83c8fad9-7ab4-4c29-adc7-1a6e6407d831 3 4 5 decimal

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

Rolling the Torus

M Unwrap V M 1 col : i 2 M cols range V V M i col stack : for V 31 line :

i 030 range :

j 018 range :

radius 20 :

p π - i π * 15 / + :

q π - j π * 9 / + :

k 0 radius range u k 1 + el 1 radius k - 1 k + / + : for

i 1 u rows range R i el 5 u i el * 625 i - / + : for

i 1 u rows range r i el 1.5 u i el * 425 i - / + : for

The 3 vectors [u,R,r] ... The 3 matrices [X,Y,Z] do sanity check Mathcad 11.

X i 131 range j 119 range mx i j el R t el p i el u t el / sin * r t el 1 q j el u t el / cos - (* p i el u t el / sin * - eval : for for mx 21 line :

Y i 131 range j 119 range my i j el r t el q j el u t el / sin * eval : for for my 21 line :

Z i 131 range j 119 range mz i j el R t el 1 p i el u t el / cos - (* r t el 1 q j el u t el / cos - (* p i el u t el / cos * + eval : for for mz 21 line :

0 0.4226 - 0.9063 - 1000 0.9063 - 0.4226 33 mat

γ 0 0.7071 - 0.7071 0.7071 - 0.5 0.5 0.7071 - 0.5 - 0.5 - 33 mat :

Thanks Ber7 for speeding/inverse roll the Torus. I have not found the same Mathcad orientation. Interesting about plotting and rotate the "scatter vectors" system.

t Torus X Unwrap Y Unwrap Z Unwrap augment γ * eval :

N 20 :

step1 12 N * 10 + 1 - (range :

@ Frame rate [10], rolling is pleasnt observing

t N > ( t N 10 + ≤ ( & t N : 1 1 line t N 10 + > t 2 N * 10 + t - : t t : if if t Torus 2 1 line

XYZ 11 Torus :

Isolate a specific "roll"

XYZ γ *

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

<= roll [roulis]

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

<= pitch [azimuth]

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

<= yaw [horizon]

α 180 ° * :

β 135 ° * :

φ 270 ° * :

γ α Ω1 β Ω2 * φ Ω3 * eval : 1.2989 1016 -^* 0.7071 - 0.7071 1.0000 1.8369 1016 -^* 0.0000 1.2989 1016 -^* - 0.7071 0.7071 33 mat

R 1 : r 0.5 : 12 mat

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

<= clear 'u' from previous discrete

Torus equation

u Clear 1

x,y mesh = even, practical limit 30 x 30 ... 40 x 40 M u v f π - π π - π 3030 CreateMesh :

x c x 1014 -^- ≤ :

3D plot in Smath: Smath 3D is a "scatter " plot.

A scatter plot is the side by side augment(x,y,z) ... | | |

In the above Torus we unwrap the matrices in their

corresponding vectors. This style allows only to plot by

points, unless a means of clearing the carriage return is

designed. Unlike the "Matrix Unwrap Style", Smath has

its own CreateMesh which allows 'plot mesh'. At right,

we clean for the low residuals that effectively worth '0'.

CreateMesh is specially coded for f(u,v) vector function.

Can't explain the quadrant plot.

Can't explain the folding of axes system.

]]>

N x y z 13 mat M M c * vectorize stack :

N x y z 0.5 - 00 0.489 - 0.104 - 0 0.5 - 0.106 - 0.104 - 0.511 - 0 0.104 - 0.5 - 00 0.489 - 0.104 - 0 0.457 - 0.203 - 0 0.467 - 0.208 - 0.104 - 0.5 - 0.106 - 0.104 - 103 mat

M N

M rows M M c * vectorize rows 21 mat 5370537021 mat

Collect the 2D surface plots u v U R r u cos * + (v cos * (: u v V R r u cos * + (v sin * : u v Z r u sin * : 31 line

The 2D surface plot does NOT save settings when closing.

]]> U V Z