74e43036-0856-46f6-8e55-b22d04db720e 19 15 12 decimal

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

Spline_interpolate & CreateMesh surface plot

t0 1 time :

In this demo, we must prepare the data set by bordering the left 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). 1. linterp responds equivalent quadratic. The project is concluded by the surface "CreateMesh"

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

Add the "interpolating metric" ... here, we just invent Don't invent steam tables !

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 12345678910110 mat

data R 0123451 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 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

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

<= Introduce the 3D surface plot u v f u v u v New Spline 31 mat :

mesh by multiples rows/cols M u v f 4 - 410 - 10820 CreateMesh :

M 1 col length 952

Sanity check

col row New Spline

33 New Spline 0.07

11 New Spline 0.01

44 New Spline 0.01 -

22 New Spline 0.06

104 New Spline 0.08 -

Finaly, interpolation is realised between the "metric" rows/cols

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

6.25 3.5 New Spline 0.01

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 0.75 π * :

φ 0.125 - (π * :

X 0 :

m 5 :

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

<= turn OFF the rotation angle. θ 0 π * :

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 :

plot XY 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) [plot] is an export. 3. The 3D 'M' above, a 3 cols vector ... ... much different than Mathcad 11 that is a matrix. Smath "CreateMesh" is specific to Smath. 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.

Animate angular rotation

angularAnim 12 :

θ π angularAnim / :

θ ° 15

γ α Ω1 β Ω2 * φ Ω3 * eval :

step1 02 angularAnim * range :

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

plot M t f :

Angular animation wrt: θ, step1 range

plot

Notes: 1. In there, 't' is a "silent scalar" a scalar reserved for animation that must NOT be used otherwise ! 2. From the animation menu, animate 'f' algo by deault. 3. On the first "animation scan", all the FRAMES in 'step1' are calculated. 4. Once [3] has calculated all the angular animation FRAMES, the system will repeat indefinitely or as spec'd @ the FRAME rate/sec. 5. This hyper short Smath code is in some way "similar" to Mathcad 11 "spin"

1 time t0 - 41 s *

Create the bi-linear matrix ... interpolate

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

Ω 16 T : X 0.1 Ω 1 col (* : Y 0.1 Ω 1 row (* : N x/y X stack Y Ω stack augment : 41 line

In real work, the metric directions X,Y are from the data source [Steam Tables]. collected respectively in order !

The function bilinear(x,y,M) is universal. It demonstrates the construction and use. This demo generates itself. If mother matirix 'M'... is from source => create as inspired.

The universal bi-linear, spline2D/Smath CreateMesh x y M bilinear A M 11 range 2 M cols range el : B M 2 M rows range 11 range el : C M 2 M rows range 2 M cols range el : j 1 A length 2 range CC j el B C 1 C rows range j j range el y linterp : for A transpose CC x linterp 51 line :

... Sanity check ...
interpolation is 0ver
   (row_1,col_1,New) 0.2 0.3 N bilinear 0.6 0.2 N bilinear 0.25 0.25 N bilinear 31 mat 612 6.25 31 mat

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 Ω 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 32 T 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 32 T 0 32 T * 0 32 T * 1 3 mat

N x/y 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 0.1 12345678910111213141516 0.2 2468101214161820222426283032 0.3 36912151821242730333639424548 0.4 481216202428323640444852566064 0.5 5101520253035404550556065707580 0.6 6121824303642485460667278849096 0.7 714212835424956637077849198105112 0.8 81624324048566472808896104112120128 0.9 9182736455463728190991081171261351441102030405060708090100110120130140150160 1.1 112233445566778899110121132143154165176 1.2 1224364860728496108120132144156168180192 1.3 13263952657891104117130143156169182195208 1.4 14284256708498112126140154168182196210224 1.5 153045607590105120135150165180195210225240 1.6 1632486480961121281441601761922082242402561717 mat

x y b x y N bilinear :