Given a sample of points from the boundary of an object IR3, we construct a representation of the object as a union of balls. We use many fewer balls than previous constructions, but our shape representation is better. We bound the distance from the surface of the union to the original object surface, and show that when the sampling is sufficiently dense the two are homeomorphic. This implies a topolgical relationship between the true medial axis of the object and both the medial axis, and the α-shape, of the union of balls. We show that the set of ball centers in our construction converges to the true medial axis as the sampling density increases.