Data set

123c4c0f-4916-4d7c-8be2-32508dcd9692 36 6 9 decimal

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

Incremental analytical spline derivative

Data set

data 0 0.195162 0.05 0.199478 0.1 0.208073 0.15 0.220832 0.2 0.23765 0.25 0.258297 0.3 0.282608 0.35 0.31025 0.4 0.341006 0.45 0.374454 0.5 0.410342 0.55 0.448173 0.6 0.487676 0.65 0.5283 0.7 0.569774 0.75 0.611512 0.8 0.653261 0.85 0.694421 0.9 0.734779 0.95 0.773736 1 0.811136 1.05 0.846402 1.1 0.879447 1.15 0.909728 1.2 0.937241 1.25 0.961489 1.3 0.982557 1.35 1 1.4 1.013997 1.45 1.024155 1.5 1.030749 1.55 1.033436 1.6 1.03258 1.65 1.027885 1.7 1.019796 1.75 1.008054 1.8 0.993176 1.85 0.974929 1.9 0.953886 1.95 0.929829 2 0.903376 2.05 0.874307 2.1 0.843271 2.15 0.810036 2.2 0.775267 2.25 0.738707 2.3 0.701026 2.35 0.661933 2.4 0.622092 2.45 0.581171 2.5 0.53982 2.55 0.497663 2.6 0.455334 2.65 0.412408 2.7 0.369501 2.75 0.326143 2.8 0.282936 2.85 0.239366 2.9 0.196024 2.95 0.152361 3 0.108965 3.05 0.06526 3.1 0.021834 3.15 0.021905 - 3.2 0.065357 - 3.25 0.109124 - 3.3 0.152587 - 3.35 0.196347 - 3.4 0.239758 - 3.45 0.283414 - 3.5 0.326638 - 3.55 0.37001 - 3.6 0.412818 - 3.65 0.455624 - 3.7 0.497683 - 3.75 0.539536 - 3.8 0.580405 - 3.85 0.620813 - 3.9 0.659954 - 3.95 0.698338 - 4 0.735134 - 4.05 0.770845 - 4.1 0.804626 - 4.15 0.836979 - 4.2 0.867054 - 4.25 0.895362 - 4.3 0.921057 - 4.35 0.944666 - 4.4 0.965357 - 4.45 0.983684 - 4.5 0.998839 - 4.55 1.011406 - 4.6 1.020609 - 4.65 1.02707 - 4.7 1.03005 - 4.75 1.030212 - 4.8 1.026857 - 4.85 1.020691 - 4.9 1.011055 - 4.95 0.998695 - 5 0.982994 - 5.05 0.964733 - 5.1 0.943331 - 5.15 0.919598 - 5.2 0.892986 - 5.25 0.864325 - 5.3 0.83309 - 5.35 0.800124 - 5.4 0.764919 - 5.45 0.728319 - 5.5 0.689824 - 5.55 0.650271 - 5.6 0.609159 - 5.65 0.56731 - 5.7 0.524216 - 5.75 0.480676 - 5.8 0.436171 - 5.85 0.39147 - 5.9 0.346041 - 5.95 0.300621 - 6 0.254662 - 6.05 0.208869 - 6.1 0.162682 - 6.15 0.116774 - 6.2 0.070577 - 6.25 0.024736 - 6.3 0.02132 6.35 0.066967 6.4 0.112773 6.45 0.158126 6.5 0.203584 6.55 0.248543 6.6 0.293537 6.65 0.337966 6.7 0.382334 6.75 0.426035 6.8 0.46954 6.85 0.512232 6.9 0.554542 6.95 0.595841 7 0.636519 7.05 0.675936 7.1 0.714439 7.15 0.751381 7.2 0.78707 7.25 0.820858 7.3 0.85302 7.35 0.882914 7.4 0.910793 7.45 0.93603 7.5 0.958868 7.55 0.978707 7.6 0.995789 7.65 1.009557 7.7 1.020264 7.75 1.027408 7.8 1.031263 7.85 1.031398 7.9 1.028119 7.95 1.02107 8 1.0106 8.05 0.99644 8.1 0.978984 8.15 0.958052 8.2 0.934088 8.25 0.907002 8.3 0.877284 8.35 0.84493 8.4 0.810473 8.45 0.773988 8.5 0.736038 8.55 0.696767 8.6 0.656758 8.65 0.616207 8.7 0.575699 8.75 0.53547 8.8 0.496088 8.85 0.457809 8.9 0.421165 8.95 0.386415 9 0.354037 9.05 0.324271 9.1 0.297526 9.15 0.274007 9.2 0.254036 9.25 0.23777 9.3 0.225431 9.35 0.217118 9.4 0.212945 1892 mat :

Recast cspline

Read this:

L 0.2 : H 9.2 : h 0.01 : 31 line

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

Observe the denominator of each spline derivative: 2nd derivative has h^2. Thus, if 'h' is too finely meshed, 2nd deriv will generate noise, though the 1rst is largely insensible. For this particular example, in that range (9.2-0.2)/0.1 we get 90 points on the canvas => more than enough for a net line/point plot. Problem is that 'h' must be quite small for Legendre Quadrature.

x φ 0.2 x ≤ (x 9.2 ≤ (& X Y x cinterp 0 if :

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 ≡

Analytical/incremental spline derivative

x φ x D1 x sin 4 1 sys

Setup dichotomy x f x φ : a 3 : b 3.25 : ε 1012 -^: xn b a + (2 / : 51 line

Dichotomy n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line

Sanity check xn xn f n 31 mat 3.125 1.031 1013 -^* - 3631 mat

Legendre wr 63 ... 64

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

0.2 3.125 6.277 9.2 41 mat

a 0.2 : b 9.2 : 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 :

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

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 11.149

Sanity check vs the closely related  sin(x).
By inspection, φ(x) is a bit longer than sin(x)
on the same range [0.2 .. 9.2]. 1 x sin x diff (2^+ sqrt x 0.2 9.2 int GraphUnit 10.862135

1 x sin x diff (2^+ sqrt x 03 π * int 11.461

Symbolic arclength gets soon complicated

1 u 2^(u diff (2^+ sqrt u int maple 2 u * 14 u 2^* + sqrt * 2 u * asinh + 4 /

1 u 3^(u diff (2^+ sqrt u int maple u 19 u 4^* + (* 6 sqrt * 1 i + (* 413 i * u 2^* - sqrt * 13 i * u 2^* + sqrt * u 6 sqrt * 1 i + (* 2 / i EllipticF * + 36 sqrt * 1 i + (* 19 u 4^* + sqrt * /

1 u sin u diff (2^+ sqrt u int maple u sin 2^sqrt u cos i EllipticE * u sin / -