<p> Euler system [radian] </p>

1b67ca4a-e5de-4b47-9171-9d6d829a3c0a 86 9 12 decimal

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

Linear interpolation/extrapolation data x Linterp vx data 1 col : vy data 2 col : x vx 1 el ≤ k 1 : x vx vx rows el ≥ k vx rows 1 - : k 1 : x vx k 1 + el ≥ k k 1 + : while 21 line if if vy k el vy k 1 + el vy k el - vx k 1 + el vx k el - / x vx k el - (* + 41 line :

XY 0.175 301.5 0.319 212.419 0.525 132.948 0.7 70.127 0.875 37.642 1.05 48.892 1.225 93.052 1.4 143.028 1.575 168.217 1.75 152.09 1.925 104.526 2.1 54.35 2.275 31.11 2.45 53.392 2.625 123.308 2.8 221.034 2.975 307.124 3.15 340.85 3.325 304.367 3.5 215.563 3.675 118.199 3.85 48.228 4.025 10.832 4.2 7.717 - 242 mat :

check extrapolation XY 0.125 Linterp XY 4.5 Linterp 21 mat 33240 - 21 mat

Smath built-in X Y x linterp

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

data point(s) sanity check XY 0.175 Linterp XY 1.4 Linterp XY 3.5 Linterp XY 4.2 Linterp 41 mat 301.5 143.028 215.563 7.717 - 41 mat

a b c spec o 5 black 13 mat : pts XY a b c spec pts : x f XY 1 col XY 2 col x ainterp : x c 1 0.175 x ≤ (x 4.2 ≤ (& cases : 41 sys

Carbon-12 α from smoothlet especially designed for this project *.sm document

pts x f x c * 2 1 sys

Spline2D . . . interpolates in data table

The matrix may be square or rectangular.

In this demo, we must prepare the data set by bordering the directions

in ascending numbers. We leave these scales in integer to avoid confusion.

The program is an algo function that carries the 3 variates (M,x,y).

1. Spline2D(M,x,y) => strictly linterp in both directions.

2. Spline2D(M,x,y) => strictly one pair at time, otherwise forLoop.

3. Linterp(data,x), linterp(X,Y,x), ainterp(X,Y,x), cinterp(X,Y,x)

interpolate strictly one value or 'x' scalar canvas at time.

]]>

M x y Spline2D 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 0.01 0.03 0.05 0.03 0.02 0.05 - 0.02 - 0.01 - 0.06 - 0.01 - 0.02 0.06 0.1 0.06 0.04 0.01 - 0.04 - 0.02 - 0.12 - 0.02 - 0.09 0.08 0.07 0.05 0.02 0.04 - 0.06 - 0.03 0.08 0.01 0.05 0.02 - 0.03 0.01 - 0.06 0.08 0.02 0.02 - 0.05 - 0.08 - 410 mat :

Prepare the data set ------------------------------------------------

data C j 1 data cols range C j el j : for C transpose 21 line :

data R i 1 data rows 1 + range R i el i 1 - : for R 21 line :

data R data C data stack augment

New data R data C data stack augment :

<= this position may be 
any alpha/numeric char. New 11 el x/y :

New x/y 123456789101 0.01 0.03 0.05 0.03 0.02 0.05 - 0.02 - 0.01 - 0.06 - 0.01 - 2 0.02 0.06 0.1 0.06 0.04 0.01 - 0.04 - 0.02 - 0.12 - 0.02 - 3 0.09 0.08 0.07 0.05 0.02 0.04 - 0.06 - 0.03 0.08 0.01 4 0.05 0.02 - 0.03 0.01 - 0.06 0.08 0.02 0.02 - 0.05 - 0.08 - 511 mat

sanity
sanity
user New 33 Spline2D New 73 Spline2D New 3.5 8.5 Spline2D 31 mat 0.07 0.06 - 0.22 - 31 mat

If we have done it right ? Answer is YES

Times Table n M i 1 n range j 1 n range M i j el i j * : for for M 21 line :

data 5 M :

data 1234524681036912154812162051015202555 mat

data C j 1 data cols range C j el j : for C transpose 21 line :

data R i 1 data rows 1 + range R i el i 1 - : for R 21 line :

New data R data C data stack augment :

<= this position may be 
any alpha/numeric char. New 11 el x/y :

New x/y 12345112345224681033691215448121620551015202566 mat

Sanity check

New 2.5 2.5 Spline2D New 1.5 1.5 Spline2D New 10 sqrt 10 sqrt Spline2D 31 mat 6.25 2.25 1031 mat

# col [X] row [Y] Spline

W3 0.02 0.17 2.79 0.07 0.72 10.6 0.14 1.49 24.4 0.23 2.5 40.4 0.32 3.8 58.4 0.42 5.17 78.4 0.54 6.7 103 0.66 8.36 133 0.8 10.3 205 0.95 12.2 253103 mat :

data C j 1 data cols range C j el j : for C transpose 21 line :

data R i 1 data rows 1 + range R i el i 1 - : for R 21 line :

New W3 R W3 C W3 stack augment :

W3 Min W3 Max 21 mat 0.02 25321 mat

<= this position may be 
any alpha/numeric char. New 11 el x/y :

.

