<content> pGrid, pMesh and pSurf </content>

0451ee08-0972-434d-8953-c5f992ad20b2 7 Alvaro Diaz Falconi 4 5 false false 0 0 decimal

Grids, Meshes and Surfaces

<content> pGrid, pMesh and pSurf </content> <content> pUtil </content>

a b c pDiag a b c 13 mat transpose diag 11 line :

Argument handling utility γ pArgγ γ length 2 ≡ γ 1 el γ 2 el pView2 pArgγ η γ 1 col γ 1 col norme / γ 2 col γ 2 col norme / augment : ζ η 32 el η 21 el * η 32 el η 11 el * - η 11 el η 22 el * sign 1 η 32 el 2^- abs sqrt * stack : η ζ augment γ 13 range 12 range el 12 mat eval 31 line if 11 line :

γ 32 el γ 21 el * γ 32 el γ 11 el * - γ 11 el γ 22 el * sign 1 γ 32 el 2^- abs sqrt * 31 mat

Argument handling utility x n pArgN ans x eval : ans sys2mat.1 stack rows 1 ≡ k 1 n range ans k el x : for 11 line if ans 31 line :

x y z pAxis 000 x 000000 y 000000 z 00073 mat :

M γ pAxis pShowAxis M cols 3 ≡ A 0 : k 13 range : 12 mat k m 00013 mat 1.05 M k col max * 3 identity k row * stack γ 13 range 12 range el * eval : A k el polyline k m 11 mat red green blue 31 mat k el solid 2 stack : eval 31 line ToDo if 01 matrix if 11 line :

r g b a rgb hex 0123456789ABCDEF : d 0 : 12 mat c H n 0255 c 0 round 12 mat min 12 mat max : m n 16 mod : 12 mat n m - 16 mod 0 ≡ hex n m - 16 / 1 + 1 substr hex m 1 + 1 substr concat hex m 1 + 1 substr if 21 line : h 255 a * H r H g H b H concat : n h strlen : 12 mat k\0324\ 1 n range d d hex h k\0324\ 1 substr findstr 1 el 1 - (16 n k\0324\ -^* + : for d 232^2 d * 232^1 - ≥ (* - eval 51 line :

C rgb C 1 el C 2 el C 3 el 1 rgb :

C a rgb C 1 el C 2 el C 3 el a rgb :

T n a pCMap C 0 : r 1 n range : c 13 range : 13 mat z r T f 02550 - n 1 - / r 1 - (* + (3 identity z 1 el row * T + a rgb eval 11 line : n 0 ≡ 0 T IsString T Jet ≡ m 0.25 n * 0 round 112 mat max eval : v z v rows 1 matrix : 12 mat x 2551 m range m / * : y 255 m / m 2 / m range (* : o 255 m 1 matrix + : 13 mat R y z o z x o y reverse stack : G y z x o x reverse y z stack : B y o x reverse o z y z stack : m R length n - : C 1 r el R r m + el G r m + el B r m + el a rgb eval : 71 line T GS ≡ R n a pCMap G n a pCMap + B n a pCMap + rgb RGB T findstr : (1 - ≡ C 1 r el T : C 1 r el rgb r 0 f : if if if zero T 0 ≡ (vectorize c el c augment eval 11 findrows 2 col eval : C 1 r el zero r T f : 21 line if if 31 line :

T x pCMap T x 1 pCMap 11 line :

T pCMap T 16 pCMap 11 line :

z a b CL pClr CL 0 ≡ (CL IsString | CL CL z a - (CL length 1 - b a - / * trunc 1 + el if 11 line :

pCMJet Jet pCMap :

pCMRed R pCMap :

pCMGreen G pCMap :

pCMBlue B pCMap :

<content> pView </content>

EulerRot(v) returns the Euler rotation matrix. At least in some convention. v pEulerRot 1000 v 1 el cos v 1 el sin - 0 v 1 el sin v 1 el cos 33 mat v 2 el cos 0 v 2 el sin 010 v 2 el sin - 0 v 2 el cos 33 mat * v 3 el cos v 3 el sin - 0 v 3 el sin v 3 el cos 000133 mat * eval 11 line :

