38ab7fd1-39dc-418d-ac09-0654b1fa5a7d 19 4 5 decimal

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

Spline_interpolate & CreateMesh surface plot

t0 1 time :

Spline_interpolate & CreateMesh surface plot In this demo, we must prepare the data set by bordering the right and upper side with ascending numbers. We leave these scales in integer to avoid confusion. The program is a dummy function that carries the 3 variates (x,y,M). The project is concluded by the surface "CreateMesh". Both directions are unique "linter" equivalent to quadratic. Spline(x,y,M) otherwise called bilinear(x,y,M)

x y M 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 49172535454101928395051120314254512233547606142638516656 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 C 12345616 mat

data R 01234561 mat

data R data C data stack augment

New data R data C data stack augment :

New 11 el x/y :

<= this position may be any alpha/numeric char.

New x/y 0.2 0.5 1 1.5 2 2.5 1.5 4917253545241019283950 2.5 5112031425445122335476056142638516667 mat :

1. Expand with the dummy directions 2. Isolate/new user support

The data set is ready for the surface plot----------------------------

u v f u v u v New Spline 31 mat :

<= Introduce the 3D surface plot

M u v f 0.2 2.5 1.5 52020 CreateMesh :

M 1 col length 2380

Sanity check

col row New Spline

2.5 4 New Spline 60

Finaly, interpolation is realised between fractional rows/cols indices

1.75 3.5 New Spline 39.5

If you are not used with that kind of maths, you may wonder if Smath is correct ? Answer is YES !

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

α 0.025 π * :

β 0.0625 π * :

φ 0.25 π * :

X 0 :

m 5 :

axes X m - 0 X 0 X 0 m + X 0 X 0 m - X 0 X m + 072 mat :

θ 0 π * :

<= turn OFF the animation angle as well as 't'.

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 :

plot M f :

Animation parameter 't' removed

M f axes 2 1 sys

Project note: 1. Generally, Engineering surfaces are linearly interpolated which in the case of splining for mesh-population is equivalent to quadratic. 2. f(M) is a display, NOT an export. The 3D export is 'M' above, a 3 cols vector ... ... much different than Mathcad 11 that is a matrix. Smath "CreateMesh" is specific to Smath. In this demo, CreateMesh is proportionally ranged, thus the surface is rectangular to image the original rectangular matrix. Smath does NOT plot matrix, thus not possible to display/superpose the original matrix.

Bi-Linear Interpolation

Brett Towler, USFWS, 1/11/2015

Bi-linear interpolation is an extension of linear interpolation for interpolating functions of two variables (e.g., x and y) on a regular 2D grid. The procedure, below, performs linear interpolation first in one direction, and then again in the other direction. Note that although each step is linear, the interpolation as a whole is not linear but instead quadratic. The code below would work for any size matrix.

x y M bilinear 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 :

1. Given the system [X, Y, XYZ]

X, Y are in ascending order of, any values representative of your system.

X 012345678910111 mat :

Y 012345678910111 mat :

XYZ 131814371440142614391437143314311422143214354781 - 305 - 490 - 727 - 199 - 864 - 259 - 140 - 2185 - 142 - 117 - 9976 - 3460 - 4357 - 5703 - 3064 - 5997 - 3463 - 2859 - 8049 - 2945 - 2803 - 14964 - 9005 - 10596 - 11972 - 8004 - 12181 - 8986 - 7414 - 13644 - 7632 - 7207 - 19953 - 16129 - 17258 - 18127 - 14822 - 18264 - 16120 - 14003 - 19128 - 14343 - 13777 - 24941 - 23501 - 23915 - 24239 - 23005 - 24278 - 23509 - 22369 - 24611 - 22715 - 22019 - 29933 - 30862 - 30649 - 30448 - 31111 - 30401 - 30858 - 31430 - 30162 - 31256 - 31698 - 35204 - 38293 - 37420 - 36747 - 39289 - 36636 - 38284 - 40228 - 35904 - 39831 - 40493 - 40403 - 45629 - 44141 - 42983 - 46735 - 42819 - 45625 - 47304 - 41594 - 47070 - 47462 - 45556 - 51682 - 50676 - 49176 - 52254 - 48957 - 51688 - 52543 - 47238 - 52429 - 52620 - 50676 - 55619 - 55266 - 54704 - 55830 - 54577 - 55631 - 55957 - 52850 - 55916 - 55983 - 1111 mat :

2. Construct the Bi-linear matrix

New x/y 0.1 X * stack 0.1 Y * transpose XYZ 0.0001 * stack augment :

Observe: Y direction ==> ... X direction [up/down] ... "x/y" any alpha numeric character New x/y 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 0.132 0.144 0.144 0.143 0.144 0.144 0.143 0.143 0.142 0.143 0.144 0.1 0.478 - 0.03 - 0.049 - 0.073 - 0.02 - 0.086 - 0.026 - 0.014 - 0.218 - 0.014 - 0.012 - 0.2 0.998 - 0.346 - 0.436 - 0.57 - 0.306 - 0.6 - 0.346 - 0.286 - 0.805 - 0.294 - 0.28 - 0.3 1.496 - 0.9 - 1.06 - 1.197 - 0.8 - 1.218 - 0.899 - 0.741 - 1.364 - 0.763 - 0.721 - 0.4 1.995 - 1.613 - 1.726 - 1.813 - 1.482 - 1.826 - 1.612 - 1.4 - 1.913 - 1.434 - 1.378 - 0.5 2.494 - 2.35 - 2.392 - 2.424 - 2.3 - 2.428 - 2.351 - 2.237 - 2.461 - 2.272 - 2.202 - 0.6 2.993 - 3.086 - 3.065 - 3.045 - 3.111 - 3.04 - 3.086 - 3.143 - 3.016 - 3.126 - 3.17 - 0.7 3.52 - 3.829 - 3.742 - 3.675 - 3.929 - 3.664 - 3.828 - 4.023 - 3.59 - 3.983 - 4.049 - 0.8 4.04 - 4.563 - 4.414 - 4.298 - 4.674 - 4.282 - 4.562 - 4.73 - 4.159 - 4.707 - 4.746 - 0.9 4.556 - 5.168 - 5.068 - 4.918 - 5.225 - 4.896 - 5.169 - 5.254 - 4.724 - 5.243 - 5.262 - 1 5.068 - 5.562 - 5.527 - 5.47 - 5.583 - 5.458 - 5.563 - 5.596 - 5.285 - 5.592 - 5.598 - 1212 mat

... Sanity check ...
interpolation is 0ver
   (row_1,col_1,New) 0.25 0.2 New bilinear 0.3 0.2 New bilinear 0.3 0.25 New bilinear 31 mat 0.503 - 0.57 - 0.884 - 31 mat

u v f u v u v New bilinear 31 mat :

M u v f 01012020 CreateMesh :

User/orient. γ 1101 - 101 - 0033 mat :

45°<= 11 - 011000033 mat

45°=> 1101 - 1000033 mat

Vertical 01010000033 mat

Horizontal 10001000033 mat

1101 - 101 - 0033 mat

plot M γ * 1 col M γ * 2 col augment :

The 3D surface plot is a freak that does not save settings neiter mesh user demand.

x y surface x y New bilinear :

plot M surface