Academic Articles & Supplements

Minimal Absorption Index for a 2-Dimensional Simplex

Alexey Ukhalov

Author

Alexey Ukhalov

Title

Minimal Absorption Index for a 2-Dimensional Simplex

Description

Wolfram Language code for finding a minimal absorption index for a 2-dimensional simplex

Category

Academic Articles & Supplements

Keywords

simplex, absorption index, convex geometry

URL

http://www.notebookarchive.org/2020-02-b23e65i/

DOI

https://notebookarchive.org/2020-02-b23e65i

Date Added

2020-02-24

Date Last Modified

2020-02-24

File Size

13.7 kilobytes

Supplements

Rights

CC BY 4.0

Minimal Absorption Index for a 2-Dimensional Simplex

Definitions, basic facts and formulas can be found in the following papers

In English:

Nevskii, M.V. & Ukhalov, A.Y. On Minimal Absorption Index for an n-Dimensional Simplex.
Automatic Control and Computer Sciences. 2018. Volume 52, Issue 7, pp 680–687.
https://doi.org/10.3103/S0146411618070209

Nevskii M., and Ukhalov A. Perfect simplices in R^5.
Beitrage zur Algebra und Geometrie / Contributions to Algebra and Geometry. 2018. Vol 59, Issue 3, pp. 501-521.

In Russian:

Nevskii M.V., Ukhalov A.Y. On Minimal Absorption Index for an n-Dimensional Simplex.
Modeling and Analysis of Information Systems. 2018. Volume 25, No 1, pp. 140-150. (In Russ.)
https://doi.org/10.18255/1818-1015-2018-1-140-150

See also book (in Russian)
Nevskii, M.\,V. Geometric estimates in polynomial interpolation. P. G. Demidov Yaroslavl State University, Yaroslavl, 2012.

A={{x1,y1,1},{x2,y2,1},{x3,y3,1}};L=Inverse[A];l[i_,x_,y_]:=L[[1]][[i]]*x+L[[2]][[i]]*y+L[[3]][[i]];T=Flatten[Table[-l[i,x,y],{i,1,3,1},{x,0,1,1},{y,0,1,1}]];ksi[x1_,y1_,x2_,y2_,x3_,y3_]:=1+3*Max[T];

res=NMinimize[{ksi[x1,y1,x2,y2,x3,y3],0≤x1&&x1≤1&&0≤y1&&y1≤1&&0≤x2&&x2≤1&&0≤y2&&y2≤1&&0≤x3&&x3≤1&&0≤y3&&y3≤1},{x1,y1,x2,y2,x3,y3}]

{2.34164,{x11.,y10.618034,x20.381966,y20.,x30.,y31.}}

ksi2=res[[1]];S={{x1,y1},{x2,y2},{x3,y3}}/.res[[2]];Ct=(S[[1]]+S[[2]]+S[[3]])/3;ksi2S={Ct+(S[[1]]-Ct)*ksi2,Ct+(S[[2]]-Ct)*ksi2,Ct+(S[[3]]-Ct)*ksi2};Graphics[{Thickness[0.001],Line[{{0,0},{1,0},{1,1},{0,1},{0,0}}],FaceForm[None],EdgeForm[Directive[Thickness[0.01],Black]],Simplex[{S[[1]],S[[2]],S[[3]]}],EdgeForm[Directive[Thickness[0.01],Red]],Simplex[{ksi2S[[1]],ksi2S[[2]],ksi2S[[3]]}]},PlotRange{{-1,2},{-1,2}},AxesTrue,Ticks{{1},{1}},TicksStyleDirective["Label",17],LabelStyleDirective[Black,17],AxesLabel{"x","y"},AxesStyleDirective[Black,Thickness[0.005]]]

Cite this as: Alexey Ukhalov, "Minimal Absorption Index for a 2-Dimensional Simplex" from the Notebook Archive (2020), https://notebookarchive.org/2020-02-b23e65i