In mathematics a generic point of a geometric object is a point with no special properties; a point which satsifies no conditions that do hold hold for every point in the space.
In algebraic geometry
In algebraic geometry, a generic point of an algebraic variety is a point for which the coordinates have the property that the only polynomial relations that hold among them are the defining equations of the variety itself.
with coefficients in the function field R(t) is generic, as it is on the circle but every polynomial relation between the coordinates is deducible from the relation X2 + Y2 = 1. On the other hand the point (3/5, 4/5) is not generic as it satisfies a rather trivial relation 5X=3 which does not hold in general.