<p> Legendre wr 63 ... 64 </p>

db3f50b5-d506-4132-a0e4-59cb94abfe54 22 3 5 auto

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

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

x y f U V M2D r 1 : i 1 U rows range c 1 : j 1 V rows range M r c el U i el V j el f eval : c c 1 + : 21 line for r r 1 + : 31 line for M 31 line :

Symbolic

M lo hi Scale hi lo - (M M min - M max M min - / (* lo + :

M mirror M 1 M rows range M cols 1 range el :

L 1 - : H 1 : N 32 : 13 mat U L H N xd : V U : 12 mat 21 line

t0 1 time :

Vectorise Diagonal Mat Dia i 1 Mat rows range Dia i el Mat i i el : for Dia 21 line :

x y f 1 x 2^y 2^+ (- 0.25 / exp * : img x y f U V M2D : 21 line

Numeric spot img 0255 Scale :

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 spot

Symmetric spot tr spot mirror tr 21 mat 2554.704 2554.704 21 mat

Win 7 Edition Starter 1.66 GHz 1 time t0 - 4.4 s *

data U spot Dia augment :

data

X data 1 col : Y data 2 col : 21 line

x φ L x ≤ (x H ≤ (& X Y x cinterp 0 if :

h 0.01 :

x D1 x 2 h * - φ x 2 h * + φ - 4 h * / - : 1rst spline derivative ≡

x D2 x 2 h * + φ - 16 x h + φ * + 30 x φ * - 16 x h - φ * + x 2 h * - φ - 12 h 2^* / : 2nd spline derivative ≡

Code the integrand x f 1 x D1 (2^+ sqrt :

x f

0 f 1

can not parse fake function
alternately use FD
FD => finite differences x f x 1 - 1 int #

<p> Legendre wr 63 ... 64 </p>

wr 0.0486909570091397 0.0243502926634244 - 0.0486909570091397 0.0243502926634244 0.0485754674415034 0.0729931217877990 - 0.0485754674415034 0.0729931217877990 0.0483447622348030 0.1214628192961206 - 0.0483447622348030 0.1214628192961206 0.0479993885964583 0.1696444204239928 - 0.0479993885964583 0.1696444204239928 0.0475401657148303 0.2174236437400071 - 0.0475401657148303 0.2174236437400071 0.0469681828162100 0.2646871622087674 - 0.0469681828162100 0.2646871622087674 0.0462847965813144 0.3113228719902110 - 0.0462847965813144 0.3113228719902110 0.0454916279274181 0.3572201583376681 - 0.0454916279274181 0.3572201583376681 0.0445905581637566 0.4022701579639916 - 0.0445905581637566 0.4022701579639916 0.0435837245293235 0.4463660172534641 - 0.0435837245293235 0.4463660172534641 0.0424735151236536 0.4894031457070530 - 0.0424735151236536 0.4894031457070530 0.0412625632426235 0.5312794640198946 - 0.0412625632426235 0.5312794640198946 0.0399537411327203 0.5718956462026340 - 0.0399537411327203 0.5718956462026340 0.0385501531786156 0.6111553551723933 - 0.0385501531786156 0.6111553551723933 0.0370551285402400 0.6489654712546573 - 0.0370551285402400 0.6489654712546573 0.0354722132568824 0.6852363130542333 - 0.0354722132568824 0.6852363130542333 0.0338051618371416 0.7198818501716109 - 0.0338051618371416 0.7198818501716109 0.0320579283548516 0.7528199072605319 - 0.0320579283548516 0.7528199072605319 0.0302346570724025 0.7839723589433414 - 0.0302346570724025 0.7839723589433414 0.0283396726142595 0.8132653151227975 - 0.0283396726142595 0.8132653151227975 0.0263774697150547 0.8406292962525803 - 0.0263774697150547 0.8406292962525803 0.0243527025687109 0.8659993981540928 - 0.0243527025687109 0.8659993981540928 0.0222701738083833 0.8893154459951141 - 0.0222701738083833 0.8893154459951141 0.0201348231535302 0.9105221370785028 - 0.0201348231535302 0.9105221370785028 0.0179517157756973 0.9295691721319396 - 0.0179517157756973 0.9295691721319396 0.0157260304760247 0.9464113748584028 - 0.0157260304760247 0.9464113748584028 0.0134630478967186 0.9610087996520538 - 0.0134630478967186 0.9610087996520538 0.0111681394601311 0.9733268277899110 - 0.0111681394601311 0.9733268277899110 0.0088467598263639 0.9833362538846260 - 0.0088467598263639 0.9833362538846260 0.0065044579689784 0.9910133714767443 - 0.0065044579689784 0.9910133714767443 0.0041470332605625 0.9963401167719553 - 0.0041470332605625 0.9963401167719553 0.0017832807216964 0.9993050417357722 - 0.0017832807216964 0.9993050417357722 642 mat :

