Numerical parameterization implicit surfaces of the form f (x, y, z) = 0

#1 Posted: 5/29/2015 8:25:23 PM

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The grid lines are found by Draghilev Method
Method 1.
Building a line of intersection of the required surface
with the plane x = 0. This line is rotated in steps of 2 * Pi / s
about a vertical axis by an angle 2 * Pi, where s is the number of steps (the grid lines)

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Davide Carpi 5/31/2015 7:19:00 AM

#2 Posted: 5/29/2015 8:39:51 PM

Fridel Selitsky

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Method 2.
The rectangular grid

ELLIPSE3d2.pdf (631 KiB) downloaded 133 time(s).

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Davide Carpi 5/31/2015 7:19:00 AM

#3 Posted: 12/17/2016 6:17:47 PM

Fridel Selitsky

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The graph an implicit surface depicted by lines
Examples of the idea of the algorithm taken from the publication of Alexey Ivanov
http://www.mapleprimes.com/questions/219995-Finding-A-Convinient-Parametrization-Of-Surfaces#answer233804

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Вячеслав Мезенцев 12/19/2016 1:43:00 AM, Davide Carpi 12/18/2016 7:04:00 PM, frapuano 12/18/2016 5:42:00 AM

#4 Posted: 1/29/2017 4:56:41 PM

Fridel Selitsky

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Two surfaces are taken from the site
http://xrt.wikidot.com/gallery:implicit
The calculation algorithm:
1.Find the line of intersection of the given surface with the plane Z=0
2.Through each point hold the plane x1 = C and find the line of intersection of that plane with a given surface.These lines represent a given surface.

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Davide Carpi 1/30/2017 9:18:00 AM

#5 Posted: 1/29/2017 11:28:16 PM

Jean Giraud

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Thanks for these two animations. In each one, the last three plots
stay red, though the animated plot works fine. A small detail:
each work sheet has three definitions left over at page 17.

Jean

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Davide Carpi 1/30/2017 9:18:00 AM

#6 Posted: 1/30/2017 3:51:39 AM

Fridel Selitsky

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Thank you Jean,
Worksheets corrected. In addition, in the algorithm for calculating along with x1 = C can be used x2 = C:

double torus x2.sm (73 KiB) downloaded 79 time(s).

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Davide Carpi 1/30/2017 9:18:00 AM, NDTM Amarasekera 1/30/2017 11:30:00 AM

#7 Posted: 1/31/2017 5:50:55 PM

Fridel Selitsky

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The tube in the form of a line of intersection of the cylinder and the torus (See post 7)
http://en.smath.info/forum/yaf_postst2307_Method-A-B--Draghilev-and-animation-spatial-mechanisms.aspx

Time calculation, 9 min.
Villarceau 9.sm (39 KiB) downloaded 69 time(s).

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Davide Carpi 2/1/2017 5:58:00 AM

#8 Posted: 3/1/2017 4:25:37 AM

Fridel Selitsky

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Helical Tube

NonCirc.sm (53 KiB) downloaded 74 time(s).
TubeCir.sm (53 KiB) downloaded 75 time(s).

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Davide Carpi 3/1/2017 6:42:00 AM

#9 Posted: 3/1/2017 4:44:07 AM

Fridel Selitsky

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The surface of the Wikipedia
https://en.wikipedia.org/wiki/File:Tri-brezel.svg

WiciSurf.sm (106 KiB) downloaded 76 time(s).

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Davide Carpi 3/1/2017 6:42:00 AM

#10 Posted: 6/14/2017 1:27:49 PM

Fridel Selitsky

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Non-circular cylindrical vessels with quasi-ellipsoidal ends
Krivoshapko S.N. Ivanov V.N,Encyclopedia of Analytical Surfaces ,Springer,2015

Non-circular cylindrical vessels.pdf (1 MiB) downloaded 86 time(s).
Non-circular cylindrical vessels.sm (209 KiB) downloaded 73 time(s).
Shell1.sm (255 KiB) downloaded 74 time(s).
Shell2.sm (33 KiB) downloaded 70 time(s).
Shell3.sm (34 KiB) downloaded 71 time(s).

#11 Posted: 6/15/2017 9:43:57 AM

Jean Giraud

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Thanks Ber7 for the work sheets [Not finished downloading].
If you visit this document, I'm wondering:
"How to add red points in this plot".

Thanks in advance ... Jean

Maths sph2xyz.sm (13 KiB) downloaded 72 time(s).

#12 Posted: 6/15/2017 12:23:51 PM

Fridel Selitsky

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Hello Jean,
Unfortunately, the 3d chart does not allow you to draw points.
Exit (hopefully temporary) -make a 3d chart on a 2d chart.
Maths sph2xyz2d.sm (21 KiB) downloaded 81 time(s).

#13 Posted: 2/6/2018 3:56:52 AM

Fridel Selitsky

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Implicit surfaces in Maxima

ImplicitMaxima.sm (363 KiB) downloaded 74 time(s).

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frapuano 2/6/2018 4:34:00 AM, Martin Kraska 2/6/2018 4:59:00 AM, Radovan Omorjan 2/6/2018 6:22:00 AM

#14 Posted: 2/6/2018 5:06:49 AM

Martin Kraska

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Nice examples!

Just some remarks for those who want to play around with Maxima Draw.

Instead of Image region with Draw() command you can use Maxima Draw3D region with interactive zoom, rotation and resize and a settings dialog.
Insert-> Maxima-> Draw3D
The performance isn't overwhelming, as the image is recalculated for each interaction.
The snippet "draw" provides help texts for the available draw commands and options.
Follow the link in my signature to see how to configure SMath with Maxima.

ImplicitMaxima_Kr.sm (492 KiB) downloaded 76 time(s).