View point at az angle from -y axis and el from +x axis. 3x3 Matrix az el pView a az : l el : 12 mat l l 180 ° * + 360 ° * mod 360 ° * + 360 ° * mod 180 ° * - : l 90 ° * > l 180 ° * l - : a a 180 ° * + : 12 mat l 90 ° * - < l 180 ° * - l - : a a 180 ° * + : 12 mat if a a 360 ° * mod 360 ° * + 360 ° * mod : a cos l sin a sin * - l cos a sin * a sin l sin a cos * l cos a cos * - 0 l cos l sin 33 mat eval 51 line :

γ 32 el γ 21 el * γ 32 el γ 11 el * - γ 11 el γ 22 el * sign 1 γ 32 el 2^- abs sqrt * 31 mat

View point for proyect into 2D plots. az el pView2 az el pView 13 range 12 range el 11 line :

P cyl2xyz P 1 el P 2 el cos * P 1 el P 2 el sin * P 3 el 31 mat :

P ρ φ z 31 mat ≡

P sph2xyz P 1 el P 3 el cos * P 2 el cos * P 1 el P 3 el cos * P 2 el sin * P 1 el P 3 el sin * 31 mat :

<content> pGrid </content>

pR(a,b,n) retunrs a vector of equally spaced n+1 points between a and b. a b n pR a b a - n trunc / 0 n trunc range * + eval 11 line :

pR(B,N) returns equally spaced vectors given a Box and a vector with the number of points. B N pR B 1 el B 2 el N 1 el pR B 3 el B 4 el N 2 el pR 12 mat eval 11 line :

Box x.1 x.2 y.1 y.2 22 mat ≡

G pGridBox C G vectorize : C min vectorize C max vectorize augment transpose eval 21 line :

X pBox B 20 matrix : k 1 X cols range B B X k col min X k col max 21 mat augment eval : for B 31 line :

pGrid(F,U,V) returns a length(U)*lenght(V) matrix with [u,v,F(u,v)] FS# X# Y# pGrid X# length 4 ≡ (Y# length 2 ≡ (& U# V# 12 mat X# Y# pR : FS# U# V# pGrid 21 line U# FS# Unknowns : FS# IsDefined ¬ (U# length 1 ≡ (& FS# num2str X# Y# pGrid r# 1 X# length range : c# 1 Y# length range : M# 0 : 13 mat FS# IsDefined FS# IsString num2str 1 ≡ XS# X# num2str vectorize : YS# Y# num2str vectorize : 12 mat str# FS# (#1#,#2#) concat : r# c# F# str# #1# XS# r# el strrep #2# YS# c# el strrep str2num str# #1# XS# r# 1 + el strrep #2# YS# c# 1 + el strrep str2num try 11 line : 31 line r# c# F# FS# r# c# el : if 11 line str# Fxy#(#1#,#2#): FS# num2str concat : str# #1# U# 1 el num2str strrep #2# U# 2 el num2str strrep str2num r c F# X# r# el Y# c# el Fxy# : 31 line if 11 F# GetType matrix ≡ M# r# c# el r# c# F# : M# r# c# el X# r# el Y# c# el r# c# F# 31 mat : if 31 line if 21 line if 11 line :

pTGrid(F,U,V) returns a [G G'] where G is the pGrid and G' is a grid with the middle points of the grid FS X Y pTGrid X length 4 ≡ (Y length 2 ≡ (& U V 12 mat X Y pR : FS U V pTGrid 21 line A pr n 1 A length 1 - (range : A' 0 : 12 mat A' n el 0.5 A n el A n 1 + el + (* : A' A length el A A length el : 31 line : FS X Y pGrid FS X pr Y pr pGrid 12 mat eval 21 line if 11 line :

