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3D Utilities

1. Sr(X,Y,Z,u,v) creates the 3 sequential vectors Packing 'Z' in matrix is a very long operation that is not retained for practical use. Instead, we use 2. Sym(f(x,y),U,V) a matrix creator. Both algorithms are paired with xd(L,H,N)

Sequential triplet : U, V, Z x y X x y Y x y Z u v Sr count 1 : i 1 u rows range j 1 v rows range U count el u i el v j el X eval : V count el u i el v j el Y eval : Z count el u i el v j el Z eval : count count 1 + : 41 line for for U V Z augment 31 line :

Mesh U, V  within the bounds L, H L H N xd U 0 : dx H L - N / : i 1 N 1 + (range U i el L dx i * dx - (+ : for U 41 line :

N 40 :

L 1 - :

H 1 :

U 1 - 1 N xd :

V U :

x y X x :

x y Y y :

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

G x y X x y Y x y Z U V Sr :

P1 G 1 col G 3 col augment :

G 1 - 1 - 2.0612 109 -^* 1 - 0.95 - 5.4645 109 -^* 1 - 0.9 - 1.3781 108 -^* 1 - 0.85 - 3.3058 108 -^* 1 - 0.8 - 7.5435 108 -^* 1 - 0.75 - 1.6374 107 -^* 1 - 0.7 - 3.3807 107 -^* 1 - 0.65 - 6.6399 107 -^* 1 - 0.6 - 1.2405 106 -^* 1 - 0.55 - 2.2045 106 -^* 1 - 0.5 - 3.7267 106 -^* 1 - 0.45 - 5.9925 106 -^* 1 - 0.4 - 9.1661 106 -^* 1 - 0.35 - 1.3337 105 -^* 1 - 0.3 - 1.8458 105 -^* 153 mat

P2 G 2 col G 3 col augment :

P1 P2 2 1 sys

θ rot θ cos θ sin - θ sin θ cos 22 mat :

θ π - 2 / :

θ 1.5708 -

Stem plot

P1 θ rot *

Zone plot

P2 θ rot *

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 U 0 : dx H L - N / : i 1 N 1 + range U i el L dx i * dx - (+ : for U 21 line 31 line :

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 :

U 0 :

V 0 :

L 1 - :

H 1 :

N 6 :

U L H N xd :

V U :

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 linterp 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 :

C C L H Scale :

R R L H Scale :

C 1 - 0.6667 - 0.3333 - 0 0.3333 0.6667 117 mat

var 1 el 0 :

R var C transpose stack :

New R C data stack augment :

New 11 el x/y :

Collect "New", plug/isolate in new matrix

Command line interpolation

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

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

1 - 1 - New Spline 2.0612 109 -^*

Isolate the matrix data

0.6667 - 0.6667 New Spline 0.0001

New evaluate, copy/paste in new variable ≡

New x/y 1 - 0.6667 - 0.3333 - 0 0.3333 0.6667 11 - 2.0612 109 -^* 5.3316 107 -^* 1.4945 105 -^* 4.54 105 -^* 1.4945 105 -^* 5.3316 107 -^* 2.0612 109 -^* 0.6667 - 5.3316 107 -^* 0.0001 0.0039 0.0117 0.0039 0.0001 5.3316 107 -^* 0.3333 - 1.4945 105 -^* 0.0039 0.1084 0.3292 0.1084 0.0039 1.4945 105 -^* 0 4.54 105 -^* 0.0117 0.3292 1 0.3292 0.0117 4.54 105 -^* 0.3333 1.4945 105 -^* 0.0039 0.1084 0.3292 0.1084 0.0039 1.4945 105 -^* 0.6667 5.3316 107 -^* 0.0001 0.0039 0.0117 0.0039 0.0001 5.3316 107 -^* 1 2.0612 109 -^* 5.3316 107 -^* 1.4945 105 -^* 4.54 105 -^* 1.4945 105 -^* 5.3316 107 -^* 2.0612 109 -^* 88 mat :

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

<= Introduce the 3D surface plot

M u v q 1 - 11 - 12121 CreateMesh :

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

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

Матрица преобразования для поворота вокруг оси z ω Ω3 ω cos ω sin - 0 ω sin ω cos 000133 mat :

α π 3 / :

β 0.0 - (π * :

φ 0.0 - (π * :

θ 0.25 π * :

<= Rotate around Y

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 :

x y XY x y / - :

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 ° 60004541 mat

| | Orientation |

<= Rotate around Y