946 0.9921 0.126 0.9876 0.1572 0.9821 0.1883 0.9757 0.2192 82 mat

x ab 1 col : y ab 2 col : 21 sys

Area Polygon wrt vertices coordinates A 12 / x i el y i 1 + el * x i 1 + el y i el * - (i 1 x rows 1 - sum * abs :

Perimeter  Polygon wrt vertices coordinates P x i 1 + el x i el - (2^y i 1 + el y i el - (2^+ sqrt i 1 x rows 1 - sum :

Center of gravity Xo 16 A * / x i el x i 1 + el + (x i el y i 1 + el * x i 1 + el y i el * - (* i 1 x rows 1 - sum * : Yo 16 A * / y i el y i 1 + el + (x i el y i 1 + el * x i 1 + el y i el * - (* i 1 x rows 1 - sum * : 21 mat

Area
Perimeter A P 21 mat 3.1413 6.2832 21 mat

Center of gravity Xo abs Yo abs 21 mat 041016 -^* 21 mat