nextstep Site Admin
Joined: 06 Jan 2007 Posts: 538

Posted: Sun Oct 12, 2014 5:59 pm Post subject: 


Hi all,
From MathMod's facebook page:
Quote:  Skeletal Graphs of Triply Periodic Surfaces(1):
"Although the notion of a skeletal graph of a surface does not have a precise mathematical definition, several experiments suggest that certain surfaces do have unique skeletal graphs. The skeletal graph can be thought of as the end result of expanding or shrinking a surface along the direction of its normal vectors, while avoiding any pinching off that would change the topology of the surface, until all that remains is a connected graph of arcs and nodes. Examples of surfaces for which there are obvious skeletal graphs are the triply periodic P, D, G, and iWp surfaces. These graphs are identical whatever the 'flavor' (minimal, CMC, or level) of the surface."
More at: https://secure.msri.org/about/sgp/jim/geom/surface/global/skeletal/index.html 
Quote:  Skeletal Graphs of Triply Periodic Surfaces(2):
David Hoffman and Jim Hoffman have demonstrated in their Scientific Graphics Project that for the Triply Periodic Minimal Surfaces (TPMS) P, G, D, and also for a fourth surface (IWP) of genus 4, there is a striking connection between the skeletal graph of the surface and a modified version of its level surface approximation.
This is a much simpler approach to approximating the skeletal graph by using level surfaces:
"The continuous level surfaces depicted on the Triply Periodic Level Surfaces page pinch off into unconnected blobs as the value of the parameter t becomes sufficiently negative or positive. At such pinch points the smaller of the volume fractions (the fraction of the total volume inside the blobs) remains considerable.
The pinching off behavior of the necks can be controlled by using slightly more complicated equations having the following form:
v = r*T1 + s*T2 + t
where r, s, and t are constants, and both T1 and T2 are one of the righthand sides of the simple level surface approximations of the G, D, P, and W surfaces, subject to translation and scaling."
More at: https://secure.msri.org/about/sgp/jim/geom/level/skeletal/index.html
The attached image shows the skeletal graphs for the TPMS P, G ,D and IWP (Primitive, Gyroid and Diamond).
The number of arcs meeting at nodes are : 3, 4, 6 and 8
MathMod script for the skeletal graphics, with eight arcs meeting at nodes.
Quote:  {
"Iso3D": {
"Component": [
"GyroidDSkelettalLatice"
],
"Fxyz": [
"10.0*(cos(x)*cos(y) + cos(y)*cos(z) + cos(z)*cos(x))5.0*(cos(x*2) + cos(y*2) + cos(z*2))14.0"
],
"Name": [
"GyroidDSkelettalLatice"
],
"Xmax": [
" 2*pi"
],
"Xmin": [
"2*pi"
],
"Ymax": [
" 2*pi"
],
"Ymin": [
"2*pi"
],
"Zmax": [
" 2*pi"
],
"Zmin": [
"2*pi"
]
}
} 

SkelatalGraphs by taha_ab, on Flickr _________________ Cheers,
Abderrahman 
