function [CH,CI,dI] = rect_hull(P)

% Converts a vertices object into a polyhedron object.
%
% Syntax:
%   "[CH,CI,dI] = rect_hull(P)"
%
% Description:
%   "rect_hull(P)" returns a polyhedron object CH as an oriented
%   hyper-rectangle constructed from the points in the vertices object "P",
%   and the matrices "CI" and "dI" that determine the inequalities of "CH".
%	 The orientation of the hyperrectangle is obtained from principal
%   component analysis.	 
%
%   The following rules determine the construction of the polyhedron object:
%   
%   - If the vertex object contains just one point, the polyhedron object
%     is n-dimensional with the length 2*vector_tol in all dimensions.
%
%   - If the vertices determine a full-dimensional hull the polyhedron
%     object is of course full dimensional.
%
%   - If the vertices determine an (n-1)-dimensional hull the polyhedron
%     object is not bloated, i.e., it is (n-1)-dimensional.
%
%   - If the convex hull of the vertices is of a dimension lower than (n-1)
%     but larger than zero, the polyhedron is bloated to full dimension,
%     i.e., this routine does never return an object that is of dimension
%     less than n-1.
%
%   Note: Since this routine accesses fields of polyhedron objects it can
%   only be used if it is located within the method of this class, i.e.,
%   if it is in the "@polyhedron" folder of the Checkmate distribution.
%
% See Also:
%   polyhedron,vertices
%
%
% --OS-- Last change: 06/21/02
%

global GLOBAL_APPROX_PARAM

CH=polyhedron;
CI=[]; dI=[]; VI=cell(1,1); ni=0; 
CE=[]; dE=[]; VE=cell(1,1); ne=0;

roundtol = GLOBAL_APPROX_PARAM.poly_vector_tol;
if (roundtol==0)
   roundtol=1e3*eps;
end

svdtol = GLOBAL_APPROX_PARAM.poly_svd_tol;
if (svdtol==0)
   svdtol=1e4*eps;
end

% Absolute value for bloating
bloattol = GLOBAL_APPROX_PARAM.poly_bloat_tol;
% must be larger than 'roundtol'!
if bloattol<=roundtol+10*eps;
   bloattol=roundtol+10*eps;
end

% Rounding and Removal of points with multiple occurrence:
verts=[];
for i=1:length(P)
   verts=[verts; roundtol*round(P(i)'/roundtol)];
end
verts=unique(verts,'rows')'; % each point contained as a column vector
n=size(verts,1); m=size(verts,2);

% Translate into the mean as new origin:
xm=sum(verts,2)/m;
tverts=[];
for i=1:m
   tverts=[tverts,verts(:,i)-xm];
end

% Singular value decomposition:
R=tverts*tverts';
[U,S,V]=svd(R);  % U=V!

% Introduce auxiliary point for distortion if two singular values are equal
ind=[];
for i=1:n
   for j=1:i-1    
      if (S(i,i)>=S(j,j)-svdtol)&(S(i,i)<=S(j,j)+svdtol)&(S(i,i)~=0)
         ind=[ind;i;j];
      end   
   end
end   
ind=unique(ind);
if length(ind)>=1
   xaux=xm;
   for i=1:length(ind)
      xaux(i)=xaux(i)+bloattol*max(verts(i,:)-xm(i));
   end
   verts=[verts,xaux];
   m=m+1; xm=sum(verts,2)/m;
   tverts=[];
   for i=1:m
      tverts=[tverts,verts(:,i)-xm];
   end
   R=tverts*tverts'; [U,S,VD]=svd(R);
end
   
% Displacement matrix (minimal/maximal values for the new coordinates):
D=[];
for i=1:n
   D=[D,[max(U(:,i)'*tverts);min(U(:,i)'*tverts)]];
if (D(2,i))>=-roundtol&(D(1,i)<=roundtol)   
      D(1,i)=bloattol; D(2,i)=-bloattol;   
   end
end

% (In)-Equalities:
CI=[U';-U'];
dI=[[D(1,:)'+U'*xm]; [-D(2,:)'-U'*xm]];

% Vertices:
VGM=[];
for j=1:n
   Vu=[VGM; [D(1,j)*ones(1,max(size(VGM,2),1))]]; Vl=[VGM; [D(2,j)*ones(1,max(size(VGM,2),1))]];
   VGM=[Vu,Vl];         
end
Vt=U*VGM;
V=[];
for i=1:size(Vt,2)
   V=[V,Vt(:,i)+xm];
end
for i=1:n
   ni=ni+1;
   Vu=[]; Vl=[];
   for j=1:size(V,2)
      if (VGM(i,j)>=D(1,i)-eps) & (VGM(i,j)<=D(1,i)+eps)
         Vu=[Vu,V(:,j)];
      else % VGM(i,j)=D(2,i)   
      	Vl=[Vl,V(:,j)];
      end
   end
   VI{ni}=vertices(Vu);
	VI{n+ni}=vertices(Vl);   
end

CH.CI=CI; CH.dI=dI; CH.VI=VI;
CH.CE=CE; CH.dE=dE; CH.VE=VE; 
CH.vtcs=vertices(V);

return