New x/y 1231 0.02 0.17 2.79 2 0.07 0.72 10.6 3 0.14 1.49 24.4 4 0.23 2.5 40.4 5 0.32 3.8 58.4 6 0.42 5.17 78.4 7 0.54 6.7 1038 0.66 8.36 1339 0.8 10.3 20510 0.95 12.2 253114 mat

Sanity check

New 37 Spline2D New 15 Spline2D New 2 sqrt 7.5 Spline2D 31 mat 103 0.32 3.47 31 mat

# col [X] row [Y] Spline2D

Remark about meshed directions . . .

On the base principle Spline2D(M,vx,vy) one pair at time,

you can mesh either or both directions at will via

forLoop . . . or nested forLoop in both directions.

Be careful not to extrapolate or not too much.

. . . See example below

]]>

How Spline2D(M,x,y) works under the hood

M x y metric 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 s 51 line :

<= metric of cols <= metric of rows <= original matrix

<= isolate 's' A transpose s x linterp

New x/y 1231 0.02 0.17 2.79 2 0.07 0.72 10.6 3 0.14 1.49 24.4 4 0.23 2.5 40.4 5 0.32 3.8 58.4 6 0.42 5.17 78.4 7 0.54 6.7 1038 0.66 8.36 1339 0.8 10.3 20510 0.95 12.2 253114 mat :

Y direction first

each col is interpolated
@ entry "Y"
'x' is ignored S New absent 5.3 metric : 0.35 4.211 64.4 31 mat

1.5

X metric of cols A New 112 New cols submatrix : 12313 mat

Y metric of rows B New 2 New rows 11 submatrix : 12345678910101 mat

5.3

1. each col in "C" is 
interpolated @ entry 
"Y" on the scan 'j' 5 0.32 3.8 58.4 5.3 0.35 4.21 64.4 6 0.42 5.17 78.4 34 mat

2. on the Transposed metric "New"
"X" is interpolated @ user  [1.5]
answer => Spline2D(New,1.5,5.3) 12331 mat 0.35 4.211 64.4 31 mat 1.5 linterp 2.28

1 0.35 2 4.211 3 64.4 32 mat 1.5 Linterp 2.2805

Transpose 'A' to match 'S' interpolated in the Y direction

modular sanity ... observe that the rows [Y]
interpolation 5.3 is included in 'S' over running  X cols A transpose S 1.5 linterp New 1.5 5.3 Spline2D 21 mat 2.28 2.28 21 mat

# X direction [cols] Y direction [rows] Spline2D

x y Ω New x y Spline2D :

Maxima 3D does not save user settings

data 0.02 0.17 2.79 0.07 0.72 10.6 0.14 1.49 24.4 0.23 2.5 40.4 0.32 3.8 58.4 0.42 5.17 78.4 0.54 6.7 103 0.66 8.36 133 0.8 10.3 205 0.95 12.2 253103 mat :

New x/y 0.3333 0.6667 1 0.1 0.02 0.17 2.79 0.2 0.07 0.72 10.6 0.3 0.14 1.49 24.4 0.4 0.23 2.5 40.4 0.5 0.32 3.8 58.4 0.6 0.42 5.17 78.4 0.7 0.54 6.7 103 0.8 0.66 8.36 133 0.9 0.8 10.3 2051 0.95 12.2 253114 mat :

sanity
sanity
user New 0.6667 0.5 Spline2D New 11 Spline2D New 0.75 0.75 Spline2D 31 mat 3.8 253 35.14 31 mat

1. Construct the 3D plot

<= 3D vector function u v q u v New u v Spline2D 31 mat :

CreateMesh paramaters L 1 - : H 1 : N 21 : 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 [radian] </p>

rad Ω1 1000 rad cos rad sin - 0 rad sin rad cos 33 mat :

Матрица преобразования для поворота вокруг оси y rad Ω2 rad cos 0 rad sin 010 rad sin - 0 rad cos 33 mat :

Матрица преобразования для поворота вокруг оси z rad Ω3 rad cos rad sin - 0 rad sin rad 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 π 4 / :

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 0 :

m 5 :

