<p> Euler system </p>

1781a992-9587-4494-837f-4c544e79b2e7 17 Ian HOOTON 4 5 decimal

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

vx 140150160170180190200225250300350400450500126124123120118115112105998778706458214 mat :

vx.1 140150160170180190200225250300350400450500114 mat :

vy 2030405060708090100110120130140150141 mat :

σ.ct 1261241231201181151121059987787064581081071051031019996908575676055508988868583817974706255504541727170696866646057504540373459585756555452494641363330274848474645444340383330272422404039383737363331282522201933333332313130282623211917162828282726262524222018161413242424232322222019171513121121212020191919171614131211101818181717171615141311109816161515151514131211109871414141313131312111098771414 mat :

Note row 4 should be 375 d.t 50050050037530030025020020091 mat :

d.dd 2.6667 5.3333 8 10.6667 13.3333 16 18.6667 21.3333 2491 mat :

vx cols 14

a b c linterp2 i c length : result i Zeros : i 1 i range result i el a b c i el linterp : for result 41 line :

vx 11 vx cols range el transpose vx 21 vx cols range el transpose d.t linterp2 5858587487879911211291 mat

Determine value of q from σct table r 1 d.dd rows range d.dd r el 20 ≤ q r el vx 11 vx cols range el transpose vx 21 vx cols range el transpose d.t linterp2 r el : q r el vx.1 vy σ.ct d.t r el d.dd r el InterpBilinear : if for

Only transposed to fit on sheet q 58585874878799 109.8667 105.6 91 mat

Proof of 2nd to last row of d.t & d.dd vx.1 vy σ.ct 200 21.3333 InterpBilinear 109.8667

Proof of last row of d.t & d.dd vx.1 vy σ.ct 20024 InterpBilinear 105.6

q 58585874878799 109.8667 105.6 91 mat :

q 58585874878799 109.8667 105.6 91 mat ≡

vx 140150160170180190200225250300350400450500114 mat :

vy 2030405060708090100110120130140150141 mat :

M 1261241231201181151121059987787064581081071051031019996908575676055508988868583817974706255504541727170696866646057504540373459585756555452494641363330274848474645444340383330272422404039383737363331282522201933333332313130282623211917162828282726262524222018161413242424232322222019171513121121212020191919171614131211101818181717171615141311109816161515151514131211109871414141313131312111098771414 mat :

create/isolate M vy stack vx M stack augment :

data 14015016017018019020022525030035040045050020126124123120118115112105998778706458301081071051031019996908575676055504089888685838179747062555045415072717069686664605750454037346059585756555452494641363330277048484746454443403833302724228040403938373736333128252220199033333332313130282623211917161002828282726262524222018161413110242424232322222019171513121112021212020191919171614131211101301818181717171615141311109814016161515151514131211109871501414141313131312111098771515 mat :

1. Technique to interpolate a data table

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M x y Spline A M 112 M cols submatrix : B M 2 M rows 11 submatrix : C M 2 M rows 2 M cols submatrix : j 1 A length 2 range s j el B C 1 C rows j j submatrix y linterp : for A transpose s x linterp 51 line :

data 25080 Spline 31

data 24075 Spline 35.3

M x y Spline A M 112 M cols submatrix : B M 2 M rows 11 submatrix : C M 2 M rows 2 M cols submatrix : j 1 A length 2 range CC j el B C 1 C rows j j submatrix y linterp : for A transpose CC x linterp 51 line :

data x/y 0.05 0.1 0.2 0.5 1245102030451 1.8 1.7 1.65 1.55 1.45 1.4 1.35 1.35 01.3 1.3 1.3 1.3 0.9 1.85 1.75 1.7 1.6 1.55 1.5 1.45 1.45 1.4 1.4 1.4 1.4 0.7 1.9 1.85 1.75 1.7 1.65 1.6 1.6 1.55 1.55 1.55 1.55 1.55 0.5 1.95 1.85 1.8 1.75 1.75 1.7 1.7 1.7 1.7 1.7 1.7 1.75 0.3 1.95 1.9 1.85 1.8 1.8 1.8 1.8 1.8 1.8 1.85 1.85 1.85 0.2 1.95 1.9 1.85 1.8 1.8 1.8 1.8 1.8 1.85 1.9 1.9 1.95 0.16 1.95 1.9 1.85 1.85 1.8 1.8 1.85 1.85 1.85 1.9 1.9 1.95 813 mat :

sanity
sanity
user data 1 0.5 Spline data 70 0.5 Spline data 0.75 0.45 Spline 31 mat 1.75 1.83 1.76 31 mat

data x/y 0.05 0.1 0.2 0.5 124510203045461 1.8 1.7 1.65 1.55 1.45 1.4 1.35 1.35 1.3 1.3 1.3 1.3 1.3 0.9 1.85 1.75 1.7 1.6 1.55 1.5 1.45 1.45 1.4 1.4 1.4 1.4 1.4 0.7 1.9 1.85 1.75 1.7 1.65 1.6 1.6 1.55 1.55 1.55 1.55 1.55 1.55 0.5 1.95 1.85 1.8 1.75 1.75 1.7 1.7 1.7 1.7 1.7 1.7 1.75 1.75 0.3 1.95 1.9 1.85 1.8 1.8 1.8 1.8 1.8 1.8 1.85 1.85 1.85 1.85 0.2 1.95 1.9 1.85 1.8 1.8 1.8 1.8 1.8 1.85 1.9 1.9 1.95 1.95 0.16 1.95 1.9 1.85 1.85 1.8 1.8 1.85 1.85 1.85 1.9 1.9 1.95 1.95 814 mat :

sanity
sanity
user data 1 0.5 Spline data 70 0.5 Spline data 0.75 0.45 Spline 31 mat 1.75 1.75 1.76 31 mat

1. Construct the 3D plot

<= Introduce the 3D surface plot u v q u v data u v Spline 31 mat :

CreateMesh paramaters L 1 - : H 1 : N 31 : 31 sys

N mesh within the bounds [L, H] in each directions M u v q L H L H N N CreateMesh :

<p> Euler system </p>

Roll angle Ω1 1000 angle cos angle sin - 0 angle sin angle cos 33 mat :

Pitch angle Ω2 angle cos 0 angle sin 010 angle sin - 0 angle cos 33 mat :

Yaw angle Ω3 angle cos angle sin - 0 angle sin angle cos 000133 mat :

α π 4 / :

α β φ 13 mat π 6 / - 0 π * 0.2 π * 13 mat ≡

β 0.0 - (π * :

φ 0.0 - (π * :

<= Rotate around Y θ 0.5 :

2. Construct the 2D interpolated surface plot

Euler combined γ α Ω1 β Ω2 * φ Ω3 * eval :

M f B M θ Ω3 * γ * eval : B 1 col B 2 col augment 21 line :

x y axes x y / - :

plot the 2D surface plot plot M f :

plot axes 2 1 sys