<content> pMesh </content>

pMesh returns a matrix suitable for 3D plots. G# h pMesh tri G# length 2 ≡ eval : G G# 1 el G# 1 el G# 1 el cols col augment eval : G' G# 2 el eval : 12 mat G G# eval : if nu G rows 1 - : nv G cols 1 - : M 03 matrix eval : r' 0 : 14 mat nh h cols : nh' nh tri + : r.nh 1 nh range : r.nv 1 nv 1 - (range : c 13 range : 15 mat u 1 nu range v 1 nv range r M rows : k r r.nh + : r' r' 1 + : 13 mat M k c el G h 1 k r - el u + h 2 k r - el v + el c el : tri v nv ≡ M r nh' + c el M r nh + c el : M r nh' + c el G' r' el c el : if if 31 line for r M rows : k r r.nv + : 12 mat M k c el M u 1 - (nh' nv * nv + 1 - (* nh' v 1 - (* + k r - 1 - (nh' * - tri - c el : 31 line for M 51 line :

Mesh of a Grid G# pMesh G# 001100110025 mat pMesh 11 line :

Triangular Mesh of a Grid G# pTMesh G# 001010110025 mat pMesh 11 line :

Utill: pMesh(FS,X,Y) Mesh from the function or        pMesh(M,CL,γ) Coloring a Mesh for a XYPlot M# γ CL pMesh M# IsString M# IsDefined 1 ≡ num2str (¬ (| M# γ CL pGrid pMesh γo γ2 12 mat γ pArgγ : M M# eval : M2 M γ2 * eval : k 2 M rows (range : L 0 : 14 mat a b 12 mat M 3 col min M 3 col max 1010 -^+ 12 mat eval : M pBox γ pAxis L k el M2 k row M2 k 1 - row stack M k 3 el a b CL pClr solid 10 pLine : stack 11 mat eval 41 line if 11 line :

<content> pSurf </content>

P cl ls w cs pLine cs 0 ≡ polyline P cl ls w 51 mat polygon P cl ls w cs 61 mat if 11 line :

X pCenter X 1 col sum X 2 col sum X 3 col sum 13 mat X rows / eval 11 line :

0.5 M 1 c el M ν c el + (* eval

nG G# γ CL CS h wm LS d pSurf c 13 range : n 0 : S 0 : N 0 : MG 03 matrix eval : γo γ2 12 mat γ pArgγ : 16 mat s 1 nG range G# s el cols 2 ≡ MG MG G# s el 1 el G# s el 2 el stack eval pGridBox stack eval : MG MG G# s el eval pGridBox stack eval : if 11 line for MG MG pBox : a b 12 mat MG 1 d el MG 2 d el 1010 -^+ 12 mat eval : 12 mat Vo 100 MG normi max + 13 matrix + (γo * eval : s 1 nG range tri G# s el cols 2 ≡ eval : G G' 12 mat G# s el 1 el G# s el 2 el 12 mat eval : G G# s el eval : if nu nv 12 mat G rows G cols 12 mat eval : nh mh ν w 14 mat h cols h max h cols 312 mat min wm CL 0 ≠ (* 14 mat eval : M 0 : r 1 nh range : nj 1 tri 0 nh 2 - (range * + eval : 13 mat u 1 nu mh - (range v 1 nv mh - (range j nj tri M G' u v el G u h 1 j el + v h 2 j el + el G u h 1 j 1 + el + v h 2 j 1 + el + el augment transpose eval : M r c el G u h 1 r el + v h 2 r el + el c el : if Mo M pCenter : N n n 1 + : el Mo Vo - norme eval : S n el M γ2 * Mo d el a b CL s el pClr LS w Mo d el a b CS s el pClr pLine eval : 41 line for for for 51 line for N 1 n range N augment eval 2 csort 1 col reverse eval : MG γ2 pAxis S N el stack 11 mat eval 71 line :

CL 0 ≡ (CL IsString | zo CL CL : nL CL length eval : zo CL CL zo a - (nL 1 - b a - / * trunc 1 + el 11 line : 21 line if CS 0 ≡ (CS IsString | zo CS CS : nS CS length eval : zo CS CS zo a - (nS 1 - b a - / * trunc 1 + el 11 line : 21 line if 12 mat

0.5 M 1 c el M ν c el + (* eval

Surface plot M γ CL CS pSurf M sys2mat.1 M ≡ G M 021 mat : n 1 : 12 mat G M : n G sys2mat.1 stack rows : 12 mat if n G γ CL n pArgN CS n pArgN 001100110025 mat 1 solid 3 pSurf 21 line :

pShowAxis 1 :

