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Guenther Gsaller

Animating Mathematica surfaces with Blender: Morphing & complex functions

Guenther Gsaller, Retired

Posted 4 months ago

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MODERATOR NOTE: part of this work was submitted to Computational Art Contest 2022. Check out the contest and other submissions here: https://wolfr.am/CompArt-22

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Animating Mathematica surfaces with Blender: Morphing & complex functions
by Guenther Gsaller

This is a continuation to the following posts:
Animating Mathematica surfaces with Blender: Surface creation & Ex-/Import
https://community.wolfram.com/groups/-/m/t/2293492
Animating Mathematica surfaces with Blender: Modified parametric surfaces
https://community.wolfram.com/groups/-/m/t/2328875
Animating Mathematica surfaces with Blender: Enhanced prism & spiral shapes
https://community.wolfram.com/groups/-/m/t/2335677
Animating Mathematica surfaces with Blender: Pasta- & Limão-surfaces
https://community.wolfram.com/groups/-/m/t/2346332
Animating Mathematica surfaces with Blender: Ruled & rotational surfaces
https://community.wolfram.com/groups/-/m/t/2366292
Animating Mathematica surfaces with Blender: Complex analysis
https://community.wolfram.com/groups/-/m/t/2382132
Animating Mathematica surfaces with Blender: Transformations & cayley tree
https://community.wolfram.com/groups/-/m/t/2412951

Abstract

The Wolfram Language was used for video processing, to create prototypes and prepare surfaces. In Example 1 cube, sphere and octahedron were morphed. Example 2 uses two complex functions and their 2d/3d-plots. The creation of a video-wall with a random collection of animations was done as bonus via Mathematica.

Experiments

Example 1: Morphing II

Cube, Sphere & Octahedron / Cube - Octahedron

In[]:=

SetDirectory[NotebookDirectory[]<>If[$OperatingSystem=="Windows","\\data","/data"]]

Out[]=

C:\Users\ggoff\Desktop\ams8pub\ams8a\data

A 3D-plot for an ellipsoid was created:

In[]:=

shape0=x^2+y^2+z^2==1

Out[]=

1

In[]:=

c=1;ContourPlot3D[Evaluate[shape0],{x,-c,c},{y,-c,c},{z,-c,c}]

Out[]=

The left-hand-side of the above expression was squared and the expression expanded:

In[]:=

shape1=(x^2+y^2+z^2)^2==1//Expand

Out[]=

1

In[]:=

c=1;ContourPlot3D[Evaluate[shape1],{x,-c,c},{y,-c,c},{z,-c,c}]

Out[]=

A reorganized form of the expression with parameter a:

In[]:=

a=.shape2=

+y^4+

+2a(

)==1

Out[]=

+2a(

)1

A list of values for parameter a was chosen to form a rounded cuboid, a spherical and an octahedral object (including the in-between stages). The objects were resized to have similar dimensions.

In[]:=

b=4;c=1;cp=Table[ContourPlot3D[Evaluate[shape2],{x,-c,c},{y,-c,c},{z,-c,c},Axes->False,Boxed->False],{a,0.001,b,b/10}];dg=DiscretizeGraphics[#]&/@cp;rb=RegionBounds[#]&/@dg;rm=RegionMeasure[#]&/@dg;borders=Reverse[(First[rm]/rm)^(1/3)];cp2={};alist=Table[a,{a,0.001,b,b/10}];Do[c=borders[[i]];AppendTo[cp2,{ContourPlot3D[Evaluate[shape2/.{a->alist[[i]]}],{x,-c,c},{y,-c,c},{z,-c,c},MeshNone,ContourStyleDirective[Orange,Opacity[0.3],Specularity[White,30]],Axes->False,Boxed->False],ContourPlot3D[Evaluate[shape2/.{a->alist[[i]]}],{x,-c,c},{y,-c,c},{z,-c,c},Axes->False,Boxed->False]}],{i,1,10}]cp3=Transpose[cp2][[1]][[{1,3,10}]]cp4=Transpose[cp2][[2]][[{1,3,10}]]

Out[]=

In[]:=

cp5=Show[cp3]

Out[]=

In[]:=

Show[cp5,ViewPoint{0,-∞,0}]

Out[]=

Three objects were exported as dxf-files:

In[]:=

Do[Export[ToString[i]<>".dxf",cp4[[i]]],{i,1,3}]

A ListAnimation was created to visualize the Morphing:

