<p> Bernstein procedure </p>

8b3ea9d5-305a-4dad-b1dd-e1d2aed1c50e 21 4 5 decimal

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

Gray C:\SmathFile\OVNI.bmp READBMP :

3 appVersion 0.98.6179 ≡

Digitize/Area/Arc lenght/CG ... all modules Numeric

λ 85 :

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 Gray

Convert existing gray pixels to black N i 1 Gray rows range j 1 Gray cols range N i j el 0 Gray i j el λ ≤ 255 cases : for for N 21 line :

isolate N N :

Extract black pixels r N rows : c N cols : k 1 : i 1 c range j 1 r range N j i el 1 ≤ X k el i : Y k el r j - : k k 1 + : 31 line No pixel if for for 41 line

Gray Min Gray Max 21 mat 8525521 mat

XY X Y augment :

Raw extraction to be cleaned

XY 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 N

data StripX strip/nest repeated X u data 1 col : v data 2 col : k 0 : i 1 data rows 1 - (range u i 1 + el u i el - 0 ≠ R k k 1 + : el u i el v i el 12 mat : 11 line StripX done if 11 line for Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 91 line :

stripped of repeated X xy XY StripX :

XY rows xy rows 21 mat 36630121 mat

XY 484746454443520519518517516515122 mat

xy 4359621729832936103911421246134814501552122 mat

Digitized ... cleaned

143.4 45.1 + 50 blue 1 5 mat xy 2 1 sys

143.4 45.1 12 mat

x xy 1 col : y xy 2 col : 12 mat

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

Center of gravity Xo abs Yo abs 21 mat 143.4 45.1 21 mat

cleaned-ordered A P 21 mat 2335241121 mat

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 002005469001092109217 mat transpose : Y 016020522216070017 mat transpose : 21 sys

XY X Y augment : Cntrl XY + 15 black plot : 21 line

n X rows 1 - :

n 6

Wy 111111117 mat transpose :

zero 1015 -^eval :

Wx 111111117 mat transpose :

t zero 1 0.05 range :

<p> Bernstein procedure </p>

Bézier-Bernstein procedure j n t Q n! j! (n j - (! (* / t j^* 1 t - (n j -^* :

t QX j n t Q X j 1 + el * Wx j 1 + el * (j 0 n sum 11 line :

t QY j n t Q Y j 1 + el * Wy j 1 + el * (j 0 n sum 11 line :

BezBern i 1 t rows range qX i el t i el QX eval : qY i el t i el QY eval : 21 line for qX qY augment 21 line :

BX BezBern 1 col : BY BezBern 2 col : 12 mat BXY BX BY augment : data BXY o 5 black plot : 31 sys

Bézier-Bernstein control points & weighting [Wx, Wy]

XY BezBern data Cntrl 4 1 sys

p 0.4121 :

xo 1 :

d 5.557141 - 4.935261 5.278708 - 4.708095 41 mat :

x d Q d n el x n 1 -^* n 1 d rows sum 11 line :

x c 10 x ≤ (x 1 ≤ (& cases :

x d F x x xo - (* x d Q * (p^:

M Low High Scale High Low - (M M Min - M Max M Min - 1012 -^+ / (* Low + :

vx BX 01 Scale : vy BY 01 Scale : xy vx vy augment : support xy o 5 red plot : 41 line

About this project:

The final fit F(x,d) was done for Russian Collab.

Apparently it was to shape in some ways an UFO ! ?

The purpose of the exercise was to represent

in maths format for analytical suite ...

normal, tangent, revolution ... etc

]]> x d F x c * support 2 1 sys

xy 00 0.0067 0.2438 0.0264 0.4459 0.0579 0.6115 0.1002 0.7444 0.152 0.8475 0.212 0.923 0.2785 0.9727 0.3501 0.998 0.4251 1 0.5017 0.9801 0.5783 0.9396 0.6532 0.8799 0.7246 0.803 0.7908 0.7108 0.8504 0.6059 0.9017 0.4911 0.9434 0.3695 0.9744 0.2446 0.9935 0.1201 1 3.3137 1014 -^* 212 mat

x f x d F :

h 0.01 :

x D 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* :

zero x D 0 ≡ x 0.3 0.5 solve : 0.417

zero f 0.996

delete blue WinGrab/Capture/Select/Crop/Save as *.BMP

0 BezBern 2 1 sys x D x c * zero 8 zero 8 - 2 2 mat 2 1 sys

XY 00 0.0067 0.2438 0.0264 0.4459 0.0579 0.6115 0.1002 0.7444 0.152 0.8475 0.212 0.923 0.2785 0.9727 0.3501 0.998 0.4251 1 0.5017 0.9801 0.5783 0.9396 0.6532 0.8799 0.7246 0.803 0.7908 0.7108 0.8504 0.6059 0.9017 0.4911 0.9434 0.3695 0.9744 0.2446 0.9935 0.1201 10212 mat :

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 XY 1 col : Y XY 2 col : 12 mat

support XY o 5 black plot :

x OVNI X Y x ainterp :

support x OVNI 2 1 sys