wr 0.0494723666239310 0.0000000000000000 0.0494118330399182 0.0494521871161596 - 0.0494118330399182 0.0494521871161596 0.0492303804237476 0.0987833564469453 - 0.0492303804237476 0.0987833564469453 0.0489284528205120 0.1478727863578720 - 0.0489284528205120 0.1478727863578720 0.0485067890978838 0.1966003467915067 - 0.0485067890978838 0.1966003467915067 0.0479664211379951 0.2448467932459534 - 0.0479664211379951 0.2448467932459534 0.0473086713122689 0.2924940585862514 - 0.0473086713122689 0.2924940585862514 0.0465351492453837 0.3394255419745844 - 0.0465351492453837 0.3394255419745844 0.0456477478762926 0.3855263942122479 - 0.0456477478762926 0.3855263942122479 0.0446486388259414 0.4306837987951116 - 0.0446486388259414 0.4306837987951116 0.0435402670830276 0.4747872479948044 - 0.0435402670830276 0.4747872479948044 0.0423253450208158 0.5177288132900333 - 0.0423253450208158 0.5177288132900333 0.0410068457596664 0.5594034094862850 - 0.0410068457596664 0.5594034094862850 0.0395879958915441 0.5997090518776252 - 0.0395879958915441 0.5997090518776252 0.0380722675843496 0.6385471058213654 - 0.0380722675843496 0.6385471058213654 0.0364633700854573 0.6758225281149861 - 0.0364633700854573 0.6758225281149861 0.0347652406453559 0.7114440995848458 - 0.0347652406453559 0.7114440995848458 0.0329820348837793 0.7453246483178474 - 0.0329820348837793 0.7453246483178474 0.0311181166222198 0.7773812629903724 - 0.0311181166222198 0.7773812629903724 0.0291780472082805 0.8075354957734567 - 0.0291780472082805 0.8075354957734567 0.0271665743590979 0.8357135543195029 - 0.0271665743590979 0.8357135543195029 0.0250886205533450 0.8618464823641238 - 0.0250886205533450 0.8618464823641238 0.0229492710048899 0.8858703285078534 - 0.0229492710048899 0.8858703285078534 0.0207537612580391 0.9077263027785316 - 0.0207537612580391 0.9077263027785316 0.0185074644601613 0.9273609206218432 - 0.0185074644601613 0.9273609206218432 0.0162158784103383 0.9447261340410098 - 0.0162158784103383 0.9447261340410098 0.0138846126161156 0.9597794497589419 - 0.0138846126161156 0.9597794497589419 0.0115193760768800 0.9724840346975701 - 0.0115193760768800 0.9724840346975701 0.0091259686763267 0.9828088105937273 - 0.0091259686763267 0.9828088105937273 0.0067102917659601 0.9907285468921895 - 0.0067102917659601 0.9907285468921895 0.0042785083468638 0.9962240127779701 - 0.0042785083468638 0.9962240127779701 0.0018398745955771 0.9992829840291237 - 0.0018398745955771 0.9992829840291237 632 mat :

This page was compiled by Mike "Pomax" Kamermans, on June 5th, 2011, as a resource for the primer on Bezier curves, specifically the section that deals with arc length computation. You are free to do what you like with this data. Go make the world a better place than it was when you found it.

w wr 1 col : weight, wi ≡ x wr 2 col : abscissea,xi ≡ 21 line

a 1 - : b 1 : m wr rows : 31 line

1 x q x diff (2^+ sqrt x int b a - 2 / (w i el y i el fnct * (i 1 m sum eval * abs ≡

y i 1 m range y i el b a - 2 / (x i el * b a + 2 / (+ : for y 21 line :

GraphUnit 1 :

Cumulative arc length of φ(x). Note that for the quadrature,
φ(x) must have the default numerical value '0' not text. ArcLength b a - 2 / (w i el y i el f * (i 1 m sum eval * : GraphUnit 508