<content> pGrid </content>

Grided Domains

The domain is given by a box and the number of subintervals or by the discrete points in two equally spaced vectors

153822 mat 2321 mat pR 13531 mat 3 4.6667 6.3333 841 mat 12 mat

U 132 pR : 12331 mat

V 583 pR : 567841 mat

Box x.1 y.1 x.2 y.2 22 mat ≡

f B N pGrid

are used for evaluating the function over it's domain. The version with U, V vectors could be used for avoid discontinuites or another specific domain.

f U V pGrid

f B N pTGrid

with the same syntax, pTGrid returns the same grid and another with the middle points.

x y f x 0.1 y * + : B 153822 mat : N 2312 mat : G f B N pGrid : G G' 12 mat f B N pTGrid : 51 line

G 1.0 3.0 1.3 31 mat 1.0 4.7 1.5 31 mat 1.0 6.3 1.6 31 mat 1.0 8.0 1.8 31 mat 3.0 3.0 3.3 31 mat 3.0 4.7 3.5 31 mat 3.0 6.3 3.6 31 mat 3.0 8.0 3.8 31 mat 5.0 3.0 5.3 31 mat 5.0 4.7 5.5 31 mat 5.0 6.3 5.6 31 mat 5.0 8.0 5.8 31 mat 34 mat

G' 2.0 3.8 2.4 31 mat 2.0 5.5 2.6 31 mat 2.0 7.2 2.7 31 mat 2.0 8.0 2.8 31 mat 4.0 3.8 4.4 31 mat 4.0 5.5 4.6 31 mat 4.0 7.2 4.7 31 mat 4.0 8.0 4.8 31 mat 5.0 3.8 5.4 31 mat 5.0 5.5 5.6 31 mat 5.0 7.2 5.7 31 mat 5.0 8.0 5.8 31 mat 34 mat

Vectorizing mat sorts the matrix values in a [X Y Z] matrix

G vectorize 1.0 1.0 1.0 1.0 3.0 3.0 3.0 3.0 5.0 5.0 5.0 5.0 34 mat 3.0 4.7 6.3 8.0 3.0 4.7 6.3 8.0 3.0 4.7 6.3 8.0 34 mat 1.3 1.5 1.6 1.8 3.3 3.5 3.6 3.8 5.3 5.5 5.6 5.8 34 mat 31 mat

G' vectorize 2.0 2.0 2.0 2.0 4.0 4.0 4.0 4.0 5.0 5.0 5.0 5.0 34 mat 3.8 5.5 7.2 8.0 3.8 5.5 7.2 8.0 3.8 5.5 7.2 8.0 34 mat 2.4 2.6 2.7 2.8 4.4 4.6 4.7 4.8 5.4 5.6 5.7 5.8 34 mat 31 mat

XX G pGridBox : 13 1.3 58 5.8 23 mat

As utility, f could be the string of the function name or the expression in two unknows. For example

x y f x x cos y * + :

f X Y pGrid x x cos y * + X Y pGrid ≡

For empirical data, pGrid sort it for use with pMesh and other functions.

In some sense, this is the most important example here

Y 0.8 1 2.3 3.2 4.7 6.1 6.5 17 mat :

X 0 1.1 2.4 3.4 4.6 5.6 6.8 7.9 81 mat :

Z 0.5 0.5 1.1 - 4.4 - 7.4 - 11.6 - 17.3 - 4.3 2.9 1.8 1 - 4.8 - 9.1 - 14 - 7 6.1 4.6 1.2 1.6 - 6.9 - 11.5 - 10.6 8.8 6.5 4 0.6 4.7 - 9.6 - 14.3 11.7 9.3 5.5 2.3 2.7 - 8 - 17 14.6 11.4 7.4 3.2 1.5 - 7.4 - 20.7 17.7 13.4 9.3 4.8 0 6.2 - 23.8 20 15.9 11.1 6.2 1.3 6 - 87 mat :

EG Z X Y pGrid :

