<p> Euler system </p>

c551ada6-f2ff-4f26-8adc-53c585e98f5c 29 6 9 decimal

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

M Min Max Scale Max Min - (M M min - M max M min - / (* Min + :

Mesh U, V  within the bounds L, H L H N xd N intervals U 0 : dx H L - N / : i 1 N 1 + range U i el L dx i * dx - (+ : for U 51 line :

L 1 - :

H 1 :

N 6 :

U L H N xd :

V U :

u v f 1 u 2^v 2^+ (- 0.1 / exp * :

Generate the symmetric square matrix x y f U V Sym r 1 : i 1 U rows range c 1 : j 1 V rows range M r c el U i el V j el f eval : c c 1 + : 21 line for r r 1 + : 31 line for M 31 line :

data x y f U V Sym :

data max 1

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 :

Border the data set: Cols(data), Rows(data) -----------------------

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 :

R data R :

C data C :

Scale both X Y border directions

C C L H Scale :

R R L H Scale :

New R C data stack augment :

Command line interpolation

Collect "New", plug/isolate in new matrix

Interpolation is within the bounds of the running --- ± x and descending | ± y

The Spline(x,y,M) algorithm works on the inner quadrant. You range CreateMesh on the border quadrant. Neither ainterp/cinterp are valid interpolators.

1 - 1 - New Spline 0

0.6667 - 0.4286 New Spline 0.0039

New 1 - 1 - 0.667 - 0.333 - 0 0.333 0.667 1 0.714 - 2.061 109 -^* 5.332 107 -^* 1.495 105 -^* 4.54 105 -^* 1.495 105 -^* 5.332 107 -^* 2.061 109 -^* 0.429 - 5.332 107 -^* 0 0.004 0.012 0.004 0 5.332 107 -^* 0.143 - 1.495 105 -^* 0.004 0.108 0.329 0.108 0.004 1.495 105 -^* 0.143 4.54 105 -^* 0.012 0.329 1 0.329 0.012 4.54 105 -^* 0.429 1.495 105 -^* 0.004 0.108 0.329 0.108 0.004 1.495 105 -^* 0.714 5.332 107 -^* 0 0.004 0.012 0.004 0 5.332 107 -^* 1 2.061 109 -^* 5.332 107 -^* 1.495 105 -^* 4.54 105 -^* 1.495 105 -^* 5.332 107 -^* 2.061 109 -^* 88 mat :

u v q u v u v New Spline 31 mat :

<= Introduce the 3D surface plot

M u v q L H L H 2121 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 * :

θ 0.125 π * :

<= Rotate around Y

X 0 :

m 2 :

γ α Ω1 β Ω2 * φ Ω3 * eval :

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 M f :

plot XY 2 1 sys

Smath does not plot matrices, only the 3 vectors of its own CreateMesh style. The Spline algo accommodates rectangular/square matrices. Smath 3D plot is not scalable and most 3D will display weird or NOT. The Spline/CreateMesh method forces the 3D plot within the XYZ axes respective range. The Smath 3D triplet system may not be recognised by others. However, the Smath 3D is in fact a 4D export wrt either of the 3 orientation + rotation. M*Ω[α,β,φ,θ]*γ

α β φ θ 41 mat ° 4500 22.5 41 mat

| | Orientation |

<= Rotate around Y

Scatter plot Mathcad  style X Y Z u v Sr count 1 : i 1 u rows range j 1 v rows range x i el u i el : y j el v i el : U count el u i el v j el X : V count el u i el v j el Y : Z count el u i el v j el Z : count count 1 + : 61 line for for U V Z augment 31 line :

L 1 - :

H 1 :

N 40 :

U L H N xd :

V U :

x y X x :

x y Y y :

x y Z 1 x 2^y 2^+ (- 0.1 / exp * :

M X Y Z U V Sr :

The module Sr(X,Y,Z,u,v) is the Mathcad code to create "Scatter plot", i.e: the 3 sequential vectors that is not an "orthogonal triplet"