2ef335a8-7776-49da-9380-3c20e15d81a7 57 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 for the data set by bording 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). Smath offers three interpolation options: 1. linterp [linear quadratic, 2. ainterp [Akima cubic spline], 3. cinterp [the more univsersal cubic spline]. The project is concluded by the surface "CreateMesh"

0.8561 - 0.3921 - 0.0719 0.536 1 0.6241 - 0.7865 - 0.7517 - 0.7169 - 0.7169 - 0.3921 - 0.7633 - 0.9722 - 1 - 0.7053 - 0.1601 - 0.7633 - 0.9675 - 0.9954 - 0.7146 - 0.0719 0.768 - 0.7517 - 0.7285 - 0.7193 - 55 mat

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 ainterp 51 line :

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

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 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----------------------------

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

<= Introduce the 3D surface plot

M u v f 4 - 410 - 102020 CreateMesh :

M 1 col length 2380

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 fractional rows/cols indices

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

9.5 3.5 New Spline 0.0025 -

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 - 5 / :

β π 8 / :

φ 0.85 π * :

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 :

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 :

coordinate axis XYZ k OX 10013 mat (γ * k col eval : k OY 01013 mat (γ * k col eval : k OZ 00113 mat (γ * k col eval : 41 sys XYZ 01 OX * - 02 OX * - 41 OX * 42 OX * 0041 OY * 42 OY * 01 OY * - 02 OY * - 0041 OZ * 42 OZ * 01 OZ * - 02 OZ * - 82 mat : TekstXYZ 00.10 41 OX * 42 OX * X 10 41 OY * 42 OY * y 10 41 OZ * 42 OZ * Z 10 45 mat : 31 sys

Animation parameter 't' removed

M f

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 disply, 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.

Animate angular rotation

plot XY 2 1 sys

angularAnim 24 :

θ π angularAnim / :

θ ° 7.5

γ α Ω1 β Ω2 * φ Ω3 * eval :

step1 02 angularAnim * range :

step1 00 range :

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

For plotting, 'step1' from plot menu [brush over click left, Animate] check step1 option.

plot M t f :

Click "Enable evaluation", click in the plot to start the animation system. check again "step1" to reanble plot:=f(M,t),,,, continuous animation starts @ ~ 200 sec. @ 15 ° angularAnim cpmpleting the circular rotation in 24 steps.

Angular animation wrt: θ, step1 range

plot XYZ TekstXYZ 3 1 sys

Notes: 1. In there, 't' is a "silent scalar" a scalar reserved for animation that must NOT be specified 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. 6. This hyper short Smath code is in some way "similar" to Mathcad 11 "spin"

1 time t0 - min 3.7