The incidence structure NQ+(3, q) has points the points not on a non-degenerate hyperbolic quadric Q+(3, q) in PG(3, q), and its lines are the lines of PG(3, q) not containing a point of Q+(3, q). It is easy to show that NQ+(3, q) is a partial linear space of order (q, q(q?1)/2). If q is odd, then moreover NQ+(3, q) satisfies the property that for each non-incident point line pair (x,L), there are either (q?1)/2 or (q+1)/2 points incident with L that are collinear with x. A partial linear space of order (s, t) satisfying this property is called a ((q?1)/2,(q+1)/2)-geometry. In this paper, we will prove the following characterization of NQ+(3,q). Let S be a ((q?1)/2,(q+1)/2)-geometry fully embedded in PG(n, q), for q odd and q>3. Then S = NQ+(3, q).