EG 0 0.8 0.5 31 mat 01 0.5 31 mat 0 2.3 1.1 - 31 mat 0 3.2 4.4 - 31 mat 0 4.7 7.4 - 31 mat 0 6.1 11.6 - 31 mat 0 6.5 17.3 - 31 mat 1.1 0.8 4.3 31 mat 1.1 1 2.9 31 mat 1.1 2.3 1.8 31 mat 1.1 3.2 1 - 31 mat 1.1 4.7 4.8 - 31 mat 1.1 6.1 9.1 - 31 mat 1.1 6.5 14 - 31 mat 2.4 0.8 731 mat 2.4 1 6.1 31 mat 2.4 2.3 4.6 31 mat 2.4 3.2 1.2 31 mat 2.4 4.7 1.6 - 31 mat 2.4 6.1 6.9 - 31 mat 2.4 6.5 11.5 - 31 mat 3.4 0.8 10.6 31 mat 3.4 1 8.8 31 mat 3.4 2.3 6.5 31 mat 3.4 3.2 431 mat 3.4 4.7 0.6 31 mat 3.4 6.1 4.7 - 31 mat 3.4 6.5 9.6 - 31 mat 4.6 0.8 14.3 31 mat 4.6 1 11.7 31 mat 4.6 2.3 9.3 31 mat 4.6 3.2 5.5 31 mat 4.6 4.7 2.3 31 mat 4.6 6.1 2.7 - 31 mat 4.6 6.5 8 - 31 mat 5.6 0.8 1731 mat 5.6 1 14.6 31 mat 5.6 2.3 11.4 31 mat 5.6 3.2 7.4 31 mat 5.6 4.7 3.2 31 mat 5.6 6.1 1.5 - 31 mat 5.6 6.5 7.4 - 31 mat 6.8 0.8 20.7 31 mat 6.8 1 17.7 31 mat 6.8 2.3 13.4 31 mat 6.8 3.2 9.3 31 mat 6.8 4.7 4.8 31 mat 6.8 6.1 031 mat 6.8 6.5 6.2 - 31 mat 7.9 0.8 23.8 31 mat 7.9 12031 mat 7.9 2.3 15.9 31 mat 7.9 3.2 11.1 31 mat 7.9 4.7 6.2 31 mat 7.9 6.1 1.3 31 mat 7.9 6.5 6 - 31 mat 87 mat

EG pMesh 30 ° * 60 ° * pView * <content> Functions </content>

Some functions and Grids for further examples

Matlab logo

x y m 31 x - (2^e x 2^y 1 + (2^+^/ * 10 x 5 / x 3^- y 5^- e x 2^y 2^+^/ * - 13 e x 1 + (2^y 2^+^* / - :

Gm'' Gm Gm' 12 mat m 3 - 33 - 322 mat 252521 mat pTGrid : (:

Up and down

x y b 8 e x 0.5 - (2^y 2^+^/ 8 e x 0.5 + (2^y 2^+^/ - :

Gb'' Gb Gb' 12 mat b 3 - 33 - 322 mat 252521 mat pTGrid : (:

Torus

R 2 : r 1.5 : 21 sys

u v torus R r u cos * + (v cos * R r u cos * + (v sin * r u sin * 31 mat :

Gt'' Gt Gt' 12 mat torus 60 ° * - 120 ° * 0 ° * 220 ° * 22 mat 152021 mat pTGrid : (:

Ellipsoid

u v ellip 1.5 u cos * v sin * 1 + 1.5 u sin * v sin * 2 + 2 v cos * 0.5 + 31 mat :

Gε'' Gε Gε' 12 mat ellip 0 ° * 360 ° * 0 ° * 180 ° * 22 mat 202021 mat pTGrid : (:

A wave

r wave 0.5 4 r * cos * : r θ wave r θ r wave 13 mat cyl2xyz : 21 line

Gw'' Gw Gw' 12 mat wave 050 ° * 270 ° * 22 mat 203021 mat pTGrid : (:

This shows or hide the axis.

pShowAxis 1 :

Some colormaps and viewpoints

CM.T 01923213 mat 641 pCMap :

γ.1 120 ° * 30 ° * pView2 :

CM.R R 641 pCMap :

γ.2 30 ° * 60 ° * pView2 :

CM.GS GS 641 pCMap :

γ.3 37.5 ° * - 30 ° * pView :

CM.J Jet 641 pCMap :

γ γ.1 :

CM CM.J :

γ2 γ 13 range 12 range el :

The goal of a mesh is to have an only one matrix with 3 columns which could be manipulated by matrix algebra. But for showing it we can use also some directives, like the line and polygon utilities in XY Plot component. The idea is preserve it as pure numerical procedure, but with the flexibility of call the function pMesh with the function expression too.