In[]:=

b=4;c=1;nn=32;cp=Table[ContourPlot3D[Evaluate[shape2],{x,-c,c},{y,-c,c},{z,-c,c},Axes->False,Boxed->False],{a,0.001,b,b/nn}];dg=DiscretizeGraphics[#]&/@cp;rb=RegionBounds[#]&/@dg;rm=RegionMeasure[#]&/@dg;factor=Reverse[(First[rm]/rm)^(1/3)];cp2={};alist=Table[a,{a,0.001,b,b/nn}];Do[c=factor[[i]];AppendTo[cp2,ContourPlot3D[Evaluate[shape2/.{a->alist[[i]]}],{x,-c,c},{y,-c,c},{z,-c,c},Axes->False,Boxed->False]],{i,1,nn}]

In[]:=

ListAnimate[cp2]

Out[]=

Blender

The three objects (cube, sphere & octahedron) were imported in dxf-format. They were subdivided and given distinct metallic materials. All of them were set slightly above ground. A shiny dark metallic plane was placed at ground level. A key-framed animation via shape-keys was implemented.

Example 2: Flow

Init

In[]:=

SetDirectory[NotebookDirectory[]<>If[$OperatingSystem=="Windows","\\data","/data"]]

Out[]=

C:\Users\ggoff\Desktop\ams8pub\ams8pub\data

Surface 1

Three plots for a complex function were created and exported:

In[]:=

fz=

z-2+2

-3+3-z

Out[]=

(-2+2)+z

(-3+3)-z

In[]:=

cp1=Show[ComplexPlot[fz,{z,-5-5I,5+5I},ColorFunctionNone],ComplexStreamPlot[fz,{z,-5-5I,5+5I},StreamStyleWhite,StreamColorFunctionNone,StreamScaleLarge],Frame->None,ImageSize->Full];Show[cp1,ImageSize->Small]

Out[]=

In[]:=

cp0=ComplexPlot[fz,{z,-5-5I,5+5I},ImageSize->Full,Frame->None];Show[cp0,ImageSize->Small]

Out[]=

In[]:=

cp2=ComplexPlot3D[fz,{z,-5-5I,5+5I},PlotPoints50,Boxed->False,Axes->False,ImageSize->Full];

In[]:=

Show[cp2,ImageSize->Small]

Out[]=

In[]:=

Export["c11.png",cp0]

Out[]=

c11.png

In[]:=

Export["c12.png",cp1]

Out[]=

c12.png

In[]:=

Export["c13.stl",cp2]

Out[]=

c13.stl

Blender

1) The first surface was imported as stl-file. It was subdivided and solidified with modifiers. To remove artifacts a Boolean difference between surface and cylinder/cube was implemented via Boolean modifier. The height of the surface was animated from 0 to 1. As textures for the surface both exported 2d-plots for the complex function 1 were imported.
2) The surface from 1) was rotated around the z-axis and the height reduced from 1 to 0 in a keyframed animation.
3) In Geometry Nodes points were distributed randomly on the plane. The surface from 1) was appended and used to instance a lot of them with different scale and rotation in an animation.

Surface 2

Three plots for a second complex function were created and exported:

In[]:=

fz=1-

z^2

Out[]=

In[]:=

Out[]=

In[]:=

cp0=ComplexPlot[fz,{z,-5-5I,5+5I},ColorFunctionNone,ImageSize->Full,Frame->None];Show[cp0,ImageSize->Small]

Out[]=

In[]:=

cp2=ComplexPlot3D[fz,{z,-5-5I,5+5I},PlotPoints50,Boxed->False,Axes->False,ImageSize->Full];Show[cp2,ImageSize->Small]

Out[]=

In[]:=

Export["cp21.png",cp0]

Out[]=

cp21.png

In[]:=

Export["cp22.png",cp1]

Out[]=

cp22.png

In[]:=

Export["cp23.stl",cp2]

Out[]=

cp23.stl

Blender
4) A staggered grid with alternating imported 2d-plots of surface 1 and 2 was placed above a plane. The surface 2 was imported as stl-file and placed above the staggered grid at the positions of the 2d-plots of surface 2. They were given a gold-like material. The golden grid was animated via rotation around the z-axis and an moving up along the z-axis.

Bonus: Video-Wall (Mathematica only)

Out[]=

Preparation: Converting 4K- to HD-videos

For the animation 54.mp4 for all published 4k-videos snippets of duration 2 seconds were created. Additionally a version was saved in gif-format. Then a random sample of the snippets was used to build the video-wall.
To reduce the amount of necessary memory and the time for calculations a sample of four videos was prepared for this notebook.

