Math Forum / Mathematics / Research / January 2007

you may write your manifold as H^T(x-x^0) =0  with a full rank matrix
H, and, equally well, after computing an orthonormal basis Y of range(H)
as Y^T(x-x^0)=0. x^0 the point in the manifold nearest to O.
now, points in the euclidean distance delta from this manifold satisfy
   ||Y^T(x-x^0)||_2^2 <= delta^2  
this is a convex inequality in R^k.
Let G be your grid, then your points are

    x \in Q \cap G and  ||Y^T(x-x^0)||_2^2 <= delta^2
but you cannot "solve" this mixed system of equalities and a nonlinear
convex inequality. things become a bit simpler if you would use the L1-distance
in which case it would suffice to use the (typically _given_) matrix H,
with column lengths normalized to one and the linear system of inequalities
   -c <= H^T(x-x^0) <= c  
with a componentwise positive vector c, the L1-distance being sum(c(i)).
but you are then in the field of integer linear programming, and
cannot hope for a simpler description of the complete feasible set.
(otherwise, integer linear programming would be a simple task, which it is not)
hth
peter