For a simplex S with vertices v(1) , v(2) , · · · , v(n+1), give a grid size 1/q, we can triangulate it  as the collection of all sub-simplices σ(y(1) , π) with vertices y(1) , · · · , y(n+1) in S such that

(i) each component of y(1) is a multiple of 1/q;

(ii) π = (π(1) , π(2) , · · · , π(n) ) is a permutation of the elements of {1,2,..,n} ;

(iii) y(i+1) = y(i) +v(π(i+1) ) − v(π(i) )/q.

Under the grid size 1/q, give a point in S, though we can check it through very troublesome method to determine which sub-simplex it belong to, so how to determine this using simple perfect method?

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