Mesh Creation

Mesh creation over a grid domain with a geometry given by a two rows matrix.

The left plots are 3D Smath plots, the middle 2D SMath plots and the right are XY Plots

S.1 Gw 001100110025 mat pMesh :

S.2 Gw 001122 mat pMesh :

S.3 Gw transpose 001122 mat pMesh :

S.1 rows 3580

S.2 rows 1780

S.3 rows 1770

Viacheslav original mesh

S.1 2 2 1.5 pDiag * γ *

Waterfall along x

S.2 γ2 *

Waterfall along y

S.3 γ CM pMesh

S.1 Gt 100122 mat pMesh :

S.2 Gt 00101123 mat pMesh :

S.3 Gt 001010110025 mat pMesh :

S.1 rows 885

S.2 rows 1185

S.3 rows 1785

fake but fast trimesh

S.1 2 2 2 pDiag * γ *

something else

S.2 γ2 *

fake but nice trimesh

S.3 γ CM pMesh

Trimesh

The first argument is now the grid of the domain points and the grid points in it's diagonal

G'' G G' 12 mat ≡

S.1 Gw'' 001010110025 mat pMesh :

S.2 Gw'' 01001123 mat pMesh :

S.3 Gw'' 010122 mat pMesh :

S.1 rows 4320

S.2 rows 3080

S.3 rows 2460

true tri mesh

S.1 2 2 2 pDiag * γ *

intermediate variation

S.2 γ2 *

fast variation

S.3 γ CM pMesh

Other notations

M γ CM pMesh

returns a color mesh for XY plots, like in the last example.

G pMesh

returns the Viacheslav's mesh using

h 001100110025 mat ≡

G pTMesh

returns the tri mesh using

h 001100110025 mat ≡

f B N pMesh

f U V pMesh

are just as shorthands for avoid griding and then meshing.

f B N pTMesh

f U V pTMesh

<content> pSurf </content>

G γ CL CS pSurf

Plots the grid G with the point view γ and colors CL for the lines and CS for surfaces.
Also, can take a set of grids.

Gε Gw'' 2 1 sys γ.1 black CM.J CM.T 2 1 sys pSurf Gm Gw 2 1 sys γ.2 CM.GS CM.T 2 1 sys CM.R pSurf Gb Gt 2 1 sys γ.3 CM.R black 2 1 sys CM.J CM.T 2 1 sys pSurf

r 2 : u v f 2 u * sin v sin (2^* u sin 2 v * cos * u cos 2 v * sin * 31 mat : G f 02 π * 02 π * 22 mat 303021 mat pTGrid : 31 line

Roman

G γ.1 black CM.J pSurf

u v f bx 6 u cos * 1 u sin + (* : by 16 u sin * : r 41 0.5 u cos * - (* : π u < bx r v π + cos * + by r v sin * stack bx by 031 mat r u v 31 mat sph2xyz + if 41 line : G f 02 π * 02 π * 22 mat 252521 mat pGrid : 21 line

Klein Bottle

G γ.2 black CM.T pSurf

r 2 : t s f r s t 2 / cos * + (t cos * r s t 2 / cos * + (t sin * s t 2 / sin * 31 mat : G f 02 π * 1 - 122 mat 201021 mat pGrid : 31 line

Möbius band

G γ.3 CM.R CM.J pSurf

x y f 1 - x 2^y 2^* x 2^y 2^+ (3^- 12 mat max 11 line : G.1 f 1 - 11 - 122 mat 121221 mat pTGrid : a 1 : u v g a v cos u cosh / * 1 + a v sin u cosh / * a u u tanh - (* 1 + 31 mat : G.2 g 0260 ° * - 220 ° * 22 mat 102021 mat pGrid : 51 line

