ordered

673e1050-3a4f-4a08-b4f6-2b24175e9258 43 15 12 decimal

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

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 :

Plot isocontour @ max [181.617] x y f x 2^y + 11 - (2^x y 2^+ 7 - (2^+ a - :

a 181.617 :

x y axes x y / - :

f axes 2 1 sys

data n Decimate collect every 'n' pairs range 1 data rows n range : k 1 : i range u k el data i 1 el : v k el data i 2 el : k k 1 + : 31 line for u v augment 31 line 31 line :

Gray C:\SmathFile\Himmelblau181617.bmp READBMP :

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 :

Gray Min Gray Max 21 mat 8525521 mat

<= Min valued gray 85

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

XY X Y augment :

xy XY 456464 range 12 range el :

XY rows 464

Raw extraction to be cleaned

XY 173 75 173 25 - 2 2 mat 2 1 sys

xy 17335173341733317332173311733017329173281732792 mat

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 N

data StripX nest stripped in 'R' 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 :

xy1 XY 1116 range 12 range el : S0 xy1 2 csort : S0 S0 23 S0 rows 12 submatrix : 31 sys

new XY 1 csort : S1 new StripX : S1 S1 4 S1 rows range 12 range el : 31 sys

profitable decimation
by pure coincidence.
Can't explain [9] ? XY XY 9 Decimate : S2 XY 16 XY rows range 12 range el : 21 sys

S3 1732712 mat :

S S0 5 Decimate S1 5 Decimate S2 S3 stack :

S rows 110

S2 S2 rows row 1733412 mat

isocontour S o 5 red plot :

10 - 14 20 14 2 2 mat isocontour 3 32 3 10 - 2 2 mat 3 1 sys

S 4103144185226268301134133882 mat

ordered S0 S2 S3 S1 S1 rows 1 range 12 range el stack :

Comments . . .

Intuitively and from other easier closed contour

this project should have as well 4 segments.

The project is on image pixel scale ...

NOT so easy to scale on the function isocurve

whereas no maths can do quick.

]]>

Isocurve ordered clockwise ...

ordered isocontour 2 1 sys 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 N ordered

ordered 410411412313314315316417418419420421522523624625626727728829830931103210331134113512361337133814391440154115421643174417451746184718481949195019512052205320542055205620572058205920602061206219631964196519661967186818691870177117721773177416751676167716781579158015811582158315841585158615871588158915901591169216931794179518961997209821992299231002410025101281023210336103401044410448104521035610360102641026810172101761008010084998899929896981009810498108981129811698120981249812898132981369714097144961489515293156911608716382166771677016863169561704917143172381733417327172231712017019169181681716716166151651516414163141621316113160121591215812157121561215511154111531115211151111501114911148111471114611145111441114311142111411114012139121381213712136121351213413133131321313113130131291412814127141261412515124151231512216121161201611917118171171711618115181141811319112191112011020109201082110721106221052210423103231022310124100249925982597269626952794279327922891289028892888288728862885278427832682268125802479247823772276217521742073207219711870186917681767166615651564146314621361136012591258115711561055105410539529518508498487477467456446436425415405394384374364354343333323313303292282272262252242232222212202192182172162152143133123114104958576673052 mat :

ordered axes 2 1 sys

x ordered 1 col : y ordered 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 :

native ordered A P 21 mat 1317450321 mat

for continuous junction S1 S1 S1 rows 1 range 12 range el :

n 5 :

iso S0 n Decimate S2 2 Decimate S3 S1 n Decimate stack :

X ordered 1 col : Y ordered 2 col : 21 sys

Xo X Mean 1 round : Yo Y Mean 1 round : 21 sys

isodata iso o 4 red plot :

centered iso 1 col Xo - iso 2 col Yo - augment : isocenter centered o 4 red plot : 21 sys

isodata isocenter 1 1 sys Iso centered

iso 65 - 28 - 66 - 24 - 65 - 20 - 64 - 16 - 63 - 12 - 61 - 8 - 58 - 4 - 56 - 054 - 452 - 850 - 1249 - 1649 - 2049 - 2450 - 2851 - 3252 - 3653 - 4054 - 4454 - 4854 - 5252 - 5649 - 6045 - 6241 - 6437 - 6533 - 6529 - 6625 - 6621 - 6617 - 6513 - 659 - 645 - 641 - 6336376211621561196123602760316035603960436047605160556059606360675971597558795783558753914994449739983299251001810111102510301044 - 10411 - 10315 - 9921 - 9524 - 9126 - 8726 - 8327 - 7927 - 7527 - 7126 - 6726 - 6325 - 5924 - 5523 - 5122 - 4720 - 4319 - 3917 - 3515 - 3114 - 2712 - 2310 - 1910 - 1511 - 1114 - 717 - 319 - 1 - 21 - 5 - 24 - 9 - 26 - 13 - 28 - 17 - 29 - 21 - 31 - 25 - 32 - 29 - 33 - 33 - 34 - 37 - 35 - 41 - 36 - 45 - 36 - 49 - 36 - 53 - 36 - 57 - 35 - 61 - 33 - 1102 mat :

x iso 1 col : y iso 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 :

decimated iso centered A P 21 mat 1293148421 mat

native ordered 1317450321 mat

Universal version  [Smath 6179]. It supports "data" sets from 
source ... data sets subject to be decimated from auxilliary
module Decimate(data,n)... reasonably fast on primitive PC. data char size clr plotG k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

