Very good job, Alvaro. Now gamma is an explicit function in Azimuth-Elevation coordinates.
I will use this function as it allows you to change the viewpoint in azimuth and elevation.
Thank you very much.
Fridel.

Hmm. Someone told about parameterize equation.
System from #78
plos.avi (3,28 MiB) scaricato 559 volta(e).

Wrote

Distance and projection of a point onto a parametric surface
...

Hi. Maybe it's cosmetic, or not, but the rotation matrix don't need the last column for the 2D projection. So, you don't have to use col (X*lambda, n), just multiply by lambda. That's clean up a lot the code. Also add your azimuth and elevation, which it's 145,48 degrees, without the need of Euler angles.

ParSur.sm (82,76 KiB) scaricato 1718 volta(e).

Best regards.
Alvaro.

Thank you Alvaro. Two columns instead of three in the rotation matrix make it easy to convert a 3D plot to 2D.

I could rotate with breaking the equation. Is this natural?
rot.avi (2,93 MiB) scaricato 574 volta(e).

Some ellipse
Ref
el.avi (1,14 MiB) scaricato 554 volta(e).

Wrote

Distance and projection of a point onto a parametric surface


ParSur.sm (74,98 KiB) scaricato 1878 volta(e).
ParSur.pdf (619,79 KiB) scaricato 1243 volta(e).

Projection of a curve onto a surface
Computing the projection of a point onto a surface is to find a closest point
on the surface, and projection of a curve onto a surface is the locus of all
points on the curve project onto the surface.


projection.sm (22,03 KiB) scaricato 1695 volta(e).
projectionB.sm (34,01 KiB) scaricato 1709 volta(e).

Wow.
In fact, this whole series on the Draghilev method, 3D to 2D projections etc. is one of the most fascinating of this forum...

Тhank you for your interest in the Topic.
Remarks:
1,Unlike the previous examples, the last file does not use dragilev's method.
2.Bar structure analogy.A heavy hoop is pivotally attached to weightless rods
that rest on an uneven surface.The slope of each bar coincides with the surface
ormal at the foothold point. Consequently, the rods perceive only normal (compressive) load.

Hi Ber. As usual, your samples are amazing.

Wrote

...
1,Unlike the previous examples, the last file does not use dragilev's method.

It could be interesting convert it to Dragilev's method. I don't have enough time right now.

Wrote

2.Bar structure analogy.A heavy hoop is pivotally attached to weightless rods
that rest on an uneven surface.The slope of each bar coincides with the surface
ormal at the foothold point. Consequently, the rods perceive only normal (compressive) load.

Oh! I try to understand that, but can take a while. Well, I understand now why it could be important. In the mean time, here a faster version, with some margin notes.

projection.sm (62,32 KiB) scaricato 1766 volta(e).
projection.pdf (840,02 KiB) scaricato 1132 volta(e).

Best regards.
Alvaro.

Hi,Alvaro.In my version 0.99.7610 your file is not loading. I will study your program
on PDF. your wonderful examples are very expressive and aesthetic . Thank you ,Fridel.

Wrote

Hi,Alvaro.In my version 0.99.7610 your file is not loading. I will study your program
on PDF. your wonderful examples are very expressive and aesthetic . Thank you ,Fridel.

Hi Fridel. I check the upload file, it's ok for me. But have something wrong, because I can't upload it to the cloud version: just it do nothing. I delete the text regions with math, and substitute them with the usual math region with a line, and then can upload: https://en.smath.com/cloud/worksheet/mgSbWNUw (obviously doesn't work there because the al_nleqsol plugin fails). Here the file:

projection_without_txt_region.sm (57,05 KiB) scaricato 1740 volta(e).

Best regards.
Alvaro.

The file has loaded and is working . Thank you.

IHi,Alvaro .In my version 0.99.7610 green dots are displayed
as unicode characters.Until I understand what is the reason.
Fridel.

Wrote

IHi,Alvaro .In my version 0.99.7610 green dots are displayed
as unicode characters.Until I understand what is the reason.
Fridel.

Hi Friedel. Nice, you're looking the matrix as hexadecimal. It could be better than reading it in binary, like in the movie. That, or this bug:

test.sm (2,5 KiB) scaricato 1776 volta(e).



Best regards.
Alvaro.

Draghilev method with variation. Also working superb
dragvar.avi (4,89 MiB) scaricato 570 volta(e).

Maple
1. 2

Метод чувствителен к выбору начального положения. Существуют области устойчивости и неустойчивости. Области зависят от выбранной точности. Не все координаты из областей устойчивости дают все решения при смене знака у дополнительно введенного параметра.
[ENG]
The method is sensitive to the choice of initial position. There are regions of stability and instability. The regions depend on the chosen accuracy. Not all coordinates from the regions of stability give all solutions when the sign of the additionally introduced parameter changes.
На рис. pol1 представлены две кривые и точки. Красным и зеленым цветом показаны точки начального положения для расчета пересечения по методу Драгилева. Красный цвет означает, что в результате решения значение достигли Nan (т.е. бесконечности); зеленый, что решения не достигли Nan. На рис. pol2 представлено для красной точки А расчеты с достижением Nan.
[ENG]
Figure pol1 shows the two curves and points. Red and green show the initial position points for calculating the intersection by the Dragilev method. Red indicates that the solutions reached Nan (i.e., infinity); green indicates that the solutions did not reach Nan. Figure pol2 shows for red point A the calculations with reaching Nan.


На рис. pol3 представлены три кривые и точки. Третья кривая построена через зеленую точку B. Видно что она пересекает две начальные кривые в точках пересечения 1,2,3,4
[ENG]
Figure pol3 shows three curves and points. The third curve is drawn through the green point B. You can see that it intersects the two initial curves at the intersection points 1,2,3,4

На анимации представлено изменение кривой 2 и области устойчивости и неустойчивости
[ENG]
The animation shows the change of curve 2 and the region of stability and instability
pol.avi (3,81 MiB) scaricato 573 volta(e).

It turned out that Draghilev's method is capable of solving Diophantine equations.
Can be viewed here.