convex hull higher dimension

hi,

I am working on 6 dimension W=[x y z tx ty tz] lets say 48 inputs points are given with the following coordinates (x y z tx ty tz)

i used

[P,v]=convhulln(W,{'Qt','Qx'})

and i got length(p)=1006 (facets) and 6 dimension

Questions

1. the triangulation method gives 3 sets of vertex for a facet for 3d but for 6d it gives 6 sets of vertices. how it works?
2. how to calculate the hyperplane which has the shortest distance from the origin to convex hull?
3. the facets are in 6 dimension how to calculate the normal of this facet. since the cross product is valid for 3 dimensions. moreover i need to use this way to calculate the minimum distance from origin.

BR

Raj

Answer 1

[Matt J]

If your polytope is bounded, you can use VERT2LCON to obtain the linear (in)equalities representing it

 [A,b,Aeq,beq]=vert2lcon(Points);

Assuming the region is solid in R^6, Aeq and beq will be empty and the rows of A will be the facet normals.

To find the minimum distance hyperplane, assuming the origin is outside the hull, use lsqlin

 hyperplaneNormal=lsqlin(speye(6),zeros(6,1),A,b,Aeq,beq);