2D surface axes system XY X m - X X X X X X m + 00 m 0 m - 0027 mat transpose :

γ 1000 0.707 0.707 - 0 0.707 0.707 33 mat

plot the 2D surface plot plot M f :

click left in plot, move [X,Y] by mouse ... resize the plot frame.

plot XY 2 1 sys

Engineering application

T e m p e r a t u r e [°C]

P r e s s u r e [ata]

Steam T/P 1 1.1 1.2 1.3 950 5.754 5.232 4.797 4.427 960 5.801 5.275 4.836 4.463 970 5.848 5.318 4.875 4.499 980 5.895 5.360 4.914 4.535 990 5.942 5.402 4.953 4.571 1000 5.989 5.444 4.992 4.607 75 mat :

Engineering app's => will be "linterp" Steam 1970 Spline2D Steam 1.15 975 Spline2D 21 mat 5.848 5.117 21 mat

Steam P °C Spline2D

Obvious preference . . .

For Engineering applications, copy/paste

algo Spline2D(M,x,y) . . . construct the

image Engineering data table => interpolate.

]]>

X 1 1.1 1.2 1.3 14 mat :

Y 950960970980990100061 mat :

M 5.754 5.232 4.797 4.427 5.801 5.275 4.836 4.463 5.848 5.318 4.875 4.499 5.895 5.36 4.914 4.535 5.942 5.402 4.953 4.571 5.989 5.444 4.992 4.607 64 mat :

X Y M 1.15 975 InterpBilinear 5.11675

X Y M x col y row InterpBilinear

X 1 1.1 1.2 1.3 14 mat :

Y 950960970980990100061 mat :

M 5.754 5.232 4.797 4.427 5.801 5.275 4.836 4.463 5.848 5.318 4.875 4.499 5.895 5.36 4.914 4.535 5.942 5.402 4.953 4.571 5.989 5.444 4.992 4.607 64 mat :

Spline-mesh an Engineering table . . .

The procedure is demo only for convenience.

No gain in accuracy, just populated. May be useful with

the search-in-table tools [not offered]. At higher mesh,

table kernel may be gray scaled [0..255] for rendering

a zone image [think in term of geographical °C]

]]>

populate the cols
populate the rows vx 1 1.3 1.025 range : vy 9501000955 range : 21 sys

M x y Spline2D 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 :

Steam T/P 1 1.1 1.2 1.3 950 5.754 5.232 4.797 4.427 960 5.801 5.275 4.836 4.463 970 5.848 5.318 4.875 4.499 980 5.895 5.360 4.914 4.535 990 5.942 5.402 4.953 4.571 1000 5.989 5.444 4.992 4.607 75 mat :

Optimiz Numeric Steam 1970 Spline2D Steam 1.15 975 Spline2D 21 mat 5.848 5.11675 21 mat

Table i 1 vx rows range j 1 vy rows range New i j el Steam vx i el vy j el Spline2D : for for x/y vy stack vx transpose New transpose stack augment 21 line :

... set decimals copy/paste/export

Table 11 mat x/y 1 1.02 1.05 1.08 1.1 1.12 1.15 1.18 1.2 1.23 1.25 1.27 1.3 950 5.75 5.62 5.49 5.36 5.23 5.12 5.01 4.91 4.8 4.7 4.61 4.52 4.43 955 5.78 5.65 5.52 5.38 5.25 5.14 5.04 4.93 4.82 4.72 4.63 4.54 4.44 960 5.8 5.67 5.54 5.41 5.28 5.17 5.06 4.95 4.84 4.74 4.65 4.56 4.46 965 5.82 5.69 5.56 5.43 5.3 5.19 5.08 4.97 4.86 4.76 4.67 4.57 4.48 970 5.85 5.72 5.58 5.45 5.32 5.21 5.1 4.99 4.88 4.78 4.69 4.59 4.5 975 5.87 5.74 5.61 5.47 5.34 5.23 5.12 5.01 4.89 4.8 4.71 4.61 4.52 980 5.9 5.76 5.63 5.49 5.36 5.25 5.14 5.03 4.91 4.82 4.72 4.63 4.54 985 5.92 5.78 5.65 5.52 5.38 5.27 5.16 5.05 4.93 4.84 4.74 4.65 4.55 990 5.94 5.81 5.67 5.54 5.4 5.29 5.18 5.07 4.95 4.86 4.76 4.67 4.57 995 5.97 5.83 5.69 5.56 5.42 5.31 5.2 5.09 4.97 4.88 4.78 4.68 4.59 1000 5.99 5.85 5.72 5.58 5.44 5.33 5.22 5.1 4.99 4.9 4.8 4.7 4.61 1214 mat 11 mat