<p> Euler system </p>

f6c3e3f0-80e5-496b-8561-73ee60b68c0a 30 4 5 decimal

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

2D rectangular spline on METRIC DOMAIN

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 :

Lo 0 : Hi 1 : 12 mat

User raw matrix raw 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 :

rangeX i 1 raw cols range vx i el i Hi Lo - raw cols / * : for vx 21 line :

rangeY i 1 raw rows range vy i 1 + el i Hi Lo - raw rows / * : for vy 21 line :

New rangeY rangeX transpose raw stack augment : 0 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

raw length 30

New 11 el x/y :

Explain this Smath document:

1. A square/rectangular matrix is from user.

2. The matrix is metric in the domain [X & Y directions].

3. Can be used to interpolate within the [X, Y] domain.

4. The Spline(New,x,y) algo is passed in the vector function

q(u,v) that will produce the Smath style 3 vectors 3D

CreateMesh(q(u,v),L,H,L,H,N,N)

5. It is subjected to Euler pre-assigned rotation system.

6. f(M) extracts the 2D surface plot for the X_Y plot.

]]>

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

For this demo, the user raw matrix is 
auto-constructed in the metric domain.
copy/paste in 'New' 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 :

strictly interpolate within the respective domains: X [1/3 ... 1],,, Y[0.1 ..1]

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

# col[X] row[Y] Spline

1. Construct the 3D plot

<= Introduce the 3D surface plot u v q u v New u v Spline 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 </p>

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

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

Матрица преобразования для поворота вокруг оси z 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 0.125 π * :

2. Construct the 2D interpolated surface plot

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

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 :

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

plot the 2D surface plot plot M f :

plot XY 2 1 sys

rotate/zoom by mouse. 'γ' rotation added to calm down the mouse "rotate/zoom"

M γ * 10 *

rangeX i 1 raw cols range vx i el i Hi Lo - raw cols / * : for vx 21 line : rangeY i 1 raw rows range vy i el i Hi Lo - raw rows / * : for vy 21 line : 21 line

x/y rangeX transpose rangeY raw 22 mat x/y 0.3333 0.6667 113 mat 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1101 mat 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 22 mat

Nicolas specific 'x' scale x h 50400500031 mat 13 / 23 / 131 mat x linterp :

x 200 h : 0.4762

sanity
sanity
user ...
user ... New 0.6667 0.5 Spline New 11 Spline New 0.75 0.75 Spline New x 0.75 Spline 41 mat 3.8 253 35.14 3.57 41 mat

W1 x/y 504005000 0.1 0.02 0.16 1.65 0.2 0.08 0.71 6.83 0.3 0.16 1.55 15.2 0.4 0.26 2.57 25.4 0.5 0.37 3.75 37.7 0.6 0.48 5.05 52 0.7 0.62 6.49 66.1 0.8 0.76 8.09 83.1 0.9 0.82 9.84 1031 1.09 11.8 156114 mat :

W2 x/y 504005000 0.1 0.02 0.18 2.18 0.2 0.08 0.73 10.6 0.3 0.16 1.5 19.1 0.4 0.26 2.54 31.7 0.5 0.37 3.86 45.9 0.6 0.47 5.22 61.5 0.7 0.61 6.77 81.1 0.8 0.75 8.47 104 0.9 0.9 10.4 1611 1.07 12.3 198114 mat :

W3 x/y 504005000 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 :

otherwise in the original domain W3 400 0.6 Spline W3 5000 0.3 Spline W3 200 0.75 Spline 31 mat 5.17 24.4 3.57 31 mat

W1 W2 W3 31 mat 200 0.75 Spline vectorize 3.52 3.65 3.57 31 mat

Naturally, there is no limit to "user interpolation" A reasonable amount of "user" could be looped through.