Algebraic Surface and a pseudosphere

G.1 G.2 2 1 sys γ.1 black CM.J pSurf

Ruled Surfaces: Hyperboloid

The ruled surfaces of directrix b and director curve δ have the form

u v f u b v u δ * + :

a b c 13 mat 12313 mat : u v f a u cos * b u sin * 031 mat v a u sin * - b u cos * c 31 mat * + : U 0 ° * 360 ° * 20 pR : V 2 - 220 pR : GH f U V pGrid : 51 line

GH γ.2 CM.R CM.T pSurf

Frenet Tube for a Knot

Frenet Tube of the path r with radious R r# R# t# θ# FrenetTube r'# r# tt# t# : diff vectorize : r''# r'# tt# t# : diff vectorize : 12 mat B# r''# r'# † r''# r'# † norme / : N# B# r'# † r'# norme / : r# N# R# * θ# cos * + B# R# * θ# sin * + 41 line :

A tube for a knot

k2 23 t * cos + (2 t * cos * 23 t * cos + (2 t * sin * 3 t * sin 31 mat : U 0360 ° * 60 pR : V 0360 ° * 10 pR : t θ f k2 0.4 t θ FrenetTube : GT f U V pGrid : 51 line

GT γ.1 black CM.J pSurf

Untwisted version with variable radious

k2 t sin 22 t * sin * + t cos 22 t * cos * - 3 t * sin - 31 mat : R 0.2 t cos * 0.5 + : t θ f k2 R t θ FrenetTube : GT2 f U V pGrid : 41 line

GT2 γ.3 black CM.J pSurf

Discontinous function

x y f x 2^y 2^+ sqrt sin x 2^y 2^+ sqrt / : G f 31 - 11 - 122 mat * 201121 mat * pGrid : 21 line

G γ.3 lightgreen CM.J pSurf <content> pCMap </content>

pCMap is for creating color maps. It take as arguments a SMath color string, or "Jet", "R", "G" and "B". or an array [r g b] with one of the r, g or b equals zero.

The other arguments are the number of the colors and the transparency as a number between 0 and 1.

CM.T 01923213 mat 641 pCMap :

Makes a colormap looking
at the zero value

from

01923213 mat 1 rgb CM 1 el ≡

2551923213 mat 1 rgb CM 32 el ≡

CM.R R 641 pCMap :

Makes a colormap with a red scale. Similar for "G" and "B"

B 90 pCMap 032649612815919122325519 mat

CM.GS GS 641 pCMap :

Makes a colormap with 64 gray values

CM.J Jet 641 pCMap :

The Jet color map, from NASA.

x y cm 0 x y stack : n CM cm 1 - 0 range n + 164 range pGrid : cm 1 CM 2 CM 3 CM 4 CM 41 sys 90 ° * 0 ° * pView 0 CM.T CM.J CM.GS CM.R 41 sys pSurf : 31 line

cm <content> pView </content>

Examples of view

az el pView

CM Jet 321 pCMap : az el view Gm az el pView black CM pSurf 11 line : 21 line

returns a rotation matrix for the given azimuth and elevation

Default Matlab view

2D view

First column view

37.5 ° * - 30 ° * view 1 1 line 0 ° * 90 ° * view 1 1 line 0 ° * 0 ° * view 1 1 line

Fridel usual view

For az = 180 is behind the matrix

Rotated first column

145 ° * 48 ° * view 1 1 line 180 ° * 30 ° * view 1 1 line 30 ° * 0 ° * view 1 1 line <content> Final example </content>

Gm'' m 3 - 33 - 322 mat 505021 mat pTGrid : Gt'' torus 60 ° * - 120 ° * 0 ° * 220 ° * 22 mat 205021 mat pGrid : Gε'' ellip 0 ° * 360 ° * 0 ° * 180 ° * 22 mat 404021 mat pGrid : 31 sys

Gm'' Gt'' Gε'' 3 1 sys γ.1 black darkgreen blue 3 1 sys CM.J pSurf

Alvaro

4 appVersion 1.0.8348.30405