In[]:=

(*path1=SetDirectory[NotebookDirectory[]<>"\\data"]fns0={"dual.mp4","around.mp4","interlocked.mp4","hope.mp4"};VideoReduced0[vlist_]:=Block[{vt},ParallelDo[vt=VideoTranscode[Video[vlist[[i]],AudioTrackSelectionNone],RasterSize1280];Export[FileBaseName[vlist[[i]]]<>"1.mp4",vt],{i,1,4}]//AbsoluteTimings]VideoReduced0[fns0]*)

Converting and trimming HD-videos

In this part four videos in HD-format were converted to 1/4 of the size, trimmed to 2 seconds snippets and saved.

In[]:=

SetDirectory[NotebookDirectory[]<>If[$OperatingSystem=="Windows","\\data","/data"]]

Out[]=

C:\Users\ggoff\Desktop\ams8pub\ams8a\data

In[]:=

fns={"dual.mp4","around.mp4","interlocked.mp4","hope.mp4"};

In[]:=

VideoReduced[vlist_,tlist_]:=Block[{vt,sv},ParallelDo[vt=VideoTranscode[Video[FileBaseName[vlist[[i]]]<>"1.mp4",AudioTrackSelectionNone],RasterSize360];sv=SnippetsVideo[vt,tlist[[i,2]],DefaultDuration->2];Export[FileBaseName[vlist[[i]]]<>"2.mp4",sv],{i,1,4}]//AbsoluteTiming]

In[]:=

props={{"du",{4},2},{"aro",{4},2},{"in",{4},2},{"hop",{2},2}};

In[]:=

VideoReduced[fns,props]

Out[]=

{9.02727,Null}

Building a video wall

Finally the saved videos from above were collected in one video.

In[]:=

VideoWall[vlist_,n_]:=Block[{vsize,gv,vef},vsize=360;gv=GridVideo[Video[#]&/@vlist,Background->Black,Spacings->5,PaddingSize->5];Export["4.mp4",gv];vef=VideoExtractFrames[gv,1];{vef,ImageDimensions[gv]}]

The snippets were selected:

In[]:=

fns=Select[#&/@FileNames["*.mp4"],StringTake[#,{-5}]=="2"&]

Out[]=

{around2.mp4,dual2.mp4,hope2.mp4,interlocked2.mp4}

Shown were the first frame, the calculation time and the dimensions of the first frame.

In[]:=

AbsoluteTiming[VideoWall[fns,4]]

Out[]=

4.31082,

,{736,420}

References

Collection of animations

https://vimeopro.com/orbs78/animations

References

Morphing II

Scene: Modified Cube, Sphere & Octahedron

Klaus, Stephan. “Cutting, Gluing, Squeezing, and Twisting: Visual Design of Real Algebraic Surfaces.” In Handbook of the Mathematics of the Arts and Sciences, edited by Bharath Sriraman, 863–77. Cham: Springer International Publishing, 2021. https://doi.org/10.1063/5.0006002.

Audio:
Stock Music & Sound Effects - Royalty Free Audio - Storyblocks
https://www.storyblocks.com/audio (accessed 2021-11-24)
Gribanova, Valentina. Cinematic Logo. Accessed November 24, 2021. Asset ID: SBA-300418822.

Flow

Scene : Modified Analytical Surface

"ComplexStreamPlot—Wolfram Language Documentation ." Accessed December 5, 2021. https://reference.wolfram.com/language/ref/ComplexStreamPlot.html?q=ComplexStreamPlot .

"Complex Visualization—Wolfram Language Documentation ." Accessed December 5, 2021. https://reference.wolfram.com/language/guide/ComplexVisualization.html .

Audio : Stock Music & Sound Effects - Royalty Free Audio - Storyblocks
https://storyblocks.com/audio (accessed 2021 - 12 - 05)
MEDIA MUSIC GROUP . Inspiring Comforting Indie Folk Acoustic Guitar 30 Sec . Accessed December 6, 2021. Asset ID : SBA - 346460058

54

Scene: 54 clips (each 2 s) delivered as a video wall.

“Video Processing—Wolfram Language Documentation.” Accessed December 14, 2021. https://reference.wolfram.com/language/guide/VideoProcessing.html.

Audio:
Stock Music & Sound Effects - Royalty Free Audio - Storyblocks
-. Indian Flute Signal. Accessed December 14, 2021. Asset ID: SBA-300055037.

POSTED BY: Guenther Gsaller

2 Replies

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Sander Huisman

Sander Huisman, University of Twente

Posted 4 months ago

Thanks for sharing! Very neat!

POSTED BY: Sander Huisman

Moderation Team

Moderation Team, WOLFRAM

Posted 4 months ago

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POSTED BY: Moderation Team

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