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

#1 Posted: 10 years ago

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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 10 years ago

#2 Posted: 10 years ago

Fridel Selitsky

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Method 2.
The rectangular grid

ELLIPSE3d2.pdf (631 KiB) downloaded 181 time(s).

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Davide Carpi 10 years ago

#3 Posted: 8 years ago

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 8 years ago, Davide Carpi 8 years ago, frapuano 8 years ago

#4 Posted: 8 years ago

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 8 years ago

#5 Posted: 8 years ago

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 8 years ago

#6 Posted: 8 years ago

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 119 time(s).

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Davide Carpi 8 years ago, NDTM Amarasekera 8 years ago

#7 Posted: 8 years ago

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 109 time(s).

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Davide Carpi 8 years ago

#8 Posted: 8 years ago

Fridel Selitsky

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Helical Tube

NonCirc.sm (53 KiB) downloaded 110 time(s).
TubeCir.sm (53 KiB) downloaded 113 time(s).

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Davide Carpi 8 years ago

#9 Posted: 8 years ago

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 119 time(s).

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Davide Carpi 8 years ago

#10 Posted: 8 years ago

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 130 time(s).
Non-circular cylindrical vessels.sm (209 KiB) downloaded 114 time(s).
Shell1.sm (255 KiB) downloaded 118 time(s).
Shell2.sm (33 KiB) downloaded 117 time(s).
Shell3.sm (34 KiB) downloaded 114 time(s).

#11 Posted: 8 years ago

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 114 time(s).

#12 Posted: 8 years ago

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 122 time(s).

#13 Posted: 7 years ago

Fridel Selitsky

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Implicit surfaces in Maxima

ImplicitMaxima.sm (363 KiB) downloaded 115 time(s).

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frapuano 7 years ago, Martin Kraska 7 years ago, Radovan Omorjan 7 years ago

#14 Posted: 7 years ago

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 119 time(s).