P iso : X P 1 col : Y P 2 col : XY P o 4 red plotG : 41 line

t0 1 time :

For 'u' & cinterp X X X X stack : Y Y Y Y stack : 21 line

N 1000 :

Linear vector 'u' for cspline S n P rows : u 1 el 0 : i 2 n 2 n * + (range u i el u i 1 - el X i el X i 1 - el - Y i el Y i 1 - el - 22 mat det + : for |,| determinant uN u n 1 + el u 1 el - N / : (increments ≡ i 1 N range ux i el i uN * : for i 1 N range Sx i 1 + el u X ux i el linterp : Sy i 1 + el u Y ux i el linterp : Sx 1 el P 11 el : Sy 1 el P 12 el : 12 mat 31 line for Sx Sy augment 31 line eval 61 line :

S S :

S 2 1 sys

"Esplina": courtesy Mathcad 11. Permission granted from Mathsoft Collab AI2000. Smath code is demonstrative ... NOT 1/1 MCD

The cspline proceeds in sorted order of 'vx', thus excluding splining closed data sets. The vector 'u' is created from the descending vector of the determinants |X,Y|... thus: directly parent to X,Y. This condition is necessary and sufficient to spline a closed data set in Cartesian coordinates system. 'uN' [incremental] determines the spacing of the interpolated points over the N points.

x S 1 col : y S 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 :

A P 21 mat 1314549221 mat

1 time t0 - 17.8 s *

Populate Hermite . . . not successful the flat S2 segment direction X

presumably a non flat region will populate for other closed contour project.

]]> Iso centered

iso 65 - 28 - 66 - 24 - 65 - 20 - 64 - 16 - 63 - 12 - 61 - 8 - 58 - 4 - 56 - 054 - 452 - 850 - 1249 - 1649 - 2049 - 2450 - 2851 - 3252 - 3653 - 4054 - 4454 - 4854 - 5252 - 5649 - 6045 - 6241 - 6437 - 6533 - 6529 - 6625 - 6621 - 6617 - 6513 - 659 - 645 - 641 - 6336376211621561196123602760316035603960436047605160556059606360675971597558795783558753914994449739983299251001810111102510301044 - 10411 - 10315 - 9921 - 9524 - 9126 - 8726 - 8327 - 7927 - 7527 - 7126 - 6726 - 6325 - 5924 - 5523 - 5122 - 4720 - 4319 - 3917 - 3515 - 3114 - 2712 - 2310 - 1910 - 1511 - 1114 - 717 - 319 - 1 - 21 - 5 - 24 - 9 - 26 - 13 - 28 - 17 - 29 - 21 - 31 - 25 - 32 - 29 - 33 - 33 - 34 - 37 - 35 - 41 - 36 - 45 - 36 - 49 - 36 - 53 - 36 - 57 - 35 - 61 - 33 - 1102 mat :

plotG, RepEnd, Unwrap, Strip

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

z n RepEnd m z rows : i 1 n range w i el z 1 el : for i n 1 + (m n + (range w i el z i n - el : for i m n + 1 + (m 2 n * + (range w i el z m el : for w 51 line :

data Strip nest stripped in 'R' u data 1 col : v data 2 col : k 0 : i 1 data rows 1 - range v i 1 + el v i el ≤ (v i 1 + el v i el ≥ (¤ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for 21 line Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 81 line :

M Unwrap V M 1 col : i 2 M cols range V V M i col stack : for V 31 line :

XW iso :

r 2 :

Hermite Polyline Project

t0 1 time :

X XW 1 col r RepEnd : Y XW 2 col r RepEnd : XY X Y augment : 31 line

X XY 1 col : Y XY 2 col : Cntrl X Y . 14 black plotG : 31 line

α 16 / :

at α = 1/6 first & last points join, other α we have a zoom ± effect

Hermite PolyLine t P α t 3^- 3 t 2^* + 3 t * - 1 + (3 t 3^* 6 t 2^* - 4 + (3 t 3^* - 3 t 2^* + 3 t * + 1 + (t 3^41 mat * :

Each "j" segment is collected in matrix of 't' rows x 'j' cols. Each U,V is Unwrapped in column, augmented in 'xy', then stripped of repeated pairs for export.

t 01 0.25 range :

<= populate at will

Hermite Procedure

t j XP μ j 11 line : t P 1 el X μ el * t P 2 el X μ 1 + el * + t P 3 el X μ 2 + el * + t P 4 el X μ 3 + el * + 21 line :

t j YP μ j 11 line : t P 1 el Y μ el * t P 2 el Y μ 1 + el * + t P 3 el Y μ 2 + el * + t P 4 el Y μ 3 + el * + 21 line :

j runs 1..rows(X)-(r+1)

U j 1 X rows r 1 + (- range : ct 1 : j j i 1 t rows range Hx i ct el t i el j XP eval : for ct ct 1 + : 21 line for Hx 41 line :

V j 1 X rows r 1 + (- range : ct 1 : j j i 1 t rows range Hy i ct el t i el j YP eval : for ct ct 1 + : 21 line for Hy 41 line :

xy U Unwrap V Unwrap augment :

xy xy Strip :

polygon XW 1 col XW 2 col . 10 black plotG :

Hermite Polyline to Esplina

xy 1 1 sys

Hermite PolyLine 0...1

x P 1 el x P 2 el x P 3 el x P 4 el 4 1 sys

xy XW polygon 31 sys

1 time t0 - min 0.5